geir ellingsrud and john christian ottem october 15, 2018€¦ · between algebra and geometry: a...
TRANSCRIPT
Introduction to Schemes
Geir Ellingsrud and John Christian Ottem
October 15, 2018
Contents
0.1 Introduction to the schemes formalism . . . . . . . . . . . . . . 11
0.1.1 Nilpotents and zero-divisors . . . . . . . . . . . . . . . . 11
0.1.2 Gluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
0.1.3 General base rings . . . . . . . . . . . . . . . . . . . . . 12
0.1.4 Prime ideals, rather than maximal ideals. . . . . . . . . . 13
1 Sheaves 15
1.1 Sheaves and presheaves . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.1 Presheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.1.2 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.1.3 Subsheaves and saturation of subpresheaves . . . . . . . 18
1.2 A bunch of examples . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 Stalks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 The germ of a section . . . . . . . . . . . . . . . . . . . . 22
1.3.2 A primer on limits . . . . . . . . . . . . . . . . . . . . . 22
1.4 Morphisms between (pre)sheaves . . . . . . . . . . . . . . . . . 23
1.5 Kernels and images . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6 Exact sequences of sheaves. . . . . . . . . . . . . . . . . . . . . 27
1.7 B-sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.8 A family of examples – Godement sheaves . . . . . . . . . . . . 31
1.8.1 Skyscraper sheaves. . . . . . . . . . . . . . . . . . . . . . 32
1.8.2 The Godement sheaf associated with a presheaf . . . . . 33
1.9 Sheafification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.10 Quotient sheaves and cokernels . . . . . . . . . . . . . . . . . . 36
1.11 The pushforward of a sheaf . . . . . . . . . . . . . . . . . . . . . 36
1.12 Inverse image sheaf . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.12.1 Adjoint property . . . . . . . . . . . . . . . . . . . . . . 38
2 Schemes 41
2.1 The spectrum of a ring . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.1 Generic points . . . . . . . . . . . . . . . . . . . . . . . . 44
2.1.2 Irreducible closed subsets . . . . . . . . . . . . . . . . . . 44
2.2 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
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2.3 First examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.3.1 The world of schemes vs the world of varieties . . . . . . 48
2.4 Distinguished open sets . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.1 Basic properties of distinguished sets . . . . . . . . . . . 50
2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 The structure sheaf on SpecA . . . . . . . . . . . . . . . . . . . 53
2.6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.6.2 Definition of the structure sheaf OSpecA . . . . . . . . . . 54
2.6.3 Maps between the structure sheaves of two spectra . . . 57
2.7 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.8 Open immersions and open subschemes . . . . . . . . . . . . . . 60
2.9 Residue fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.10 Relative schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Gluing 65
3.1 Gluing sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Gluing maps of sheaves . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 The gluing of schemes . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Global sections of glued schemes . . . . . . . . . . . . . . . . . . 69
3.5 The gluing of morphisms . . . . . . . . . . . . . . . . . . . . . . 70
3.6 The category of affine schemes . . . . . . . . . . . . . . . . . . . 70
3.7 Universal properties of maps into affine schemes . . . . . . . . . 71
3.8 The functor from varieties to schemes . . . . . . . . . . . . . . . 73
4 More examples 75
4.1 A scheme that is not affine . . . . . . . . . . . . . . . . . . . . . 75
4.2 The projective line . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.1 Affine line a doubled origin . . . . . . . . . . . . . . . . . 77
4.3 The blow-up of the affine plane . . . . . . . . . . . . . . . . . . 77
4.3.1 Blow-up as a variety . . . . . . . . . . . . . . . . . . . . 77
4.3.2 The blow-up as a scheme . . . . . . . . . . . . . . . . . . 78
4.4 Hyperelliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 80
5 Geometric properties of schemes 81
5.1 Some topological properties . . . . . . . . . . . . . . . . . . . . 81
5.2 Properties of the scheme structure . . . . . . . . . . . . . . . . . 82
5.2.1 Integral schemes . . . . . . . . . . . . . . . . . . . . . . 83
5.2.2 Noetherian spaces and quasi-compactness . . . . . . . . . 84
5.3 The dimension of a scheme . . . . . . . . . . . . . . . . . . . . . 86
5.4 More examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4.1 A more fancy example . . . . . . . . . . . . . . . . . . . 88
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6 Projective schemes 91
6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2 Basic remarks on graded rings . . . . . . . . . . . . . . . . . . . 92
6.2.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.3 The Proj construction . . . . . . . . . . . . . . . . . . . . . . . 93
6.3.1 ProjR as a scheme . . . . . . . . . . . . . . . . . . . . . 96
6.4 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.4.1 Closed immersions . . . . . . . . . . . . . . . . . . . . . 99
6.5 The Veronese embedding . . . . . . . . . . . . . . . . . . . . . . 99
6.5.1 Remark on rings generated in degree 1 . . . . . . . . . . 100
6.5.2 The Blow-up as a Proj . . . . . . . . . . . . . . . . . . . 100
7 Fiber products 103
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
7.1.1 Fiber products of sets. . . . . . . . . . . . . . . . . . . . 103
7.1.2 The fiber product in general categories . . . . . . . . . . 104
7.2 Products of affine schemes . . . . . . . . . . . . . . . . . . . . . 105
7.3 A useful lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
7.4 The gluing process . . . . . . . . . . . . . . . . . . . . . . . . . 108
7.5 The final reduction . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.6 Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.7.1 Varieties versus Schemes . . . . . . . . . . . . . . . . . . 110
7.7.2 Non-algebraically closed fields . . . . . . . . . . . . . . . 111
7.8 Base change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.9 Scheme theoretic fibres . . . . . . . . . . . . . . . . . . . . . . . 114
7.10 Separated schemes . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.10.1 The diagonal . . . . . . . . . . . . . . . . . . . . . . . . 117
7.10.2 Separated schemes . . . . . . . . . . . . . . . . . . . . . 118
7.11 The valuative criterion . . . . . . . . . . . . . . . . . . . . . . . 119
7.11.1 Varieties vs schemes again . . . . . . . . . . . . . . . . . 121
8 Quasi-coherent sheaves 123
8.1 Sheaves of modules . . . . . . . . . . . . . . . . . . . . . . . . . 124
8.1.1 The support of a sheaf . . . . . . . . . . . . . . . . . . . 127
8.2 Pushforward and Pullback of OX-modules . . . . . . . . . . . . 127
8.2.1 Pushforward . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2.2 Pullback . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
8.2.3 Adjoint properties of f∗, f∗ . . . . . . . . . . . . . . . . . 128
8.2.4 Pullback of sections . . . . . . . . . . . . . . . . . . . . . 128
8.2.5 Tensor products . . . . . . . . . . . . . . . . . . . . . . . 129
8.3 Quasi-coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . 129
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8.4 Quasi-coherent sheaves on affine schemes . . . . . . . . . . . . . 130
8.4.1 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . 133
8.5 Quasi-coherent sheaves on general schemes . . . . . . . . . . . . 134
8.6 Coherent sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . 137
8.7 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.7.1 Quasi-coherence of pullbacks . . . . . . . . . . . . . . . . 139
8.7.2 Quasi-coherence of the direct image . . . . . . . . . . . . 140
8.7.3 Coherence of the direct image . . . . . . . . . . . . . . . 142
8.8 The categories of coherent and quasi-coherent sheaves . . . . . . 142
8.9 Closed immersions and subschemes . . . . . . . . . . . . . . . . 143
8.9.1 Morphisms to a closed subscheme . . . . . . . . . . . . . 145
8.9.2 Induced reduced scheme structure . . . . . . . . . . . . . 145
9 Vector bundles and locally free sheaves 147
9.1 Locally free sheaves and projective modules (work in progress) . 149
9.2 Invertible sheaves and the Picard group . . . . . . . . . . . . . . 149
9.2.1 Operations on locally free sheaves . . . . . . . . . . . . . 151
9.3 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
9.3.1 The sheaf of sections of a vector bundle . . . . . . . . . . 154
9.3.2 Equivalence between vector bundles and locally free sheaves154
9.4 Extended example: The tautological bundle on Pnk . . . . . . . . 155
10 Sheaves on projective schemes 159
10.1 The graded ∼-functor . . . . . . . . . . . . . . . . . . . . . . . . 160
10.2 Serre’s twisting sheaf O(1) . . . . . . . . . . . . . . . . . . . . . 161
10.3 Extending sections of invertible sheaves . . . . . . . . . . . . . . 163
10.4 The associated graded module . . . . . . . . . . . . . . . . . . . 164
10.4.1 Sections of the structure sheaf . . . . . . . . . . . . . . . 165
10.4.2 The homomorphism α. . . . . . . . . . . . . . . . . . . . 166
10.4.3 The map β . . . . . . . . . . . . . . . . . . . . . . . . . 167
10.5 Closed subschemes of ProjR . . . . . . . . . . . . . . . . . . . . 169
10.6 The Segre embedding . . . . . . . . . . . . . . . . . . . . . . . . 170
10.7 Two important exact sequences . . . . . . . . . . . . . . . . . . 172
10.7.1 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . 172
10.7.2 Complete intersections . . . . . . . . . . . . . . . . . . . 172
11 Differentials 175
11.1 Tangent vectors and derivations . . . . . . . . . . . . . . . . . . 175
11.2 Regular schemes and the Zariski tangent space . . . . . . . . . . 177
11.2.1 Zariski tangent space and the ring of dual numbers . . . 178
11.3 Derivations and Kahler differentials . . . . . . . . . . . . . . . . 179
11.3.1 The module of Kahler differentials . . . . . . . . . . . . . 180
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11.4 Properties of Kahler differentials . . . . . . . . . . . . . . . . . . 181
11.4.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . 182
11.4.2 Base change . . . . . . . . . . . . . . . . . . . . . . . . . 183
11.4.3 Exact sequences . . . . . . . . . . . . . . . . . . . . . . . 183
11.4.4 The diagonal and ΩB/A . . . . . . . . . . . . . . . . . . . 184
11.5 Some explicit computations . . . . . . . . . . . . . . . . . . . . 185
11.6 The sheaf of differentials . . . . . . . . . . . . . . . . . . . . . . 187
11.7 The sheaf of differentials of PnA . . . . . . . . . . . . . . . . . . . 189
11.8 Relation with the Zariski tangent space . . . . . . . . . . . . . . 190
11.8.1 Smooth morphisms . . . . . . . . . . . . . . . . . . . . . 193
11.9 The conormal sheaf . . . . . . . . . . . . . . . . . . . . . . . . . 194
12 First steps in sheaf cohomology 195
12.1 Some homological algebra . . . . . . . . . . . . . . . . . . . . . 196
12.2 Cech cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 198
12.3 Cohomology of sheaves on affine schemes . . . . . . . . . . . . . 202
12.3.1 Cech cohomology and affine coverings . . . . . . . . . . . 205
12.4 Chomology and dimension . . . . . . . . . . . . . . . . . . . . . 205
12.5 Cohomology of sheaves on projective space . . . . . . . . . . . . 206
12.6 Extended example: Plane curves . . . . . . . . . . . . . . . . . . 209
12.7 Extended example: The twisted cubic in P3 . . . . . . . . . . . 210
13 More on sheaf cohomology 213
13.1 Flasque sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
13.2 The Godement resolution . . . . . . . . . . . . . . . . . . . . . . 215
13.2.1 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . 216
13.2.2 Sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . 217
13.2.3 Acyclic sheaves . . . . . . . . . . . . . . . . . . . . . . . 220
13.2.4 The Cech resolution . . . . . . . . . . . . . . . . . . . . 222
13.3 Godement vs. Cech . . . . . . . . . . . . . . . . . . . . . . . . 223
14 Divisors and linear systems 225
14.1 Weil divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
14.1.1 The divisor of a rational function . . . . . . . . . . . . . 228
14.1.2 The divisor of a section of an invertible sheaf . . . . . . . 229
14.1.3 The class group . . . . . . . . . . . . . . . . . . . . . . . 230
14.1.4 Projective space . . . . . . . . . . . . . . . . . . . . . . . 230
14.1.5 Weil divisors and invertible sheaves . . . . . . . . . . . . 231
14.1.6 The class group and unique factorization domains . . . . 232
14.1.7 A useful exact sequence . . . . . . . . . . . . . . . . . . 233
14.2 Cartier divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
14.2.1 Cartier divisors and invertible sheaves . . . . . . . . . . 235
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14.2.2 CaCl(X) and Pic(X) . . . . . . . . . . . . . . . . . . . . 236
14.2.3 From Cartier to Weil . . . . . . . . . . . . . . . . . . . . 237
14.2.4 Projective space . . . . . . . . . . . . . . . . . . . . . . . 238
14.3 Effective divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 239
14.3.1 Linear systems . . . . . . . . . . . . . . . . . . . . . . . 240
14.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
14.4.1 The quadric surface . . . . . . . . . . . . . . . . . . . . . 241
14.4.2 The quadric cone . . . . . . . . . . . . . . . . . . . . . . 243
14.4.3 Quadric hypersurfaces in higher dimension . . . . . . . . 245
14.4.4 Hirzebruch surfaces . . . . . . . . . . . . . . . . . . . . . 246
15 Two applications of cohomology 249
15.1 Vector bundles on the projective line . . . . . . . . . . . . . . . 249
15.2 Non-split vector bundles on Pn . . . . . . . . . . . . . . . . . . . 251
16 Maps to projective space 253
16.1 Globally generated sheaves . . . . . . . . . . . . . . . . . . . . . 254
16.1.1 Pullbacks and pushforwards . . . . . . . . . . . . . . . . 254
16.2 Morphisms to projective space . . . . . . . . . . . . . . . . . . . 255
16.2.1 Example: The twisted cubic . . . . . . . . . . . . . . . . 255
16.2.2 Morphisms and globally generated invertible sheaves . . 256
16.3 Application: Automorphisms of Pn . . . . . . . . . . . . . . . . 258
16.4 Projective embeddings . . . . . . . . . . . . . . . . . . . . . . . 259
16.5 Projective space as a functor . . . . . . . . . . . . . . . . . . . . 261
16.5.1 Generalizations . . . . . . . . . . . . . . . . . . . . . . . 262
17 More on projective schemes 265
17.1 Ample invertible sheaves and Serre’s theorems . . . . . . . . . . 265
17.1.1 Euler characteristic . . . . . . . . . . . . . . . . . . . . . 267
17.2 Proper and projective schemes . . . . . . . . . . . . . . . . . . . 267
18 The Riemann–Roch theorem and Serre duality 269
18.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
18.2 Divisors on Curves . . . . . . . . . . . . . . . . . . . . . . . . . 269
18.2.1 The genus of a curve . . . . . . . . . . . . . . . . . . . . 270
18.3 The canonical divisor . . . . . . . . . . . . . . . . . . . . . . . . 271
18.4 Hyperelliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . 271
18.5 The Riemann Roch theorem for curves . . . . . . . . . . . . . . 274
18.6 Proof of Serre duality for curves . . . . . . . . . . . . . . . . . . 277
18.6.1 Laurent tails . . . . . . . . . . . . . . . . . . . . . . . . . 280
18.6.2 Residues on A . . . . . . . . . . . . . . . . . . . . . . . . 280
18.6.3 Residues on X . . . . . . . . . . . . . . . . . . . . . . . 281
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18.6.4 The final part of the proof . . . . . . . . . . . . . . . . . 283
19 Applications of the Riemann–Roch theorem 287
19.1 Morphisms of curves . . . . . . . . . . . . . . . . . . . . . . . . 287
19.1.1 Morphisms of curves are finite . . . . . . . . . . . . . . . 288
19.1.2 Pullbacks of divisors . . . . . . . . . . . . . . . . . . . . 289
19.1.3 Morphisms to P1 . . . . . . . . . . . . . . . . . . . . . . 290
19.2 Very ampleness criteria . . . . . . . . . . . . . . . . . . . . . . . 291
19.3 Curves on P1 × P1 . . . . . . . . . . . . . . . . . . . . . . . . . 292
19.4 Curves of genus 0 . . . . . . . . . . . . . . . . . . . . . . . . . . 293
19.5 Curves of genus 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 294
19.5.1 Divisors on X . . . . . . . . . . . . . . . . . . . . . . . . 295
19.6 Curves of genus 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 297
19.7 Curves of genus 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 298
19.8 Curves of Genus 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 301
19.8.1 Classifying curves of genus 4 . . . . . . . . . . . . . . . . 301
A To be deleted 303
A.1 L.E.S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
A.2 Pic(X) and H1(X,O×X) . . . . . . . . . . . . . . . . . . . . . . . 307
Introduction
0.1 Introduction to the schemes formalism
If X is an affine variety over an algebraically closed field k, then it has an
affine coordinate ring A(X) of regular functions X → k, and A(X) completely
characterises X up to isomorphism. This sets up a bijective correspondence
between algebra and geometry: affine varieties correspond to finitely generated
k-algebras without zero-divisors, and morphism between varieties corresponds to
k-algebra homomorphisms. The modern formulation of algebraic geometry, due
to Grothendieck, vastly generalizes this picture, replacing the k-algebras without
zero-divisors with the more general category of commutative rings.
This has the tremendous advantage that many of the arguments and intuition
carry over from the classical setting, and can be used to study more general
problems in other branches of mathematics. For instance, studying diophantine
equations over non-closed fields (such as Q) is clearly of interest in number
theory.
We should emphasise that the main goal of algebraic geometry is still to
understand algebraic varieties over a field. However, enlarging our category to
schemes offers a tremendous flexibility. Let us give a few examples of this.
0.1.1 Nilpotents and zero-divisors
Nilpotents and zero-divisors appear already in Bezout’s theorem: The conic
C = Z(y − x2) intersects a general line in two points, but the line y = 0 only at
the origin (with ‘multiplicity’ 2). Looking at the ideals, it is intuitive that the
intersection should correspond to the ideal (y − x2, y) = (x2, y) which gives us
the intersection number (’multiplicity 2’), but that ideal does not correspond to
an algebraic variety, as it is not radical.
Similar phenomena appear naturally when you take ‘limits’ (or ‘deformations’)
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0.1. Introduction to the schemes formalism
of algebraic varieties. As a basic example, consider the family of curves in A2
given by Vt = Z(y2− tx). Then for t 6= 0, Vt is a smooth conic, whereas for t = 0
it is a ‘double line’. The limit as t→ 0 is problematic; if you insist that it should
correspond to the affine variety Z(y), which has degree 1, then the degree of Vtis not a continuous function in t.
These problems vanish when passing to schemes: it is completely unproblem-
atic to work with geometric objects (affine schemes) associated to the coordinate
rings k[x, y]/(x2, y) or k[x, y]/(y2) respectively. Note however that we have to
adjust our viewpoint about regular functions: For instance, in the second example
the element ‘y’ is non-zero in the ring k[x, y]/(y2) - but it takes the value 0 at
each of the closed points of the underlying topological space (the x-axis).
0.1.2 Gluing
Just like in differential geometry, we do not like to consider manifolds as embedded
in some Rn (although Whitney’s theorem says that such an embedding exists). It
is much more fruitful to study manifolds M locally, using how it is glued together
by a bunch of open subsets in Rn. For instance, the concept of a tangent vector of
M at a point p can be defined using equivalence classes of locally parameterized
curves passing through p; glueing these sets together for each p, we obtain the
tangent bundle of M .
In a similar fashion, we will see that we get a lot of flexibility by talking
about varieties or schemes as glued objects rather than embedded objects in
some An or Pn. We can define the normalization of a variety by normalizing
each chart; we can talk about tangent and cotangent bundles by gluing together
local trivializations.
Moreover, by gluing together affine schemes we obtain much more interesting
spaces which may indeed fail to be embeddable in any Pn. And a posterori
non-quasi-projective varieties are not so obscure as one would think - they really
do occur in many contexts in algebraic geometry.
0.1.3 General base rings
Working with a non-algebraically closed base field is obviously important in
number theory: If we want to talk about rational points on the curve given
by the equation x3 + y3 = z3 we need to be able to think of elliptic curves as
schemes over the field Q. There is a naive fix, to just go to the algebraic closure,
but this misses the main points: The curves x3 + y3 = z3 and x3 + y3 = 9z3 are
isomorphic over Q, but the first has only a few trivial points over Q and the
second has infinitely many.
Going from fields to more general rings offers even greater flexibility. For
instance, the variety Z(y2 − x(x − 1)(x − t)) can be viewed simultaneously a
12
Contents
surface S in A3k and a curve C of genus 1 curve over the ring k[t]. The k[t]-points
here correspond to rational curves on S. This sort of interplay is very useful in
many situations.
0.1.4 Prime ideals, rather than maximal ideals.
By the Nullstellensatz, the points of an affine variety are in bijection with the
maximal ideals of its coordinate ring. We will see that it is in fact much more
natural to include all the prime ideals of this ring. This is something that is
almost forced on us by functoriality: Think about the inclusion k[x] → k(x).
The maximal ideal (0) of k(x) pulls back to a prime ideal of k[x] which is not
maximal. Including the set of all prime ideals, we obtain a very nice category
which behaves essentially just like the category of commutative rings.
Beside of these categorical advantages, there are also a number of places in
more classical settings where thinking of points defined by prime ideals provides
an extra conceptual clarity. We will try to highlight a few examples where this
happens in Chapter 2.
13
Chapter 1
Sheaves
The concept of a sheaf was conceived in the German camp for prisoners of war
called Oflag XVII where French officers taken captive during the fighting in
France in the spring 1940 were imprisoned. Among them was the mathematician
and lieutenant Jean Leray. In the camp he gave a course in algebraic topology(!!)
during which he introduced some version of the theory of sheaves. He was aimed
at calculating the cohomology of a total space of a fibration in terms of invariants
of the base and the fibres (and naturally the fibration). To achieve this, in
addition to the concept of sheaves, he also invented spectral sequences.
After the war, Henri Cartan and Jean Pierre Serre developed the theory
further, and finally the theory was brought to the state as we know it today by
Alexandre Grothendieck.
1.1 Sheaves and presheaves
A common theme in mathematics is to study spaces by describing them through
their local properties. A manifold is a space which looks locally like Euclidean
space; a complex manifold is a space which is locally biholomorphic to Cn;
an algebraic variety is a space that looks locally like the zero set of a set
of polynomials. Here it is clear that point set topology alone is not enough
to fully capture these three notions: In each case, the space comes equipped
with a natural set of functions: C∞-functions, holomorphic functions, and the
polynomials respectively, and these are preserved under the local identifications.
Sheaves are introduced to make these ideas precise. To explain this better,
let us consider the primary example of a sheaf: The sheaf of continuous maps on
15
1.1. Sheaves and presheaves
a topological space X. By definition, X comes with a collection of ‘open sets’,
and these encode what it means for a map f : X → Y to another topological
space Y to be continuous: For every open U ⊆ Y , the set f−1(U) should be
open in X. For two topological spaces X and Y , we can define for each open
U ⊆ X, a set of continuous maps
C(U, Y ) = f : U → Y | f is continuous.
Note that if V ⊆ U is open, then the restriction f |V of a continuous function f
to V is again continuous, so we obtain a map
ρUV : C(U, Y )→ C(V, Y )
f 7→ f |V .
Moreover, note that if W ⊆ V ⊆ U , we can restrict to W by first restricting to
V , and so ρUW = ρVW ρUV . The collection of the sets C(U, Y ) together with
their restriction maps ρUV constitutes the sheaf of continuous maps from X to
Y .
An essential feature of continuity is that it is a local property; f is continuous
if and only it is continuous in a neighbourhood of every point, and of course
two continuous maps that are equal in a neighbourhood of every point, have to
be equal everywhere. Moreover, continuous functions can be glued: Given an
open covering Uii∈I of an open set U , and continuous functions fi ∈ C(Ui, Y ),
so that for each x ∈ X the value fi(x) does not depend on i, we can patch the
maps fi together to form a continuous map f : U → Y , so that f |Ui = fi for
each i: Just define f(x) = fi(x) for any i such that x ∈ Ui.Essentially, a sheaf on a topological space X is a structure that encodes these
properties. On a topological space, a sheaf can be thought of as a distinguished
set of functions, but they can also be more general objects that behave as sets
of functions. The main aspect is that we want the distinguished properties to
be preserved under restrictions to open sets, and that they are determined from
their local properties.
1.1.1 Presheaves
We begin with the definition of a presheaf.
Definition 1.1. Let X be a topological space. A presheaf of abelian groups Fon X consists of the following two sets of data:
(i) for each open U ⊆ X, an abelian group F(U);
16
Chapter 1. Sheaves
(ii) for each pair of nested opens V ⊆ U a restriction map
ρUV : F(U)→ F(V ).
The restriction maps must furthermore satisfy the following two conditions
(a) for any open U ⊆ X, one has ρUU = idF(U);
(b) for any nested opens W ⊆ V ⊆ U , one has ρUW = ρVW ρUV .
We will usually write s|V for ρUV (s) when s ∈ F(U). The elements of F(U) are
usually called ‘sections’ (or ‘sections over U ’). You may perhaps think of them
as ‘abelian group-valued functions on U ’.
We will often write Γ(U,F) for the group F(U); here Γ is the ‘global sections’-
functor (it is functorial in both U and F).
Of course the notion of a presheaf is not confined to presheaves of abelian
groups. One may speak about presheaves of sets, rings, vector spaces or whatever
you want: Indeed, for any category C one may define presheaves with values
in C . The definition goes just like for abelian groups, the only difference being
that one requires the ‘spaces’ of sections F(U) over open sets U to be objects
in the category C and no longer abelian groups, and of course the restriction
maps are all required to be morphisms in C . One may phrase this definition
purely in categorical terms by introducing the small category openX of open sets
in X whose objects are the open sets, and the morphisms are the inclusion maps
between open sets. With that definition up our sleeve, a presheaf with values in
the category C is just a contravariant functor
F : openX → C .
We are certainly going to meet sheaves with a lot more structure than the mere
structure of abelian groups e.g., like sheaves of rings, but they will all have an
underlying structure of abelian group, so we start with those. That being said,
sheaves of sets play a great role in mathematics, and in algebraic geometry, so
we should not completely wipe them under the rug. Most results we establish
for sheaves of abelian groups can be proved mutatis mutandis for sheaves of sets
as well, as long as they can be formulated in terms of sets.
1.1.2 Sheaves
We are now ready to give the main definition of this chapter:
Definition 1.2. A presheaf F is a sheaf if it satisfies the two conditions:
17
1.1. Sheaves and presheaves
(i) (Locality axiom) Given an open subset U ⊆ X with an open covering
U = Uii∈I , and a section s ∈ F(U), then if s|Ui = 0 for all i, then
s = 0 ∈ F(U).
(ii) (Gluing axiom) If U and U are as in (i), and if si ∈ F(Ui) is a collection of
matching on the overlaps, i.e.,
si|Ui∩Uj = sj|Ui∩Uj ∀i, j ∈ I,
then there exists a section s ∈ F(U) so that s|Ui = si for all i.
Note that the condition (i) says that sections are uniquely determined from
their restrictions to smaller open sets. The condition (ii) says that you can patch
together local sections to a global one, provided they agree on overlaps, just like
in the example of continuous functions.
Remark 1.3. There is a nice concise way of formulating the two sheaf axioms
at once. For each open over U = Ui of an open set U ⊆ X there is a sequence
0→ F(U)α−→∏i
F(Ui)β−→∏i,j
F(Ui ∩ Uj)
where the maps α, β are defined by α(s) = (s|Ui)i, and β(si) = (si|Ui∩Uj −sj|Ui∩Uj)i,j, Then F is a sheaf if and only if these sequences are exact. Indeed,
you should check that the Locality axiom is equivalent to α being injective, and
the Gluing axiom to ker β = imα. This reformulation is sometimes handy when
proving that a given presheaf is a sheaf.
1.1.3 Subsheaves and saturation of subpresheaves
If F is a presheaf on X, a subpresheaf G is a presheaf such that G(U) ⊆ F(U)
for every open U , and such that the restriction maps of G are induced by those
of F . If F and G are sheaves, of course G is called a subsheaf.
Let F be a sheaf on X and G ⊆ F a subpresheaf. We say that a section
s ∈ F(U) locally lies in G if for some open covering Ui i∈I one has s|Ui ∈ G(Ui)
for each i.
Definition 1.4. We define the sheaf saturation G of G in F by letting the
sections of G over U be the sections of F over U that locally lie in G.
The sheaf saturation G is again a subpresheaf of F (with restriction maps
being the ones induced from F). In fact, G is, almost by definition, a sheaf. The
locality axiom is satisfied for G, since it holds for F . Furthermore, given a set
of patching data for G; that is, an open covering Uii∈I of an open set U and
18
Chapter 1. Sheaves
sections si over Ui matching on intersections, the si’s can be glued together in
F since F is a sheaf, and since they are born to locally lie in G, the patch lies
locally in G as well and is a section of G over U . So also the Gluing axiom holds
for G.
So G is basically the smallest subsheaf of F which contains G. If G already is
a sheaf, we don’t get anything new, so that G = G.
1.2 A bunch of examples
Example 1.5. Take X = Rn and let C(X,R) be the sheaf whose sections over
an open set U is the ring of continuous real valued functions on U , and the
restriction maps ρUV are just the good old restriction of functions. Then C(X,R)
is a sheaf of rings (functions can be added and multiplied), and both the sheaf
axioms are satisfied. You should convince yourself that this is true.
Example 1.6. For a second familiar example let X ⊆ C be any open set. On
X one has the sheaf OX of holomorphic functions. That is, for any open U ⊆ X
the sections OX(U) is the ring of holomorphic (i.e., complex analytic) functions
on U . One can relax the condition of holomorphy to get the larger sheaf KX
of meromorphic functions on X. This sheaf contains OX , and the sections over
an open U are the meromorphic functions on U . In a similar way, one can
get smaller sheaves contained in OX by imposing vanishing conditions on the
functions. For example if p ∈ X is any point, one has the sheaf denoted mp of
holomorphic functions vanishing at p. As the name indicates the sections of mp
over U are holomorphic functions in U , and if p ∈ U , one requires additionally
that they should vanish at p. Convince yourself that this indeed is a subsheaf of
OX .
Exercise 1. Let X ⊆ C be an open set, and assume a1, . . . , an are distinct
points in X and n1, . . . , nr be natural numbers. Define F(U) to be the set of
those meromorphic functions f ∈ Γ(U,KX) holomorphic away from the ai’s and
having a pole order bounded by ni at ai. Show that F is a sheaf of abelian
groups. Is it a sheaf of rings?
Example 1.7 (A presheaf that is not a sheaf). The sheaf C(X,R) from Example
1.5 has a subpresheaf F which is not a sheaf: For each open U let F(U) =
Cb(U,R), the group of bounded continuous functions. F is not a sheaf, because
you can’t glue; e.g., the function f(x) = x is bounded on any open ball Br(0) =
x ∈ Rn|‖x‖ < r, but the restrictions f |Bn do not give you a bounded continuous
function on all of⋃n≥0Bn = Rn.
Example 1.8. (Constant presheaf) For any space X and any abelian group A
one has the constant presheaf whose group of sections over any nonempty open
19
1.2. A bunch of examples
set U equals A and equals 0 if U = ∅. This is not a sheaf, since if U ∪ U ′ is a
disjoint union, any choice of elements a, a′ ∈ A will give sections over U and
U ′ respectively, and they match on the empty intersection! But if a 6= a′, they
cannot be glued. In fact, this is a sheaf if and only if any two non-empty open
subsets of X have non-empty intersection. Varieties with Zariski topology are
examples of such spaces.
Example 1.9 (Skyscraper sheaf). Let A be a group. For x ∈ X define a presheaf
Ax by Ax(U) = A if x ∈ U and Ax(U) = 0 otherwise. This is a sheaf, usually
called a ‘skyscraper’ sheaf.
The two next examples suggest another way of thinking about sheaves: They
are ‘sections’ of maps Y → X into X:
Example 1.10 (Tautological sheaf on projective space). Let P (V ) ' Pn be the
projective space of lines in an (n+ 1)-dimensional vector space V . Let
L ⊆ P (V )× V
be the set of pairs ([l], v) where [l] denotes the class of a line l ⊆ V and v ∈ l.Let π be the projection to the first factor. Note that the fiber over any point
p ∈ P (V ) is a 1-dimensional vector space (π−1(p) is the line in V corresponding
to p). This is a basic example of a non-trivial vector bundle. We will discuss
these later in the course.
For now, we can use it to define the so-called tautological sheaf on P (V ): For
any U ⊆ P (V ), let
F(U) = σ : U → L|π σ = idU
This is indeed a sheaf, usually denoted by O(−1).
Example 1.11. Here is a related example from topology. Consider the Mobius
strip M with its projection π : M → S1 to S1. This is an example of a fiber
bundle; any fiber of π is homeomorphic to the unit interval [0, 1]. The sheaf of
sections of π is given over an open set U ⊆ S1 by
F(U) = s : U →M |π s = idU
Again, this is a sheaf.
There is also the trivial fiber bundle π : S1×[0, 1]→ S1 given by the projection
to the first factor. This is not homeomorphic to the bundle above. [Brain teaser:
Prove this.]
Our main interest in this course will still be the following:
20
Chapter 1. Sheaves
Example 1.12. Let V be an algebraic variety (quasi-projective, as in Hartshorne
Ch. I) with the Zariski topology. For each open U ⊆ X, define OV (U) to be
the ring of regular functions U → k. This is certainly a presheaf, and in fact, a
sheaf.
1.3 Stalks
Given a presheaf F of abelian groups on X. With every point x ∈ X there is an
associated abelian group Fx called the stalk 1 of F at x. The elements of Fx are
called germs of sections2 near x.
The definition goes as follows: We begin with the disjoint union∐
x∈U F(U)
whose elements we think of as pairs (s, U) where U is any open neighbourhood of
x and s is a section of F over U . We want to identify sections that coincide near x;
that is, we declare (s, U) and (s′, U ′) to be equivalent, and write (s, U) ∼ (s′, U ′),
if there is an open V ⊆ U ∩ U ′ with x ∈ V such that s and s′ coincide on V ;
that is one has
s|V = s′|V .
This is clearly a reflexive and symmetric relation. And it is transitive as well.
Indeed, if (s, U) ∼ (s′, U ′) and (s′, U ′) ∼ (s′′, U ′′), one may find open neighbour-
hoods V ⊆ U ∩ U ′ and V ′ ⊆ U ′ ∩ U ′′ of x over which s and s′, respectively s′
and s′′, coincide. Clearly s and s′′ then coincide over the intersection V ′ ∩ V ,
and the relation ∼ is an equivalence relation.
Definition 1.13. The stalk Fx at x ∈ X is by definition the set of equivalence
classes
Fx =∐x∈U
F(U)/ ∼ .
In case F is a sheaf of abelian groups, the stalks Fx are all abelian groups.
This is not a priori obvious, since sections over different open sets can not be
added. However if (s, U) and (s′, U ′) are given, the restrictions s|V and s′|V to
1Norsk: ‘stilk’2Norsk: ‘seksjonskimer’ eller bare ‘kimer’
21
1.3. Stalks
any open V ⊆ U ∩ U ′ can be added, and this suffices to define an abelian group
structure on the stalks.
1.3.1 The germ of a section
For any neighbourhood U of x ∈ X, there is a map F(U) → Fx sending a
section s to the equivalence class where the pair (s, U) belongs. This class is
called the germ of s at x, and a common notation for it is sx. The map is
a homomorphism of abelian groups (rings, modules,..) as one easily verifies.
Clearly one has sx = (s|V )x for any other open neighbourhood V of x contained
in U—or expressed in the lingo of diagrams—the following diagram commutes:
F(U) //
ρUV
Fx
F(V ).
<<
When working with sheaves and stalks, it is important to remember the three
following properties; the third one is easily deduced from the two first.
• The germ sx of s vanishes if and only s vanishes on a neighbourhood of x,
i.e., there is an open neighbourhood U of x with s|U = 0.
• All elements of the stalk Fx are germs, i.e., of the shape sx for some section
s over an open neighbourhood of x.
• The abelian sheaf F is the zero sheaf if and only if all stalks are zero, i.e.,
Fx = 0 for all x ∈ X.
Example 1.14. Let X = C, and let OX(U) be the ring of holomorphic functions
on U . If f and g are two sections of OX over a neigbourhood of p having the
same germ at p, i.e., f = g ∈ OX,p, then f = g where they are both defined.
If you replace ‘holomorphic’ by C∞, then f and g having the same germ at p
only implies that the derivatives of f and g at p of all orders of coincide, i.e.,
f (n)(p) = g(n)(p) for all n ≥ 0, but not much else.
1.3.2 A primer on limits
Another notation for the stalk of F at x is
Fx = lim−→U3xF(U).
22
Chapter 1. Sheaves
This is the direct limit (or ‘colimit’) of all F(U) when U runs over the partially
ordered set of open sets containing x. Taking the direct limit is a general
construction in algebra, so let us give a few more details here for future reference.
A directed set I is a partially ordered set such that for each pair of elements
i, j ∈ I there is a third element k such that i < k and j < k. If I is a directed
set and C is a category, a directed system of objects in C is a collection Gii∈Iof objects in C, such that for all i < j there is a morphism fij : Gi → Gj, with
fii = id and fjk fij = fik.
Direct limits
The direct limit of Gi, denoted by G = lim−→i∈I Gi is an object in C, equipped
with morphisms gi : Gi → G which satisfy the following universal property: For
any object H ∈ C with maps hi : Gi → H such that hi = hj fij for each i ≤ j,
there is a unique map h : G→ H making the following diagram commute:
Gihi //
fi
H
Gh
>>
Heuristically, two elements in the direct limit represent the same element in the
direct limit if they are ‘eventually equal.’
If the Gi are sets (or groups, rings, ..), an explicit construction for this is
the quotient∐
i∈I Gi/ ∼, where and g ∼ h, with g ∈ Gi and h ∈ Gj, if there
exists a k ∈ I such that fik(g) = fjk(h). In the case I is the set of opens, and
GU = F(U), we recover the previous definition of the stalk Fx.
Inverse limits
By reversing all the arrows, we can define the ‘inverse limit’ or ‘projective limit’
of a directed system Gi is defined as above. That is, the maps Gi → Gj are
defined for j < i, and lim←−i∈I Gi is an element of C equipped with universal maps
to each of the Gi.
1.4 Morphisms between (pre)sheaves
A morphism φ : F → G of (pre)sheaves on a space X is collection of maps
φU : F(U)→ G(U) so that the following diagram commutes for each inclusion
23
1.4. Morphisms between (pre)sheaves
V ⊆ U :
F(U)φU //
ρUV
G(U)
ρUV
F(V )φV // G(V )
In this way the abelian sheaves on X form a category AbShX whose objects are
the abelian sheaves on X and the morphisms being the maps between them.
The composition of two maps of sheaves is defined in the obvious way as the
composition of the maps on sections.
Remark 1.15. If one considers the sheaves F and G as contravariant func-
tors on the category openX , a map between them is what is called a natural
transformation (naturlig transformasjon) between the two functors.
As usual, a map φ between two (pre)sheaves F and G is an isomorphism
if it has a two-sided inverse, i.e., a map ψ : G → F such that φ ψ = idG and
ψ φ = idF .
A map φ : F → G between two presheaves F and G induces for every point
x ∈ X a map φx : Fx → Gx between the stalks. Indeed, one may send a pair
(s, U) to the pair (φU (s), U), and since φ behaves well with respect to restrictions,
this assignment is compatible with the equivalence relations; that is, if (s, U)
and (s′, U ′) are equivalent and s and s′ coincide on an open set V ⊆ U ∩U ′, one
has
φU(s)|V = φV (s|V ) = φV (s′|V ) = φU ′(s′)|V .
Obviously (φ ψ)x = φx ψx and (idF)x = idFx , so the assignment φ→ φx is a
functor from the category of abelian sheaves to the category of abelian groups.
Example 1.16. In the case X = R let Cr(X) be the subsheaf of C(X,R)
consisting of functions being r times continuously differentiable (check that
this is indeed a subsheaf !). The differential operator D = d/dx defines a map
D : Cr → Cr−1.
Exercise 2. In the same vein, the differential operator gives a mapD : OX → OX ,
where as previously X ⊆ C is an open set. Show that the assignment
A (U) = f ∈ OX(U) | Df = 0
defines a subsheaf A of OX . Show that if U is a connected open subset of X,
ones has A (U) = C. In general for a not necessarily connected set U , show
that A (U) =∏
π0UC where the product is taken over the set π0U of connected
components of U .
24
Chapter 1. Sheaves
1.5 Kernels and images
Let φ : F → G be a map between two abelian sheaves on X.
Definition 1.17. The kernel kerφ of φ is a subsheaf of F whose sections over
U are just kerφU , i.e., the sections in F(U) mapping to zero under φU : F(U)→G(U).
The requirement in the definition is compatible with the restriction maps
since φV (s|V ) = φU(s)|V , for any section s over the open set U and any open
V ⊆ U . Thus we have defined a subpresheaf of F which, indeed, is a subsheaf:
If si i are gluing data for the kernel, one may glue the si’s to a section s of
F over U . One has φ(s)|Ui = φ(s|Ui) = φ(si) = 0, and from the locality axiom
for G it follows that φ(s) = 0. We leave it to the reader to verify that the stalk
(kerφ)x of kerφ at x equals kerφx. We have proven
Lemma 1.18. Let φ : F → G be a map of abelian sheaves. The kernel kerφ is
a subsheaf of F having the two properties
• Taking the kernel commutes with taking sections: Γ(U, kerφ) = kerφU ,
• Forming the kernel commutes with forming stalks: (kerφ)x = kerφx.
One says that the map φ is injective if kerφ = 0. This is, in view of the
previous lemma, equivalent to the condition kerφx = 0 for all x, i.e., that all φxare injective. One often expresses this in a slightly imprecise manner by saying
that φ is injective on all stalks.
When it comes to images the situation is not as nice as for kernels. One
defines the image presheaf contained in G by letting the sections over U be equal
to imφU . However this is not necessarily a sheaf. If si = φUi(ti) are gluing data
for the image presheaf there is no reason for the ti’s to match on the intersections
Uij = Uj ∩ Uj even if the si’s do; the differences ti|Uij − tj|Uij may very well
be non-zero sections of the kernel of φUij ! In fact, we will see several explicit
examples of this later.
To remedy this situation, we simply make the following definition:
Definition 1.19. For a morphism φ : F → G we define the sheaf imφ to be
the saturation of the image presheaf U 7→ imφU , i.e., the smallest subsheaf
containing the images.
Forming the image of a map of sheaves does not always commute with taking
sections, but as we shall verify in the upcoming lemma, forming images commutes
with forming stalks. One has
25
1.5. Kernels and images
Lemma 1.20. Let φ : F → G be a map of abelian sheaves. The image imφ is a
subsheaf of G.
• For all open subsets U of X one has imφU ⊆ Γ(U, imφ).
• For all x ∈ X one has (imφ)x = imφx.
Proof. We only have to verify the last statement, so let tx ∈ imφx and pick
an sx ∈ Fx with φx(sx) = tx. This means that over some open neighbourhood
V , one has φV (s) = t, and t is a section of imφ over V . This shows that
imφx ⊆ (imφ)x. The other way around, if t is a section of G over an open U
containing x locally lying in image presheaf, the restriction t ⊆ V lies in imφVfor some smaller neighbourhood V of x, hence the germ tx lies in imφx.
The map φ : F → G is said to be surjective if the image sheaf imφ = G. This
is equivalent to all the stalk-maps φx being surjective (one says φ is surjective on
stalks), but it does not imply that all maps φU on sections are surjective. Here
is a counterexample:
Example 1.21. Let OX be the sheaf of holomorphic functions on X = CP1
and let p, q ∈ X be two distinct points. Let OX(−p) and OX(−q) be the two
subsheaves of OX of holomorphic functions vanishing at p and q respectively and
consider the map
φ : OX(−p)⊕ OX(−q)→ OX
defined by (f, g) 7→ f + g. Then φx is surjective for every x ∈ X [Check this!],
but φX cannot be surjective, since by Liouville’s theorem, the only globally
homolophic functions are the constants, and so OX(−p)(X) = OX(−q)(X) = 0
and OX(X) = C.
In particular, since imφ(X) = 0, this shows that the naive image presheaf
F(U) = imφU is not a sheaf.
However, for the map φ to be an isomorphism, one has the following
Proposition 1.22. Let φ : F → G be a map of abelian sheaves. Then the
following four conditions are equivalent
• The map φ is an isomorphism.
• For every x ∈ X the map on stalks φx : Fx → Gx is an isomorphism,
• One has kerφ = 0 and imφ = G,
• For all open subsets U ⊆ X the map on sections φU : F(U)→ G(U) is an
isomorphism.
26
Chapter 1. Sheaves
Proof. Most of the implications are straightforward from what we have done so
far and are left to student. We comment just on the two most the salient points.
Firstly, assume that all the stalk maps φx : Fx → Gx are isomorphism, and
let us deduce from this that all the maps φU : F(U) → G(U) on sections are
isomorphisms. It is clear that φU is injective since forming kernels commute with
taking sections as in lemma 1.18 on page 25. So assume that s ∈ G(U). For
each x ∈ U there is a germ t(x) induced by a section tx of F over some open
neighbourhood Ux of x satisfying φx(t(x)) = s(x). Let sx = s|Ux .After possibly having shrunken the neighbourhood Ux, one has φUx(tx) = sx.
The sx’s match on the intersections Ux ∩ Ux′—they are all restrictions of the
section s—and therefore the tx’s match as well because φUx is injective as we
just observed. Hence, the tx’s patch together to a section t of F that must map
to s since it does so locally.
Secondly, if all the φU ’s are isomorphisms, we have all the inverse maps φ−1U
at our disposal. They commute with restrictions since the maps φU do. Indeed,
from φV ρUV = ρUV φU one obtains ρUV φ−1U = φ−1
V ρUV and the φ−1U ’s
thus define a map φ−1 : G → F of sheaves, which of course, is inverse to φ. We
conclude that φ is an isomorphism.
1.6 Exact sequences of sheaves.
Given a sequence
. . .φi−2 // Fi−1
φi−1 // Fiφi // Fi+1
φi+1 // . . .
of maps of abelian sheaves, we say that the sequence is exact at stage i if
kerφi = imφi−1. The short exact sequences are the ones one most frequently
encounters. They are sequences of the form
0 // F1φ1 // F2
φ2 // F3// 0 (1.6.1)
that are exact at each stage. This is just another and very convenient way
of simultaneously saying that φ1 is injective, that φ2 is surjective and that
imφ1 = kerφ2. Forming the sections of the short exact sequence (1.6.1) over
some open set U one finds the sequence
0 // Γ(U,F1) // Γ(U,F2) // Γ(U,F3) (1.6.2)
which is exact at the three leftmost stages, but the map Γ(U,F2) // Γ(U,F3)
to the right is not necessarily surjective (although it of course can be). One
way of phrasing this is to say that taking sections over an open set U—that is
27
1.6. Exact sequences of sheaves.
Γ(U, ∗)—is a left exact functor (venstreeksakt funktor). However this is not right
exact in general. The defect of this lacking surjectivity is a fundamental problem
in every part of mathematics where sheaf theory is used, and to cope with it one
has cohomology!
Proposition 1.23. For a short exact sequence 0 → F ′ φ−→ F ψ−→ F ′′ → 0, we
have the following induced exact sequence
0 −→ Γ(U,F ′)φU−→ Γ(U,F )
ψU−→ Γ(U,F ′′),
Proof. The map φ is injective as a map of sheaves, hence injective on all open
sets U , so the sequence above is exact at Γ(U,F1). To see that it is also exact in
the middle, we show thatker(ψU) = im(φU). (The image presheaf is then given
by the kernel of a morphism of sheaves, which is indeed a sheaf.)
Consider for each x ∈ U the the diagram
0 F ′(U) F(U) F ′′(U) 0
0 F ′x Fx F ′′x 0
φU ψU
φx ψx
the bottom row is exact, since the above sequence is exact.
ker(ψU) ⊇ im(φU): Let s ∈ Γ(U,F1) and consider for each x ∈ U the germ
of ψU(φU(s)) in the stalk F ′′x :
(ψU(φU(s)))x = ψx(φx(sx)).
But by exactness, ψx(φx(sx)) = 0 for all x ∈ U . Hence ψU(φU(s)) = 0, so
im(φU) ⊆ ker(ψU).
ker(ψU) ⊆ im(φU) : Let t ∈ ker(ψU), so ψU(t) = 0. Then for all x ∈ U
we have that ψx(tx) = (ψU(t))x = 0, so the germ of t at x is an element in
ker(ψx) = im(φx) by exactness again. Hence for every x ∈ U there is a s′x ∈ F ′x,
say of the form [(s′(x), V(x))] for some open neighborhood V(x) ⊆ U of x and
s′(x) ∈ Γ(V(x),F′), such that φx(s
′x) = tx. Then we have that for x, y ∈ U
φV(x)∩V(y)(s′(x)|V(x)∩V(y)) = t|V(x)∩V(y) = φV(x)∩V(y)(s
′(y)|V(x)∩V(y)),
so that by the injectivity of φV(x)∩V(y) (already proved), we get the required
condition
s′(x)|V(x)∩V(y) = s′(y)|V(x)∩V(y)
for the gluing of the s′(x) for x ∈ U . Therefore we have a section s ∈ Γ(U,F )
28
Chapter 1. Sheaves
with the property that for all x ∈ U
s|V(x) = s′(x).
Now we can conclude that for every x ∈ U
(φU(s))x = φx(sx) = φx(s′x) = tx,
since sx = s′x, which gives φU(s) = t as desired.
Let us give a few examples where the surjectivity on the right fails:
Example 1.24 (The exponential sequence). Let X = C − 0. The non-
vanishing holomorphic functions in an open set U ⊆ X form a multiplicative
group, and there is a sheaf O∗X with these groups as sections. For any f
holomorphic in U the exponential exp f(z) is a section of O∗X . Hence there is an
exact sequence
0 // Z // OXexp // O∗X // 0,
since non-vanishing functions locally have logarithms. However unless U is
simply connected, exp(U) is not surjective. In that case there will always be
some non-vanishing function in U without a logarithm defined throughout U .
Example 1.25 (Differential operators). Let X = C and recall the sheaf OX of
holomorphic functions and the map D : OX → OX sending f(z) to the derivative
f ′(z). There is an exact sequence
0 // C // OXD // OX
// 0.
This hinges on the two following facts. Firstly, a function whose derivative
vanishes identically is locally constant; hence the kernel kerD equals the constant
sheaf CX . Secondly, in small open disks any holomorphic function has a primitive
function, i.e., is a derivative, e.g., if f(z) =∑
n≥0 an(z − a)n in a small disk
around a, the function g(z) =∑
n≥0 an(n+ 1)−1(z− a)n+1 has f(z) as derivative.
However, taking sections over open sets U we merely obtain the sequence
0 // Γ(U,C) // Γ(U,OX)DU // Γ(U,OX).
Whether DU is surjective or not, depends on the topology of U . If U is simply
connected, one deduces from Cauchy’s integral theorem that every holomorphic
function in U is a derivative, so in that case DU is surjective. On the other
hand, if U not simply connected, DU is not surjective; e.g., if U = C− 0 , the
function z−1 is not a derivative in U .
29
1.7. B-sheaves
Example 1.26. Consider an algebraic variety X with the sheaf OX of regular
functions on X. For any point x ∈ X let k(x) denote the skyscraper sheaf (see
Example 1.9) whose only non-zero stalk is the field k located at x. There is a map
of sheaves evx : OX → k(x) sending a function that is regular in a neighbourhood
of x to the value it takes at x. This maps sits in the exact sequence of sheaves
0 // mx// OX
evx // k(x) // 0,
where mx by definition is the kernel of evx (the sections of mx are the functions
vanishing at x). Taking two distinct points x and y in X we find a similar exact
sequence
0 // Ix,y// OX
evx,y // k(x)⊕ k(y) // 0,
where Ix,y is the sheaf of functions vanishing on x and y, and
evx,y(f) =
(0, 0) if x, y /∈ U,(f(x), 0) if x ∈ U but y /∈ U,(0, f(y)) if y ∈ U but x /∈ U,(f(x), f(y)) if x, y ∈ U.
If for example X = P1 (or any other complete variety), there are no global
regular functions on X other than the constants, and hence Γ(X,OX) = k. But
of course, Γ(P1, k(x)⊕ k(y)) = k ⊕ k, so the map evx,y can not be surjective on
global sections.
1.7 B-sheaves
Recall that a basis for a topology on X is a collection of open subsets B such
that any open set of X can be written as a union of elements of B. In many
situations it turns out to be convenient to define a sheaf by saying what it should
be on a specific basis for the topology on X. In this section we will see that
there exists a unique way to produce a sheaf, given sections that glue over open
subsets in B.
Let us first make the following definition:
Definition 1.27. A B-presheaf F consists of the following data: (i) For each
U ∈ B, an abelian group F(U); and (ii) for all U ⊆ V , with U, V ∈ B, a
restriction map ρUV : F(U) → F(V ). As usual, these are required to satisfy
the relations ρUU = idF(U) and ρWU = ρV U ρWV . A B-sheaf is a B-presheaf
satisfying the Locality and Gluing axioms for open sets in B.
30
Chapter 1. Sheaves
Proposition 1.28. Let X be a topological space and let B be a basis for the
topology on X. Then
(i) Every B-sheaf F extends uniquely to a sheaf on X.
(ii) If φ : F → G is a morphism of B-sheaves, then φ extends uniquely to a
morphism between the corresponding sheaves.
Indeed, for any open set U ⊆ X, we can write U as a union of open sets
Ui ∈ B, and then we can define F(U) to be the set of elements si ∈∏
iF(Ui)
such that si|Ui∩Uj = sj|Ui∩Uj . This is indeed gives a well-defined sheaf, and it
must be unique, since we can see stalks using open sets in B.
1.8 A family of examples – Godement sheaves
We will consider a class of rather peculiar sheaves, called Godement sheaves, to
demonstrate the versatility and the generality of the notion of sheaves.
Let X any topological space. Assume that we for each point x ∈ X are given
an abelian group Ax. The groups Ax can be chosen in a completely arbitrary
way, at random if you will. The choice of these groups gives rise to a sheaf Aon X whose sections over an open set U ⊆ X are given as
Γ(U,A ) =∏x∈U
Ax,
and whose restriction maps are defined as the natural projections
ρUV :∏x∈U
Ax →∏x∈V
Ax,
where V ⊆ U is any pair of open subsets of X. The restriction map just “throws
away” the components at points in U not lying in V .
Proposition 1.29. A is a sheaf.
Proof. The Locality condition holds since if the family Ui i of open sets covers
U , any point x0 ∈ U lies in some Ui0 , so if s = (ax)x∈U ∈ Γ(U,A ) is a section,
the component ax0 survives in the projection onto Γ(Ui,A ) =∏
x∈Ui Ax. Hence
if s|Ui = 0 for all i, it follows that s = 0.
The Gluing condition holds: Assume given an open cover Ui i of U and
sections si ∈ Ui matching on the intersections Ui ∩ Uj. The matching conditions
mean that the component of si at a point x is the same whatever i is as long as
x ∈ Ui. Hence we get a section s of A over U by using this common component
as the component of s at x. It is clear that s|Ui = si.
31
1.8. A family of examples – Godement sheaves
Definition 1.30. The sheaf A is sometimes called the Godement sheaf of the
collection Ax .
The construction is not confined to abelian groups, but works for any category
where general products exist; like sets, rings, etc.
So what is the stalk of A at a point x? Don’t ask that question! The group
can be fairly complicated. Of course there is a map Ax → Ax, but that is in
general the best you can say. For example, suppose Ay = Z/2 for all y ∈ X, and
that x = limxn with xn 6= xm for all n 6= m. Then Ax maps onto the (infinite)
set of tails of sequences of 0s and 1s. Namely, every open neighbourhood of x
contains almost all of the xn. On the other hand, if every neighbourhood of x
contains a point y such that Ay = 0, then Ax = 0.
1.8.1 Skyscraper sheaves.
This is a very special instance of a Godement sheaf where all the abelian groups
Ax are zero except for one. So we fix just one abelian group A, and we choose a
point x ∈ X. The corresponding Godement sheaf is denoted A(x) and is called
a skyscraper sheaf (skyskraperknippe). The sections are described by
Γ(U,A(x)) =
A if x ∈ U,0 otherwise .
Contrary to the general case, in this case the stalks are easy to describe. One
verifies easily that they are zero everywhere except at x, where the stalk equals
A. Indeed, if y 6= x, then y lies in the open set X − x over which all sections
of x∗A vanish.
In most spaces concerning us there are plenty of non-closed points. For such
points one still has the Godement sheaf A(x), but the argument above does not
work, and the description of the stalks are somehow more complicated. One
would hardly call A(x) a skyscraper sheaf.
Exercise 3. Assume that x is not closed and let Z = x be the closure of
the singleton x . Show that the stalks of A(x) are A(x)y = 0 if y 6∈ Z and
A(x)y = A for points y belonging to Z
Slightly generalising the construction of a skyscraper sheaf, one may form
the Godement sheaf A defined by a finite set of distinct closed points x1, . . . , xrand corresponding abelian groups A1, . . . , Ar. Then one sees, having in mind
that an empty direct sum is zero, that the sections of A over an open set U is
32
Chapter 1. Sheaves
given as Γ(U,A ) =⊕
xi∈U Ai. The stalks of A are
Ax =
0 when x 6= xi for all i
Ai when x = xi.
One is tempted to call such a sheaf a barcode sheaf (barcodeknippe)!
Example 1.31 (A set of peculiar examples). Of course if Z is a discrete subset
of X, not necessarily finite, one may associate a Godement sheaf A with any
family Az z∈Z of abelian groups indexed by Z (letting Ax = 0 if x /∈ Z) and
the sections over an open set U is by definition of a Godement sheaf given by
Γ(U,A ) =∏
z∈Z∩U
Az
Just as in the finite case the stalks will be
Ax = Ax
However if Z has accumulation points this is no longer true. Assume for simplicity
that zi is a sequence converging to z. Two sequences of elements ai and a′i are
said to define the same tail if there is some N such that ai = a′i for i ≥ N . Then
the stalk at z will be the space of tails from the Ai, that is, Az =∏
iAi/ ∼.
1.8.2 The Godement sheaf associated with a presheaf
Assume F is a given abelian presheaf on X. The stalks Fx of F of course give a
collection of abelian groups indexed by points in X, good as any other, and we
may form the corresponding Godement sheaf which we denote by Π(F). The
sections of Π(F) are given by
Γ(U,Π(F)) =∏x∈U
Fx, (1.8.1)
and the restriction maps are the projections like for any Godement sheaf. Note
that this sheaf can be very big.
There is an obvious and canonical map
κF : F → Π(F)
sending a section s ∈ F(U) to the element (sx)x∈U of the product in (1.8.1). This
map is functorial in F , for if φ : F → G is a map, one has the stalkwise maps
φx : Fx → Gx, and by taking appropriate products of these, we obtain a map
33
1.9. Sheafification
Π(φ) : Π(F)→ Π(G). This sheaf is sometimes called the sheaf of discontinuous
sections of F . On sections over U it has the effect
Π(φ)((sx)x∈U
)=(φx(sx)
)x∈U
and there is a commutative diagram
F κF //
φ
Π(F)
Π(φ)
G κG// Π(G).
(1.8.2)
It is not hard to check that Π(φ ψ) = Π(φ) Πψ and that Π(idF) = idΠ(F).
1.9 Sheafification
Given any presheaf F on X, there is a canonical way of defining a sheaf F+ that
in some sense is the sheaf that best approximates it. What prevents the presheaf
F from being a sheaf is of course the failure of one or both of the sheaves axioms.
To remedy this one must factor out all sections of F whose germs are everywhere
zero, and one has to enrich F by adding enough new sections to be able to glue
from local sections.
A nice and canonical way of doing this is by using the Godement sheaf Π(F)
associated with F . Recall the canonical map κ : F → Π(F) that sends a section s
of F over an open U to the sequence of germs (sx)x∈U ∈∏
x∈U Fx = Γ(U,Π(F)).
This map clearly kills the doomed sections, i.e., those whose germs all vanish.
Now we can get an actual sheaf, by taking the image of κ in Π(F):
Definition 1.32. For a presheaf F on X, we define its sheafification F+ as the
image sheaf imκ in Π(F) (in other words, F+ is the saturation of the subpresheaf
U 7→ imκU in Π(F)).
Abusing notation slightly, we also write κ : F → F+ for the canonical map.
So what are the sections of F+? Over an open set U ⊆ X they are charac-
terized by those locally coming from F . To be completely explicit about it, F+
is the subsheaf of Π(F) given by
F+(U) = (sx) ∈∏x∈U
Fx|(sx) locally lies in F
where, as before, the sentence in the bracket means the following: for each p ∈ U ,
there exists an open neighbourhood V ⊆ U containing p, and a section t ∈ F(V )
such that for all v ∈ V we have sv = tv in Fv.
34
Chapter 1. Sheaves
Lemma 1.33. The sheafification F+ depends functorially on F . Moreover, if
F is a sheaf, κ : F → F+ is an isomorphism, so that F and F+ are canonically
isomorphic.
Proof. Assume that φ : F → G is a map between two presheaves. Let s be
section of Π(F) over some open set U , so that s locally lies in F . In other words
there is a covering Ui of U and sections s′i of F over Ui with s|Ui = κF(s′i).
Hence by (1.8.2) one has
Π(φ)(s|Ui) = Π(φ)(κF(s′i)) = κG(φ(s′i)).
We see that Π(φ)(s) lies locally in G, and Π(φ) takes F+ into G+. There is a
commutative diagram
F //
φ
F+ //
φ+
Π(F)
Π(φ)
G // G+ // Π(G)
In case F is a sheaf, the map κF maps F injectively into Π(F) and F = imκFis its own saturation, hence κ is an isomorphism.
Proposition 1.34. Given a presheaf F on X. Then the sheaf F+ and the
natural map κ : F → F+ enjoys the universal property that any map F → Gwhere G is sheaf, factors through F+ in a unique way. This characterises F+ up
to unique isomorphisms.
Proof. If G in the diagram above is a sheaf, the map G → G+ is an isomorphism
and φ+ takes values in G and provides the wanted factorization. The uniqueness
statement is a general feature of solutions to universal problems.
From what we have proved above, we have that for a presheaf F and a sheaf
GHomPrSh(F , i(G)) = HomSh(F+,G)
where on the right hand side, i(G) denotes G but considered as a presheaf. A
fancy way of restating this is that the sheafification functor F 7→ F+ is an
adjoint to the forgetful functor i : Sh→ PrSh from sheaves to presheaves.
Lemma 1.35. Sheafification preserves stalks: Fx = (F+)x via κx.
Proof. The map κx : Fx → (F+)x is injective, since Fx → (Π(F))x is injective.
To show that it is surjective, suppose that s ∈ (F+)x. We can find an open
neighbourhood U of x such that s is the equivalence class of (s, U) with s ∈F+(U). By definition, this means there exists an open neighbourhood V ⊆ U of
x and a section t ∈ F(V ) such that s|V is the image of t in Π(F)(V ). Clearly
the class of (t, V ) defines an element of Fx mapping to s.
35
1.10. Quotient sheaves and cokernels
Example 1.36 (Constant sheaves). Recall Example 1.8 which showed that the
constant presheaf U 7→ A is usually not a sheaf (A is an abelian group). However,
the sheafification of this constant presheaf is a sheaf, and that is what we call
the constant sheaf with value A and denote by AX .
That being said, the sheaf AX is not quite worthy of the name as is is not
quite constant. One has
Γ(U,AX) =∏π0(U)
A, (1.9.1)
where π0(U) denotes the set of connected components of the open set U .
1.10 Quotient sheaves and cokernels
The main application of the sheafification process we just described is to show
that one may form quotient sheaves. So assume that G ⊆ F are sheaves,
and define a presheaf whose sections over U is the quotient F(U)/G(U). The
restriction maps of F and G respect the inclusions G(U) ⊆ F(U) and hence pass
to the quotients F(U)/G(U). We use these maps as the restriction maps for the
quotient presheaf. The quotient sheaf F/G is by definition the sheafification of
this quotient presheaf. The quotient sits in the exact sequence
0 // G // F // F/G // 0 .
The cokernel cokerφ of the map φ : F → G of abelian sheaves is then just the
quotient sheaf G/ imφ; it fits in the exact sequence
0 // kerφ // F φ // G // cokerφ // 0
Example 1.37. To illustrate why we have to sheafify in these constructions,
consider again the exponential map in Example 1.24. The naive presheaf U 7→coker exp(U) is not a sheaf: the class of the function f(z) = z restricts to 0 in
coker exp on sufficiently small open sets, but it is itself not zero (since otherwise
we would be able to define a global logarithm on C− 0)
1.11 The pushforward of a sheaf
So far we have been interested in various constructions of sheaves on a fixed space
X. These constructions become more interesting when one involve continuous
maps between spaces, and try to transfer sheaves between them.
Let X and Y be two topological spaces with a continuous map f : X → Y
between them. Assume that F is an abelian sheaf on X. This allows us to define
36
Chapter 1. Sheaves
an abelian sheaf f∗F on Y in by specifying the sections of f∗F over the open set
U ⊆ Y to be:
(f∗F)(U) = F(f−1U),
and the restriction maps are those of F .
The sheaf f∗F is called the pushforward sheaf or the direct image of F . It
is straightforward to see that f∗F is a sheaf and not just a presheaf. Indeed,
if Ui i is an open covering of U a set of patching data consists of section
si ∈ Γ(Ui, f∗F) = Γ(f−1U,F). That they match on the intersection means that
they coincide in Γ(Ui ∩ Uj, f∗F) = Γ(f−1Ui ∩ f−1Uj, F ), and therefore can be
glued to a section in Γ(f−1U,F) = Γ(U, f∗F) since F is a sheaf. The Locality
axiom also follows for f∗F because it holds on F .
If you want, the pushforward sheaf f∗F is just the restriction of F to the
subcategory of the open sets in X which are inverse images of open sets in Y .
Example 1.38. Let i : x → X be the inclusion of a point in X. If G is the
constant sheaf of a group G on x, then i∗G is the skyscraper sheaf G(x) from
Example 1.9.
Example 1.39. Consider an affine variety X ⊆ An and let i : X → An be the
inclusion. For each open U ⊆ An define IX(U) = f ∈ OAN (U)|f(x) = 0∀x ∈X. Then IX is a sheaf (of ideals) and we have an exact sequence
0→ IX → OAn → i∗OX → 0
This is a functorial construction:
Lemma 1.40. If g : X → Y and f : Y → Z are continuous maps between
topological spaces, and F is a sheaf on X, one has
(f g)∗F = f∗(g∗F).
Remark, this is indeed an equality, not merely an isomorphism.
Proof. A good exercise in manipulating sections.
Lemma 1.41. The functor f∗ is left exact. That is: Given an exact sequence of
sheaves on X
0 // F ′ // // F ′′ // 0
then the following sequence is exact
0 // f∗F ′ // f∗F // f∗F ′′.
Proof. As f∗F are sections of F over inverse images of opens in Y , the lemma
follows readily from the exactness of the sequence (1.6.2) on page 27 above.
37
1.12. Inverse image sheaf
Exercise 4. Denote by ∗ a one point set. Let f : X → ∗ be the one and
only map. Show that f∗F = Γ(X,F) (where strictly speaking Γ(X,F) stands
for the constant sheaf on ∗ ).
1.12 Inverse image sheaf
If F is a sheaf on X and U ⊆ X is an open subset, then F defines a sheaf on U
by restriction of sections. For an arbitrary subset Z ⊆ X, this naive restriction
does not give a sheaf on Z so easily, because an open subset V ⊆ Z, is usually
not open in X. However, as in the definition of stalks, we can take limits of
F(U) as U runs over smaller and smaller open subsets containing V . This gives
the following definition:
Definition 1.42. If i : Z → X is the inclusion of a closed subset Z ⊆ X, we
define the restriction F|Z as the sheafification of the following presheaf:
V 7→ lim−→U⊇VF(U)
We can extend this idea to any continuous map f : X → Y and a sheaf G on
Y . This gives the inverse image of G, denoted f−1G, which we define as follows.
As above, we know the values of G on open subsets V on Y , but we want to
define a sheaf on X. Here the image of an open set U , f(U) need not be open in
Y , but we can look at equivalence classes of G(V ) as V gets closer and closer to
f(U). This gives the following definition:
Definition 1.43. Let f : X → Y be a continuous map; G a presheaf on Y . We
define the inverse image f−1G as the sheafification of the following presheaf:
f−1p G : U 7→ lim−→
V⊇f(U)
G(V ). (1.12.1)
Note in particular that the stalk of f−1G at a point x ∈ X is isomorphic to
Gf(x). Indeed, it suffices to verify this on the level of presheaves:
(f−1p G)x = lim−→
U3xf−1p G(U) = lim−→
U3xlim−→
V⊇f(U)
G(V )
= lim−→V 3f(x)
G(V ) = Gf(x)
1.12.1 Adjoint property
The definition of f−1G is natural, but a little bit hard to work with for actual
computations, since it involves both taking a direct limit over open sets and a
38
Chapter 1. Sheaves
sheafification. What’s more important is what this sheaf does: It is the adjoint
of the functor f∗ as a functor from ShX to ShY . The precise statement is the
following:
Theorem 1.44. Let f : X → Y be a morphism, and let F be a sheaf on X and
let G be a presheaf on Y . Then we have a natural bijection
HomY (G, f∗F) ' HomX(f−1G,F)
which is functorial in F and G.
So sheaf morphisms φ : G → f∗F correspond bijectively to maps f−1G → F .
In particular, applying this to F = f−1G, and G = f∗F with the identity maps,
we see that there are canonical maps
η : G → f∗f−1G,
which is functorial in G, and
ν : f−1f∗F → F
which is functorial in F .
To prove this theorem, it will be convenient to introduce some notation. Let
us define an f -map Λ : G → F to be a collection of maps ΛV : G(V )→ F(f−1(V ))
indexed by open subsets V ⊆ Y such that
G(V )ΛV//
ρG
F(f−1V )
ρF
G(V ′)ΛV ′ // F(f−1V ′)
commutes for all V ′ ⊆ V ⊆ Y open. With this definition in mind, we can now
prove the following lemma, which implies the above theorem:
Lemma 1.45. Let f : X → Y be a continuous map. Let F be a sheaf on X and
let G be a sheaf on Y . There are canonical bijections between the following four
sets:
(1) The set of f -maps Λ : G → F .
(2) The set of sheaf maps G → f∗F .
(3) The set of sheaf maps f−1G → F .
(4) The set of presheaf maps f−1p G → F .
39
1.12. Inverse image sheaf
Proof. The bijection between (3) and (4) follows by the adjoint property of
sheafification (becuase F is a sheaf), so it suffices to consider the sets in (1), (2)
and (4).
Let us first consider (1) and (2). If we are given a map of sheaves φ : G → f∗F ,
we have a map φV : G(V ) → f∗F(V ) = F(f−1V ) for each open set V ⊆ Y .
By the definition of a sheaf map, this commutes with the various restriction
mappings to smaller opens V ′ ⊆ V , so we get a well-defined f -map Λ : G → F .
Conversely, it is clear that any f -map Λ appears from a map of sheaves in this
manner, so we have established the desired bijection.
For the bijection between (1) and (4), suppose we are given a map of
presheaves f−1p G → F , so that we have a map
lim−→W⊇f(U)
G(W )→ F(U).
Applying this to U = f−1(V ), and compositing with the map G(V ) into the
direct limit, we get a map G(V )→ F(f−1V ). Again, this is compatible with the
restriction maps to smaller open sets V ′ ⊆ V , so we get a well-defined f -map
Λ : G → F . Conversely, it is clear that any f -map Λ arises in this manner, i.e.,
each ΛV factors as
G(V ) F(f−1V )
lim−→W⊇V G(W )
ΛV
φf−1V
for some map of presheaves φ : f−1p G → F : Just define φU for U ⊆ X by
composing ΛW with the restriction map to get a map G(W )→ F(f−1W )→ F(U)
for W ⊇ f(U) – the fact that Λ is an f -map means that we get an induced map
in the direct limit. Over U = f−1V , this φ makes the above diagram commute.
This completes the proof of the lemma.
40
Chapter 2
Schemes
Affine schemes are the building blocks in the theory of schemes. Every scheme is
locally an affine scheme, i.e., every point has a neighbourhood which is isomorphic
to an affine scheme. This is similar to the notion of a manifold, where topological
space is locally diffeomorphic to an open set in Rn. However, affine schemes are
infinitely more complicated than open sets in Rn and can have extremely bad
singularities.
The affine schemes are basically just a geometric embodiment of rings (com-
mutative with unit). There is a one-to-one correspondence between rings and
affine scheme, and even more, given two rings the ring homomorphisms between
them correspond in a one-to-one with the scheme-maps between the two corre-
sponding scheme, but the maps they change direction. The situation is in perfect
analogy with what happens for affine varieties. The same two correspondences
holds true for those as well. However, in that case there are heavy constraints on
the rings involved, they are confined to be integral domains of finite type over
an algebraically closed field.
2.1 The spectrum of a ring
Following the Grothendieck maxim of always working in the maximal generality
the theory allows, we start out working with a commutative1 ring A with a
unit element. We are going to define a scheme written SpecA, called the prime
spectrum (primspekteret) of A or just the spectrum (spekteret) of A for short.
1Many attempts have been made on defining non-commutative geometry, and there aresome versions around. None however are so far completely satisfying.
41
2.1. The spectrum of a ring
As for any scheme the structure of SpecA has two layers. There is an underlying
topological space on which lives a sheaf of rings.
Recall the situation for affine varieties and assume for a moment that A is
an algebra of finite type over the algebraically closed field k that is an integral
domain. In other words, A is the coordinate ring of a variety, say X. So, the
elements of A are the regular functions on X.
We know by Hilbert’s Nullstellensatz that points of X correspond to the
maximal ideals in A, a point x being associated with the ideal mx of regular
function vanishing at x, and this relation is biunivoque—as the french say—every
maximal ideal is the vanishing ideal of a point.
Algebraically closed sets contained in X are loci where the element of ideals a
in A vanish; i.e., they are of the shape V (a) = x ∈ X | f(x) = 0 all f ∈ a , or
with the concordance of points and maximal ideal in mind this may be phrased as
V (a) = m | m ⊆ A maximal and a ⊆ m . The variety X is supplied with the
Zariski topology whose open sets are the just the complements of the algebraically
closed sets V (a).
The guide line when generalising this is simple: Replace maximal ideals with
prime ideals! This motivates the following definition:
Definition 2.1. For a ring 2 A we define its spectrum as
SpecA = p | p ⊆ A is a prime ideal .
The set SpecA has a topology, which generalizes the Zariski topology on a
variety, and the definitions are verbatim the same. The closed sets in SpecA are
to be those of the shape
V (a) = p ∈ SpecA | p ⊇ a
where a is any ideal in A. Of course one has verify that the axioms for a topology
are satisfied. For closed sets the wording of the axioms is that the union of two
closed and the intersection of any number (finite or infinite) must be closed. And
of course, both the whole space and the empty set must be closed. The family of
subsets of the form V (a) satisfies these axioms, as follows from the next lemma:
Lemma 2.2. Let A be a commutative ring with unit and assume that ai i∈i is
a family of ideals in A. Let a and b be two ideals in A. Then the following three
statements hold true:
• V (a ∩ b) = V (a) ∪ V (b) = V (ab),
• V (∑
i ai) =⋂i V (ai),
2From now on the term ‘ring’ stands for a commutative ring having a unit element.
42
Chapter 2. Schemes
• V (A) = ∅ and V (0) = SpecA.
Proof. Prime ideals are by definition proper ideals, so V (A) = ∅, and of course
the zero-ideal is contained in any ideal so V (0) = SpecA. This proves the last
statement. The second follows as easily, the sum of a family of ideals being
contained in an ideal if and only each of the ideals is.
The first of the three statements in the lemma needs an argument. The
inclusion V (a)∪ V (b) ⊆ V (a∩ b) is clear, so our task is to show that V (a∩ b) ⊆V (a) ∪ V (b). Let p be a prime ideal such that a ∩ b ⊆ p. If b * p, there is
an element b ∈ b with b /∈ p, but since ab ∈ a ∩ b for all a ∈ a, and p contains
a ∩ b, one has ab ∈ p. The ideal p being prime, one deduces that a ∈ p, and
consequently one has the inclusion a ⊆ p.
Corollary 2.3. The set V (a) where a runs through all the ideals in A is the
family of closed sets for a topology on SpecA.
The next lemma is about what inclusions between the closed sets of SpecA
hold, and we recognise them all from the theory of varieties.
Lemma 2.4. One has
• V (a) ⊆ V (b) if and only if√a ⊇√b. In particular one has V (a) = V (
√a).
• V (a) = ∅ if and only if a = A.
• V (a) = SpecA if and only if a ⊆√A.
Proof. The main point of the proof is that radical of an ideal equals the intersec-
tion all the prime ideals containing it, or expressed with a formula:
√a =
⋂p⊇a
p. (2.1.1)
In particular one deduces that a and√a are contained in the same prime ideals
so that V (a) = V (√a). To show the first claim in the lemma we begin with
assuming that V (a) ⊆ V (b). From (2.1.1) we then obtain
√a =
⋂p∈V (a)
p ⊇⋂
p∈V (b)
p =√b.
If now p lies in V (a) and√a ⊇√b, the chain of inclusions p ⊇
√a ⊇√b ⊇ b
imply the inclusion p ∈ V (b) as well. And by that, the first claim is proved.
The second claim follows from the fact that√a = A if and only if a = A
(one has 1n = 1!), and the last holds as the radical of A by definition satisfies√A =
√(0).
43
2.1. The spectrum of a ring
Remark 2.5 (A few words about notation). Many authors, unlike Hartshorne,
like to denote points in SpecA in a way that distinguish points in SpecA from
the corresponding ideals in A, although they are identical. For example by letting
geometric points be indicated by latin letters like x and y, and the corresponding
prime ideals by px or py. This might seem superfluous, but there is a reason.
When we start working with general schemes, points do no more correspond to
prime ideals in a canonical way, so in that case its not only natural but required.
2.1.1 Generic points
The Zariski topology on SpecA differs greatly from the standard topology in
the sense that points (eg., prime ideals) can fail to be closed. In fact, the
next proposition implies that a point x ∈ SpecA is closed if and only if the
corresponding ideal is maximal.
Proposition 2.6. If p is a prime ideal of A, the closure p of the one-point
set p in SpecA equals the closed set V (p).
Proof. If the point p is contained in a smaller closed set than V (p), there is
an ideal a with p ∈ V (a) ⊆ V (p). By lemma 2.4 above this implies that√p ⊆√a ⊆ p, from which it follows that p =
√a, and hence we conclude that
V (a) = V (p).
In general there are of course typically a lot of prime ideals which are not
maximal. In fact, the rings having the property that every prime ideal is maximal
are quite special: in the noetherian case they corresponds exactly to the artinian
rings (in which case the spectrum consists of a finite set of points).
Definition 2.7. We say that a point x is a generic point of the closure V (x) =
x of the singleton x .
2.1.2 Irreducible closed subsets
Recall that a topological space X is said to be irreducible if it can not be written
as the union two proper closed subsets; that is, if X = Z∪Z ′ with Z and Z ′ both
being closed subsets, then either Z = X or Z ′ = X holds true. This concept is
well known from the theory of varieties where it is shown that a closed algebraic
subset V (a) of the affine space An(k) is irreducible if and only if the radical√a
is a prime. The corresponding statement is true in SpecA:
Lemma 2.8. A closed subset Z ⊆ SpecA is irreducible if and only if Z is of
the form Z = V (p) for a prime ideal p.
44
Chapter 2. Schemes
Proof. We have seen that the closure p is irreducible. For the reverse direction,
we let V (a) ⊆ SpecA be an irreducible closed subset. Recall that one has√a =
⋂a⊆p p, and if
√a is not prime, there are more than one prime involved in
the intersection. We may divide them into two different groups thus representing√a as the intersection
√a = b ∩ b′ where b and b′ are ideals both different from
a. One concludes that V (a) = V (b) ∪ V (b′).
2.2 Functoriality
Let A and B be two rings and assume that φ : A→ B is a ring homomorphism.
Let us check that the inverse image of a prime ideal p ⊆ B really is a prime
ideal: That ab ∈ φ−1(p) means that φ(ab) = φ(a)φ(b) ∈ p, so at least one of
φ(a) or φ(b) lies in p. Hence sending p to φ−1(p) gives us a well defined map
SpecB → SpecA,and we shall denote this map by φ∗.
Lemma 2.9. Assume that φ : A→ B is a map of rings. Then the induced map
between the ring spectra φ∗ : SpecB → SpecA is continuous.
Proof. We are to verify that inverse images of closed sets are closed, so let a ⊆ A
be an ideal. The the following equalities hold
(φ∗)−1(V (a)) = p ⊆ B | φ−1(p) ⊇ a = p ⊆ B | p ⊇ φ(a) = V (φ(a)B),
since p ⊇ φ(a) if and only if φ−1(p) ⊇ a as φ−1(φ(a)) ⊇ a.
This is also a functorial construction since clearly φ−1(ψ−1(p)) = (ψφ)−1(p)
where φ and ψ are two composable ring homomorphism, and of course one has
id∗A = idSpecA.
This sets up a contravariant functor from the category rings of rings to the
category top of topological spaces. It sends a ring A to the prime spectrum SpecA
and a map φ : A→ B of rings to the continuous map φ∗ : SpecB → SpecA.
We include two prototype examples:
Example 2.10 (Spec(A/a)). If a ⊆ A is an ideal, the ring homomorphism
A→ A/a induces a map Spec(A/a)→ SpecA. Note that prime ideals in A/a
correspond exactly to prime ideals p in A containing a: This is exactly our closed
set V (a). Thus the induced map on spectra corresponds to the inclusion of V (a)
in SpecA. This is the prototype of a closed immersion.
Example 2.11 (SpecAf ). For an element f ∈ A we can consider the localization
Af of A in which f is inverted, and the corresponding ring homomorphism
A → Af . The prime ideals in the localized ring Af stand in a natural one-
to-one correspondence with the prime ideals p of A not containing f , that is,
45
2.3. First examples
the complement D(f) = SpecA − V (f). The induced map SpecAf → SpecA
corresponds to the inclusion of the open set D(f) of SpecA. This is an example
of an open immersion.
2.3 First examples
Example 2.12 (Fields). If K is a field, the prime spectrum SpecK has only
one element the zero-ideal being the only ideal i K. This also holds true for
local rings with the property that all elements in the maximal ideal are nilpotent,
i.e., the radical√
(o) of the ring is a maximal ideal. For noetherian rings this is
equivalent to the ring being an artinian local rings.
Example 2.13. Show that the converse of the last statement in the example
above is true. That is if SpecA has just one element, A is a local ring all whose
non-units are nilpotent.
Example 2.14. The ring R = C[x]/(x2) is not a field, but it has only one prime
ideal (namely the ideal (x)). Note that (0) is not prime, since x2 = 0, but
x 6∈ (0).
Example 2.15 (Discrete valuation rings). A discrete valuation ring A has only
two prime ideals, the maximal ideal m and the zero ideal (0). So its prime
spectrum SpecA has just two points, and SpecA = η, x with x corresponding
to the maximal ideal m and η corresponding to (0). The point x is closed in
SpecA, and therefore the set η being its complement is open. So η is an open
point! The point η is the generic point of SpecA; its closure is the whole SpecA.
Certainly the prime spectrum SpecA is not Hausdorff, as η is contained in the
only open set containing x, the whole space.
The spectrum of a DVR
Example 2.16 (SpecZ). There are two types of prime ideals in Z . There
is the zero-ideal and there are the maximal ideals (p)Z, one for each prime p.
These are closed points in SpecZ, however one has V (0) = SpecZ, so the point
corresponding to the zero-ideal is a generic point, the closure is the whole of
SpecZ.
46
Chapter 2. Schemes
The spectrum SpecZ— a la Mumford
Example 2.17. If R is a domain, if an ideal of the form (r) r ∈ R is prime,
then r is irreducible (meaning, if r = ab, then either a or b is a unit). Indeed, if
r = ab, then since (r) is prime, then without loss of generality a ∈ (r), so a = xr
for some x ∈ R, and hence r = ab = xrb, so bx = 1 (since R is a domain).
The converse however is not true. Let R = Z[√−5]. Then 2 ∈ R is irreducible,
and 2 · 3 = 6 = (1 +√−5)(1−
√−5), so the ideal (2) is not prime (the factors
involving√−5 are not multiples of 2).
Example 2.18 (PIDs). If R is a principal ideal domain (PID), any ideal is of
the form (r) for some r ∈ R. Also, every PID is a unique factorization domain
(same argument as over Z). So the points of SpecR corresponds to (0) and the
set of irreducible elements (modulo units).
Example 2.19. For R = C[[x]], the non-zero ideals are (xn). R is a PID, so the
prime ideals are (0) and (x).
Example 2.20. The ring R = C[[x2, x3]] is not a PID. The non-zero ideals are
(xn + cxn+1) and (xn, xn+1). Of these, only (x2, x3) is prime (Prove this!).
Exercise 5. Let A′ → A be a surjection of commutative rings whose kernel is
nilpotent. Show that the map SpecA→ SpecA′ is a homeomorphism.
Exercise 6 (Direct products of rings). Assume that e1, . . . , er is a complete set of
orthogonal idempotents in the ring A, meaning that one has 1 = e1 +e2 + · · ·+er,
that eiej = 0 when i 6= j, and that e2i = ei. Such a set of idempotents
corresponds to a decomposition of A as the direct product A = A1 ×A2 × . . . Arwhere Ai = eiA.
(i) Let a be an ideal in A and let ai = aei. Show that ai is an ideal in Ai and
that a = a1 + a2 + · · ·+ ar.
(ii) Show that a is a prime ideal if and only ai = Ai for all but one index i0and ai0 is a prime ideal in Ai0 .
(iii) Show that SpecA = SpecA1 ∪ SpecA2 ∪ · · · ∪ SpecAr, the union being
disjoint and each SpecAi being open in SpecA.
Exercise 7. Assume that A1 and A2 are rings and let A = A1 ×A2. Show that
SpecA is not connected; i.e., SpecA = SpecA1 ∪ SpecA2 and the union being
disjoint with both SpecA1 and SpecA2 being open in SpecA.
47
2.3. First examples
Show that every ideal a in A can be written as a = a1+a2 where a1 ⊆ A1× 0 and a2 ⊆ 0 × A2. Hint: 1A = (1A1 , 0) + (0, 1A2).
Show that a is prime if and only if ai’s is prime in A1× 0 (or in 0 ×A2)
and the other one equals Aj.
Assume that A is a ring. Then SpecA is disconnected, i.e., has more than
two connected components, if and only if A has two idempotents e1 and e2 with
e1 + e2 = 1.
Exercise 8. Show that SpecA is connected if and only if the only two idempo-
tents in A are 1 and 0.
Exercise 9. Show that the closure p of a point p ∈ SpecA is given by p = V (p).
Show that a point p in SpecA is closed if and only if p is a maximal ideal.
Exercise 10. Let a ⊆ A be an ideal. Show that√a =
⋂p⊇a p. Hint: If f /∈
√a
the ideal aAf is an ideal in the localization Af , hence contained in a maximal
ideal.
Exercise 11. Let k be an algebraically closed field. Let X ⊆ Ank be the set of
closed points with the induced topology. Show that X is homeomorphic to the
variety kn, that is kn with the Zariski topology.
2.3.1 The world of schemes vs the world of varieties
Comparing the world of varieties with the world of schemes, one notices many
similarities, but there are also quite a few dramatic differences. And in In
some sense the category of varieties over an algebraically closed field k is a full
subcategory of the category of schemes, as we are going to elucidate in full later
on. Now we just give a glimpse into the situation taking a look at the simplest
case, namely the affine varieties An(k).
If one lets A = k[x1, . . . , xn] denote the ring of polynomials over k in the
variables x1, . . . , xn, one knows thanks to Hilbert’s Nullstellensatz that then
maximal ideals in A stand in a one-to-one correspondence with the points of the
affine space An(k); they are all of the form (x1 − a1, . . . , xn − an) with the ai’s
being elements in k.
The affine variety An(k) is the subset of the scheme Ank = SpecA consisting
of the closed points, or the points in SpecA corresponding to maximal ideals.
The good old Zariski topology on the variety An(k) is the induced topology.
Indeed, the closed sets of the induced topology are by definition all of the form
V (a) ∩ An(k), which clearly equals the “old” V (a) from the world of varieties.
The scheme SpecA has however many more points that are not closed. There
are a great many prime ideals in A not being maximal; at least if n ≥ 2.
48
Chapter 2. Schemes
Example 2.21 (The affine line A1k = Spec k[x]). In the polynomial ring k[x]
all ideals are principal, and all non-zero prime ideals are maximal. They are
of the form (f(x)) where f(x) is an irreducible polynomial, hence of the form
(x− a) since we have assumed that k is algebraically closed. There is only one
non-closed point in Spec k[x], the point η corresponding to the zero-ideal. The
closure η is the whole line A1k. We called η the generic point.
Example 2.22 (The affine plane A2k = Spec k[x1, x2]). In this case the maximal
ideals are of the shape (x1−a1, x2−a2) and constitute all the closed points of A2k.
Every irreducible polynomial f(x1, x2) generates a prime ideal ηf = (f) which
together with the zero ideal are all non-maximal prime ideals. In addition to the
point ηf , The members of the closed set V (f(x1, x2)) are all the closed points
corresponding to ideals (x1 − a1, x2 − a2) containing f(x1, x2). This condition
being equivalent to f(a1, a2) = 0, the closed points of V (f(x1, x2)) correspond
to what one in the world of varieties would call the curve given by the equation
f(x1, x2) = 0.
2.4 Distinguished open sets
There is no way to describe the open sets in SpecA as simply and elegantly
as the closed sets can be described. However there is a natural basis for the
topology on SpecA whose sets are easily defined, and it turns out to be very
useful. For every element f ∈ A we let D(f) be the complement of the closed
set V (f), that is
D(f) = p | f /∈ p .
These are clearly open sets and are called distinguished open sets.
Lemma 2.23. The open sets D(f) form a basis for the topology of SpecA when
f runs trough A.
Proof. We are to show that any open subset U of SpecA can be written as
the union of distinguished open sets, that is, of sets of the form D(f). Let the
complement U c of U be given as U c = V (a), and choose a set fi i of generators
for the ideal a (not necessarily a finite set in general). Then we have
U = V (a)c = V (∑i
(fi))c = (
⋂i
V (fi))c =
⋃i
D(fi).
Exercise 12. Show that D(f) = ∅ if and only if f is nilpotent. Hint: Use that√(0) =
⋂p∈SpecA p.
49
2.5. Examples
2.4.1 Basic properties of distinguished sets
We proceed by giving some of the basic properties of the distinguished open sets
needed in the theory.
Lemma 2.24. A family D(fi) i forms an open covering of SpecA if and only
if one may write 1 =∑
i aifi with the ai’s being elements from A only a finite
number of which are non-zero. The topology of SpecA is quasi-compact.
Proof. One has V (∑
i(fi))c = (
⋂i V (fi))
c =⋃iD(fi), so the open sets D(fi)
constitute a covering if and only if the ideal generated by the fi’s is the whole
ring A, that is 1 belongs there.
This shows in particular that any covering by distinguished open sets may be
reduced to a finite one. As the distinguished sets form a basis for the topology,
any open cover may be refined to one whose sets all are distinguished, and hence
it can be reduced to a finite covering. Thus SpecA is quasi-compact.3
Lemma 2.25. One has D(f) ∩D(g) = D(fg).
Proof. If p is a prime ideal, f /∈ p and g /∈ p hold trues simultaneously if and
only if fg /∈ p.
Lemma 2.26. One has D(g) ⊆ D(f) if and only if gn ∈ (f) for a suitable
natural number n. In particular one has D(f) = D(fn) for all natural numbers
n.
Proof. The inclusion D(g) ⊆ D(f) holds if and only if V (f) ⊆ V (g), and by
lemma 2.4 om page 43 this is true if and only if (g) ⊆√
(f), i.e., if and only if
gn ∈ (f) for a suitable n.
In fact, the inclusion D(g) ⊆ D(f) is equivalent to the localization map
τ : A→ Ag extending to a map Af → Ag. Indeed, τ extends if and only if τ(f),
i.e., f regarded as en element in Ag is invertible, that is to say there is an c ∈ Aand an m ∈ N with gm(fc− 1), in turn, this is equivalent to gm = cf for some c
and some m.
2.5 Examples
One may think about the elements f in A as some sort of functions on SpecA.
They are not functions in the usual sense so we speak about an analogy. If x is
a point in SpecA corresponding to the prime ideal p, one has the fraction field
3In many contexts a compact spaces is by definition also Hausdorff. In our context whereHausdorff topologies are rare, one uses the term quasi-compact for spaces such that every opencovering can be reduced to a finite one.
50
Chapter 2. Schemes
k(x) = K(A/p) of the quotient A/p, and the element f reduced modulo p gives
an element f(x) ∈ k(p), which may considered as the “value” of a at x; clearly
f(x) = 0 if and only if f ∈ p. Be aware that “values” the element f take at
points, are in lying in different fields that might vary with the point. There is
also then pathology of nilpotents. A nilpotent element f in A is contained in all
prime ideals and vanishes everywhere on SpecA, but is still a non-zero function!
Example 2.27 (SpecZ again). Assume that f ∈ SpecZ is a number. In the
analogy with functions, the “value” f takes at a prime p is the residue class of f
modulo p; that is, the of f in the finite field Fp = Z/pZ.
The reduction mod p-map Z → Fp induces a map SpecFp → SpecZ. The
one and only point in SpecFp is sent to the point in SpecZ corresponding to the
maximal ideal (p) generated by p. On the level of structure sheaves, the map is
just the restriction map.
The inclusion Z ⊆ Q of the integers in the field of rational numbers induces
likewise a map SpecQ→ SpecZ, that sends the unique point in SpecQ to the
generic point η of SpecZ.
Every ring A has (since by our convention it contains a unit element) a prime
ring, i.e., the subring generated by 1. Hence there is a canonical (and in fact,
unique) map SpecA→ SpecZ.
This canonical map factors through the map SpecFp → SpecZ described
above if and only if A is of characteristic p. This is clear if one considers the
diagram on the ring level:
Z //
Fp
A
the canonical map Z→ A goes via Fp if and only if A is of characteristic p.
Exercise 13. In the same vain, show that a ring A is Q-algebra (that is, contains
a copy of Q) is and only if the canonical map SpecA→ SpecZ factors through
the generic point SpecQ→ SpecZ.
Example 2.28 (SpecZ[i]). The inclusion Z ⊆ Z[i] induces a continuous map
φ : SpecZ[i]→ SpecZ.
We will study SpecZ[i] by studying the fibers of this map. If p ∈ Z is a prime,
the fiber over (p)Z consists of those primes that contain (p)Z[i]. These come in
three flavours:
• p stays prime in Z[i] and the fiber over (p)Z has one element, namely
(p)Z[i]. This happens if and only if p ≡ 3 mod 4.
51
2.5. Examples
• p splits into a product of two different prime ideals, and the fiber consists
of these two points. This happens if and only if p ≡ 1 mod 4.
• p factors into a product of repeated primes (such a prime is said to ‘ramify’
and only (2) does this here).
Let us explain the last point. The ideal (2)Z[i] is not radical: Note that
(2)Z[i] = (2i)Z[i] = (1 + i)2Z[i].
So the fiber consists of the single prime (1 + i)Z[i].
The following picture shows Spec(Z[i]):
The spectrum Spec(Z[i])
Now here’s where the Galois group comes into play. The group G =
Gal(Q[i]/Q) permutes the primes in Spec(Z[i]) sitting over any p in Spec(Z). In
this case
G = Z/2Z = 〈g〉
where g is the order two element given by complex conjugation. So for instance
if you look at the primes sitting over say (5), namely (2 + i) and (2− i), you see
that complex conjugation maps one into the other.
So we think of Spec(Z[i]) as sitting over Spec(Z) and G permuting the fibers
over in the base space Spec(Z).
Exercise 14. (i) Show that the fiber of φ over a prime p is homeomorphic to
SpecFp[x]/(x2 + 1)
and that dimFp Fp[x]/(x2 + 1) =2. Hint: Use that Z[i] = Z[x]/(x2 + 1).
(ii) Show that Fp[x]/(x2 + 1) is a field if and only if x2 + 1 does not have a
root in Fp.(iii) Show that Fp[x]/(x2 + 1) is a field if and only if (p)Z[i] is a prime ideal.
Example 2.29 (SpecR[t]). It is instructive to take a closer look at the prime
spectrum SpecR[t]. The ring R[t] being a principal ideal domain, any prime
52
Chapter 2. Schemes
ideal has the shape (f(t)) where f(t) is an irreducible polynomial, and we may
well assume f(t) to be monic. From the the fundamental theorem of algebra it
follows that either f is linear, that is, f(t) = t− a for α ∈ R, or f is quadratic
with to conjugate complex and non-real roots; That is, f(t) = (t − a)(t − a)
with a ∈ C but a /∈ R. This shows that the closed points in SpecR[t] may be
identified the set of pairs a, a for a ∈ C. And of course, there is the generic
point η corresponding to the zero ideal.
The Galois group G = Gal(C/R) = Z/2Z acts on C[t] via the conjugation
map σ; that is, the map that sends a polynomial f(t) =∑ait
i into∑
i aiti. The
corresponding scheme map from SpecC[t] to SpecC[t] defines an action of Z/2Zon SpecC[t], and the points of SpecR[t] are the quotient of points of SpecC[t]
by this action. We just saw this holds for closed points, and clearly the generic
point of SpecC[t] is invariant and corresponds to the generic point of SpecR[t].
The quotient map is the map SpecC[t] → SpecR[t] induced by the inclusion
R[t] ⊆ C[t].
Example 2.30. Generalizing the previous example, let K ⊆ L be a Galois
extension of fields with Galois group G; i.e., G acts on L and K = LG, the
subfield of invariant elements. This action induces and action on L[t] by letting the
action of a group element g on the polynomial f(t) =∑ait
i be g(f) =∑g(ai)t
i.
The map K[t] ⊆ L[t] induces a map π : SpecL[t] → SpecK[t]. Show that
π has the following universal property: For any affine scheme SpecA and any
invariant morphism ψ : SpecL[t]→ SpecA, i.e., a morphism such that ψ g = ψ
for all g, the map ψ factors through π; that is, there is a ψ′ : SpecK[t]→ SpecA
with ψ = ψ′ π. The diagram looks like
SpecL[t]g //
&&
SpecL[t]
xx
ψ // SpecA
SpecK[t]
44
2.6 The structure sheaf on SpecA
We have now come to point where we define a structure sheaf on the topological
space SpecA. This is a sheaf of rings OSpecA whose stalks all are local rings, so
that the pair (SpecA,OSpecA) is a ringed space.
The two paramount properties of the structure sheaf OSpecA are the following:
• Sections over distinguished opens: Γ(D(f),OSpecA) = Af .
• Stalks: OSpecA,x = Apx .
53
2.6. The structure sheaf on SpecA
These are two properties that one uses all the time when working with
the structure sheaf on an affine scheme. Moreover, as we will see, they even
characterize the structure sheaf uniquely.
2.6.1 Motivation
The model for the structure sheaf OSpecA on the prime spectrum SpecA of a
ring A is the sheaf of regular functions on an affine variety X, and when we
subsequently define it we mimic what happens on varieties, so it is worthwhile
recalling that situation.
Let X be a variety with coordinate A; that is, A is the ring of globally defined
regular functions on X. The fraction field K of A is the field of rational functions
on X, and the regular functions on any open set are elements in K.
For different open sets U the set of functions regular in U form different
subrings of K, and if V ⊆ U is an open contained in U , the ring of regular
functions Γ(V,OX) on V of course contains the ring Γ(U,OX) of those regular on
the bigger set U . The restriction map is nothing but the inclusion Γ(U,OX) ⊆Γ(V,OX); it does nothing to the functions in K; just considers the functions in
Γ(U,OX) to lie in Γ(V,OX).
Those function regular on the distinguished open set D(f) are allowed to
have powers of f in the denominator, and they constitute the subring Af ⊆ K
of elements of the form af−n. If D(g) is another distinguished open set that
is contained in D(f), i.e., D(g) ⊆ D(f), one has f = cgn, and hence Ag ⊆ Af(since f−1 = cg−n).
The general situation differs only significantly from the situation of varieties
in that the localization maps Af → Ag are no more necessarily injective, but the
formalism is the same.
The standard example of restriction maps not being injective is the union of
two varieties; to make it simple say the union of two line in the plane given by
the equation xy = 0; i.e., X = Spec k[x, y]/(xy). The regular function x vanishes
identically on the open D(y) where y 6= 0, and the regular function y vanishes
on D(x). So this is by no means a big mystery, it naturally appears once we
allow reducible spaces into the mix.
2.6.2 Definition of the structure sheaf OSpecA
One may straight away write down a definition of the structure sheaf, but many
students experience this as coming out of the blue. So we find it instructive to
make a detour and start with the definition of a B-presheaf that has a virtue of
being intuitive.
54
Chapter 2. Schemes
Definition 2.31. Let B be the collection of distinguished open subsets D(f).
We define the B-presheaf O by
O(D(f)) = Af
and for D(g) ⊆ D(f) we let the restriction map be localization map Af → Ag.
Let SD(f) be the multiplicative system s ∈ A | s 6∈ p ∀p ∈ D(f). There is
a localization map τ : Af → S−1D(f)A since f ∈ SD(f). The following lemma says
that the ring O(D(f)) is independent of the representative f for D(f):
Lemma 2.32. The map τ is an isomorphism, permitting us to identify Af =
S−1D(f)A
Proof. The point is that any element s ∈ SD(f) does not lie in p for any p ∈ D(f),
in other words, one has D(s) ⊆ D(f). This is equivalent to√
(f) ⊆√
(s), and
one may write fn = cs for an appropriate c ∈ A and n ∈ N . Assume that
af−m ∈ Af maps to zero in S−1D(f)A. This means that sa = 0 for some s ∈ SD(f).
But then fna = csa = 0, and therefore a = 0 in Af . This shows that the map τ
is injective. To see that is surjective, take any as−1 in S−1D(f)A and write is as
as−1 = ca(cfn)−1 = caf−n.
Proposition 2.33. O is a B-sheaf of rings.
The situation is as follows. We are given a distinguished set D(f) and an
open covering D(f) =⋃i∈I D(fi). Of course then D(fi) ⊆ D(f), and we have
localization maps τi : Af → Afi and τij : Afi → Afifj . The statement in the
Proposition is equivalent to the exactness of the following sequence
0 // Afα //∏
iAfiρ //∏
i,j Afifj (2.6.1)
where α(a)i = τi(a) and ρ((ai))i,j = (τij(ai)− τji(aj)). It is clear that α ρ = 0
since τij τi = τji τj.
Lemma 2.34. The sequence (2.6.1) is exact.
Proof. We start by observing that we assume A = Af—that is f = 1— indeed,
one has (Af)fi = Afi and (Af)fifj = Afifj since fnii = hif for suitable natural
numbers ni.
Then to the proof: That α(a) = 0 means that a is mapped to zero in each of
the localizations Afi , just means that a power of fi kills a. So for each i one has
fnii a = 0 for natural numbers ni. The open sets D(fi) covers D(f) which then
is covered by the D(fnii )’s as well. Thus one we may write 1 =∑
i bifnii , which
55
2.6. The structure sheaf on SpecA
upon multiplication by a gives
a =∑i
bifnii a = 0.
In down-to-earth terms, the equality ker ρ = imα means the following:
Assume given a sequence of elements ai ∈ Afi such that ai and aj are mapped
to the same element in Afifj for every pair i, j of indices. Then there ought to
be an a ∈ A, such that every ai is the image of a in Afi , i.e., τi(a) = ai.
Each ai can be written as ai = bi/fnii where bi ∈ A, and since the indices are
finite in number one may replace ni with n = maxi n. That ai and aj induce the
same element in the localization Afifj means that we have the equations
fNi fNj (bif
nj − bjfni ) = 0, (2.6.2)
where N a priori depends on i and j, but again due to there being only finitely
many indices, it can be chosen independent of the indices. Equation (2.6.2) gives
bifNi f
mj − bjfNj fmi = 0 (2.6.3)
where m = N + n. Putting b′i = bifNi we see that ai equals b′i/f
mi in Afi , and
equation (2.6.3) takes the form
b′ifmj − b′jfmi = 0. (2.6.4)
Now D(fmi ) = D(fi), and the distinguished open sets D(fmi ) form an open
covering of SpecA. Therefore we may write 1 =∑
i cifmi . Letting a =
∑i cib
′i,
we find
afmj =∑i
cib′ifmj =
∑i
cib′jf
mi = b′j
∑i
cifmi = b′j
and hence a = b′j/fmi .
Using Proposition 1.28 of Chapter 1, we may now make the following defini-
tion:
Definition 2.35. We let OSpecA be the unique sheaf extending the B-sheaf O.
To end this section, we prove that our sheaf indeed satisfies the two properties
we want:
Proposition 2.36. The sheaf OSpecA on SpecA as defined above is a sheaf of
rings satisfying the two paramount properties above, namely
• Sections over distinguished opens: Γ(D(f),OSpecA) = Af .
56
Chapter 2. Schemes
• Stalks: OSpecA,x = Apx.
In particular, Γ(X,OX) = A.
Proof. We defined O so that the first property would hold, so only the second
point needs proof. Using the first part, and the fact that you can compute stalks
by running through open sets of the form D(f), it suffices to show that the
canonical homomorphism
φ : lim−→f 6∈p
Af → Ap
isomorphism. φ is surjective: Any element in Ap is of the form a/f with f 6∈ p.
This element lies in Af and so it lies in the image of φ. φ is injective: if af−n ∈ Afis mapped to 0 in Ap, then ∃g 6∈ p such that ga = 0, hence af−n = 0 ∈ Agf and
so φ is injective.
The last statement follows by taking f = 1.
2.6.3 Maps between the structure sheaves of two spectra
Previously, we assigned to any map φ : A → B of rings a continuous map
φ∗ : SpecB → SpecA.
We climb one step in the hierarchy of structures and shall associate to φ a
map of sheaves of rings
φ# : OSpecA → φ∗OSpecB.
By a simple patching argument, it suffices to tell what φ# should do to the
sections over open sets belonging to a basis for the topology as long as this
definition is compatible with the restriction maps; and of course, the basis we
shall use is the basis of the distinguished open sets. Everything follows from the
following lemma:
Lemma 2.37. Let φ : A → B be a map of rings and let f ∈ A be an element.
Then (φ∗)−1(D(f)) = D(φ(f))
Proof. We have
(φ∗)−1(D(f)) = p ⊆ B | f /∈ φ−1(p) = p ⊆ B | φ(f) /∈ p = D(φ(f)).
This means that Γ(D(f), φ∗OSpecB) = Bφ(f), and we know that Γ(D(f),OSpecA) =
Af . The original map of rings φ : A → B now localizes to a map Af → Bφ(f),
sending af−n to φ(a)φ(f)−n, and that shall be the map φ# on sections over the
distinguished open set D(f).
57
2.7. Schemes
It is obvious, but anyhow the matter of some easy work, to check that this
definition is compatible with the restriction maps among distinguished open sets:
Indeed, if D(g) ⊆ D(f), we can as usual write gn = cf , and the localization map
Af → Ag sends af−m to acmg−nm. One has φ(g)n = φ(c)φ(f), and the diagram
below is commutative:
Af //
Ag
Bφ(f)
// Bφ(g),
which is the required compatibility.
2.7 Schemes
So far, we have defined a topological space SpecA together with a sheaf of rings
on it. This is the prototype of a ringed space:
Definition 2.38. A ringed space is a pair (X,OX) where X is a topological
space and OX is a sheaf of rings on X.
A morphism of ringed spaces is a pair (f, f#) : (X,OX) → (Y,OY ) where
f : X → Y is continuous, and
f# : OY → f∗OX
is a map of sheaves of rings on Y (so that f#(U) is a ring homomorphism for
each open U ⊆ X).
Example 2.39. Consider a manifold X with the sheaf C∞(X) of smooth func-
tions. The pair (X,C∞(X)) forms a ringed space. Indeed, any morphism
f : X → Y of manifolds is smooth if and only if for every local section h of
C∞(Y ) the composition f#(h) = h f is a local section of C∞(X). This gives a
map
f# : C∞(Y )→ f∗C∞(X)
Thus a map f gives rise in a natural way to a morphism of ringed spaces
f : (X,C∞(X)) −→ (Y,C∞(Y ))
Let us consider what happens to the germs of functions near a point: For a point
x ∈ X, the ring of germs of functions C∞(X)x is a local ring with maximal ideal
mx the functions which vanish at x (similarly for C∞(Y )y where y = f(x) ∈ Y ).
Note that the map on stalks
f# : C∞(Y )y → C∞(X)x
58
Chapter 2. Schemes
maps the maximal ideal into the maximal ideal: this is simply because f(x) = y.
This example motivates the following definition:
Definition 2.40. A locally ringed space is a pair (X,OX) as above, but with
the additional requirement that for every x ∈ X, the stalk OX,x is a local ring.
A morphism of locally ringed spaces is a pair (f, f#) : (X,OX)→ (Y,OY ) as
above, with the additional requirement that for every x ∈ X, f#x : OY,f(x) → OX,x
is a local homomorphism; that is the map
f#Y,f(x) : OY,f(x) → OX,x
satisfies
(f#Y,f(x))
−1(mx) = mf(x)
where mx ⊆ OX,x and mf(x) ⊆ OY,f(x) are the maximal ideals.
The intuition behind the local homomorphism condition comes from regarding
the sections of the structure sheaf as regular functions: we want g ∈ OY to
vanish at a point y = f(x) if and only if f#(g) vanishes at x.
Remark 2.41. Explicitly, the map f#Y,f(x) : OY,f(x) → OX,x in the definition, is
the map defined by the composition
OY,f(x) = lim−→f(x)∈V
OY (V )→ lim−→f(x)∈V
OX(f−1V )→ OX,x
Finally, we can give the formal definition of a scheme.
Definition 2.42. An affine scheme is a locally ringed space (X,OX) which is
isomorphic to (SpecA,OSpecA) for some ring A. A scheme is a locally ringed
spaced (X,OX) that is locally isomorphic to an affine scheme.
So a scheme is a topological space X and OX is a sheaf of rings such that all
stalks are local rings. A locally ringed space being locally affine means that it
possesses an open cover Ui of X with each open (Ui,OX |Ui) being isomorphic
(as locally ringed spaces) to an affine scheme.
A morphism or map for short between two schemes X and Y is simply a map
φ between X and Y regarded as locally ringed spaces. It has two components, a
continuous map, that we denote φ as well slightly abusing the language, and a
map of sheaves of rings
φ# : OY → φ∗OX
with the additional requirement that φ# induce local homomorphisms on the
stalks. This is to say, for all x ∈ Y the map φ#x : OY, φ(x) → φ∗OX,x maps the
maximal ideal in OY, x into the one in φ#x OX,x.
In this way the schemes form a category, which we denote by Sch. We denote
by AffSch the subcategory of affine schemes.
59
2.8. Open immersions and open subschemes
2.8 Open immersions and open subschemes
If X is a scheme and U ⊆ X is an open subset, there restriction OX |U is a sheaf
on U , making (U,OX |U ) into a locally ringed space. This is even a scheme, since
if X is covered by affines Vi = SpecAi, then each U ∩ Vi is open in Vi, hence
can be covered by affine schemes. It follows that there is a canonical scheme
structure on U , and we call (U,OX |U) an open subscheme of X and say that U
has the induced scheme structure.
As a special case, consider U = SpecAf ⊆ SpecA = X. It follows from the
definition of the sheaf OX that the restriction OX |U coincides with the structure
sheaf on SpecAf .
Remark 2.43. A word of warning: An open subscheme of an affine scheme
might not not itself be affine (we will see an example of this in Chapter 3).
2.9 Residue fields
Built into the definition of a scheme is the condition that the stalks OX,x are
local rings. In particular, for x ∈ X, we have a maximal ideal mx ⊆ OX,x, and a
corresponding field OX,x/mx.
Definition 2.44. We define the residue field of x to be k(x) = OX,x/mx.
If U ⊆ X is an open set containing x, and s ∈ OX(U) (or if s is an element
of OX,x), we let s(x) denote the class of s modulo mx in k(x).
The point of the above definition is this: For varieties, we construct the sheaf
OX from the regular functions on which we think of as continuous maps X → k.
However, in the world of schemes, we do not have the luxury of having a field k
to map into – all we know is that locally OX comes from a ring A. This means
that we have to tweak our notion of a ‘regular function’ on X; they are not maps
into some fixed field, but rather maps into the disjoint union∐
x∈X k(x). Thus
on X = SpecZ, the element s = 17 ∈ Z defines a function on X. Some of its
”values” are given by
s((2)) = 1, s((3)) = 1, s((7)) = 3, s((11)) = 6, s((17)) = 0, s(((19)) = 17
Of course the values have to be interpreted in the appropriate residue field Z/p.It is instructive to consider the simplest examples of affine schemes, namely
spectra of fields. If K is a field, then of course the underlying topological
SpecK is very simple – it is a singleton set corresponding to the zero ideal.
However, the structure sheaf O on SpecK carries more information: Morphisms
SpecL→ SpecK correspond exactly to field extensions L ⊇ K. In particular,
SpecK and SpecL are isomorphic if and only if K ' L.
60
Chapter 2. Schemes
More generally, for a scheme X it is interesting to study morphisms SpecK →X for fields K. Such morphisms will be referred to as K-valued points, and will
be denoted by X(K) (this jargon will be explained below). The residue fields
satisfy the following ‘universal’ property with respect to such morphisms:
Proposition 2.45. Let K be a field. Then to give a morphism of schemes
SpecK → X is equivalent to giving a point x ∈ X plus an injection k(x)→ K
Proof. Let (f, f#) : SpecK → X be the morphism and let x be the image of
f (SpecK consists of only one point, so this is a well-defined closed point of
X.) The map on stalks is just f#x : OX,x → OSpecK,0 = K, which gives a map on
residue fields k(x) = OX,x/mx → OSpecK,0 = K. As always with non-zero maps
of fields, this has to be an injection.
Conversely, let x ∈ X and k(x) → K. We construct a map of topological
spaces f : SpecK → X, which takes SpecK to x ∈ X. We also construct a map
of structure sheaves f# : OX → f∗OSpecK : for opens U ⊆ X not containing x,
f#(U) is the zero map as f−1(U) = ∅, while for opens U ⊆ X containing x,
we need maps OX(U)→ K. These maps are constructed using the given map
k(x) → K via the compositions
OX(U) OX,x k(x) K.
This yields a morphism of schemes (f, f#) : SpecK → X.
In particular, we have for X a variety over an algebraically closed field k
X(k) = closed points of X
More generally still, we make the following definition:
Definition 2.46. Let X, S be schemes. An S-valued point is a morphism
f : S → X. If S = SpecR, we call this an R-valued point. We denote the set of
such points by X(R).
The name R-valued point can be understood from the following example.
Example 2.47. Let An = SpecZ[x1, . . . , xn] and let R be a ring. What are
the R-valued points of An? Well, a morphism g : SpecR → SpecZ[x1, . . . , xn]
is determined by the n-tuple ai = g∗(x1) for i = 1, . . . , n. Hence, An(R) = Rn.
More generally, we have for any scheme S
An(S) = HomSch(S,An) = Γ(S,OS)n.
In fancy language, this says that An represents the functor taking a scheme to
n-tuples of elements of Γ(S,OS).
61
2.10. Relative schemes
Now, let
X = SpecZ[x1, . . . , xn]/I
where I = (f1, . . . , fr). The set of R-points can be found similarly: Indeed, any
morphism
g : SpecR→ SpecZ[x1, . . . , xn]/I
is determined by the n-tuple ai = g∗(x1) for i = 1, . . . , n, and those n-tuples that
occur are exactly those such that f 7→ f(a1, . . . , an) defines a homomorphism
Z[x1, . . . , xn]/I → R.
In other words, the ai are elements in R which are solutions of f1 = · · · = fr = 0.
In other words, this justifies the notation X(R) – X(R) and X(S) naturally
generalize the solution set of the polynomials f1 = · · · = fr = 0.
2.10 Relative schemes
There is also the notion of relative schemes (relative skjemaer) where a bases
scheme S is chosen. A scheme over S (et skjema over S) is scheme X with a
morphism π : X → S called the structure map. If two such schemes over S are
given, say X → S and Y → S , then a map between them is a map X → Y
compatible with the two structure maps; that is, rendering the diagram below
commutative
X //
Y
S
The schemes over S form a category Sch/S, and the set morphisms as defined
above is denoted by HomS(X, Y ). Composition of maps makes it is a functor
in both X and Y . It is a functor from Sch/S to Sets, contravariant in X and
covariant in Y . If the base scheme S is affine, say S = SpecA, a short hand
notation for Sch/Spec A is Sch/A.
As there is a canonical map from any scheme X to SpecZ, every scheme is a
Z-scheme. On the level of categories one may express this a Sch = Sch/Z.
In case of affine schemes say S = SpecA and X = SpecB, giving a map
X → S is giving a ring homomorphism A→ B, or said differently giving B the
structure of an A-algebra. So if S = SpecA the category Aff/S of affine schemes
over S is equivalent to the category Alg/A of A-algebras.
Definition 2.48. 1. A morphism f : X → Y is of locally finite type if Y has
a cover consisting of affines Vi = SpecBi such that f−1Vi can be covered
62
Chapter 2. Schemes
by affine subsets of the form SpecAij, where each Aij is finitely generated
as a Bi-module.
2. f is of finite type if, in (i), one can do with a finite number of SpecAij.
3. f is finite if f−1Vi = SpecAi, where Ai is a finite Bi-module.
We say that a scheme over k is of locally finite type (resp. of finite type, resp.
finite) over k, if the morphism X → Spec k has this property.
For an affine scheme these conditions translate into the following:
• SpecA is a scheme over k if and only if there is a homomorphism k → A,
or equivalently, that A is a k-algebra.
• SpecB → SpecA is a morphism of schemes over k if and only if the induced
map A→ B is a homorphism of k-algebras.
• SpecA is of finite type over k if and only if A is a finitely generated
k-algebra.
63
Chapter 3
Gluing
3.1 Gluing sheaves
The setting is a scheme X with an open covering Ui i∈I with a sheaf Fi on each
open subset Ui. As usual the sheaves can take values in any category, but the
principal situation we have in mind is when the sheaves are sheaves of abelian
groups. The intersections Ui ∩ Uj are denoted by Uij, and triple intersections
Ui ∩ Uj ∩ Uk are written as Uijk.
The gluing data in this case consist of isomorphisms τji : Fi|Uij → Fj|Uij . The
idea is to identify sections of Fi|Uij with Fj|Uij using the isomorphisms τij. For
the gluing process to be feasible, the τij’s must satisfy the three conditions
• τii = idFi ,
• τji = τ−1ij ,
• τki = τkj τji,
where the last identity takes place where it can; that is, on the triple intersection
Uijk. Observe that the three conditions parallel the three requirements for a
relation being an equivalence relation; the first reflects reflectivity, the second
symmetry and the third transitivity.
The third requirement is obviously necessary for gluing to be possible. A
section si of Fi|Uijk will be identified with its image sj = τij(si) in Fj|Uijk , and in
its turn, sj is going to be equal to sk = τkj(sj). Then, of course, si and sk are
enforced to be equal, which means that τki = τkj τji.
65
3.1. Gluing sheaves
Proposition 3.1. In the setting as above, there exists a unique sheaf F on X
such that there are isomorphisms υi : F |Ui → Fi satisfying υj = τji υi over the
intersections Uij.
Proof. Let V ⊆ X be an open set and let Vi = Ui ∩ V and Vij = Uij ∩ V . We
are going to define the sections of F over V , and they are of course obtained by
gluing sections of the Fi’s along Vi using the isomorphisms τij. We define
Γ(V, F ) = (si)i∈I | τji(si|Vij) = sj|Vij ⊆∏i∈I
Γ(Vi, Fi). (3.1.1)
The τij’s are maps of sheaves and therefore are compatible with all restriction
maps, so if W ⊆ V is another open set, one has τij(si|Wij) = sj|Wij
if τji(si|Vij ) =
sj|Vij . By this, the defining condition (3.1.1) is compatible with componentwise
restrictions, and they can therefore be used as the restriction maps in F . We
have thus defined a presheaf.
The first step in what remains of the proof, is to establish the isomorphisms
υi : F |Ui → Fi. To avoid getting confused by the names of the indices, we work
with a fixed index α ∈ I. Suppose V ⊆ Uα is an open set. Then naturally one
has V = Vα, and projecting from the product∏
i Γ(Vi, Fi) onto the component
Γ(Vα, Fα) = Γ(V, Fα) gives us a map of presheaves1 υα : F |Vα → Fα. This map
sends (si)i∈I to sα. We treat lovers of diagrams with a display:
Γ(V, F ) //
υα ''
∏i∈I Γ(Vi, Fi)
pα
Γ(V, Fα),
and proceed to verify that the map just defined gives us the searched for
isomorphism:
To begin with, on the intersections Vαβ the requirement in the proposition
that υβ = τβα υα is fulfilled. This follows immediately since by the second
property of the glue, one has sβ = τβα(sα)
It is surjective: Take a section σ of Fα over some V ⊆ Uα and define
s = (τiα(σ|Viα))i∈I . Then s satisfies the condition in (3.1.1), and is a legitimate
element of Γ(V, F ); indeed, by the third gluing condition we obtain
τji(τiα(σ|Vjiα)) = τjα(σ|Vjiα)
for every i, j ∈ I, and that is just the condition in (3.1.1). As ταα(σ|Vαα) = σ by
the first property of the glue, the element s projects to the section σ of Fα.
1Restrictions operating componentwise, it is straight forward to verify this map beingcompatible with restrictions.
66
Chapter 3. Gluing
It is injective: This is clear, since if sα = 0 if follows that si|Viα = τiα(sα) = 0
for all i ∈ I. Now Fα is a sheaf and the Viα constitute an open covering of Vα.
We conclude that s = 0 by the locality axiom for sheaves.
The next step is to show that F is a sheaf, and we start with the patching
axiom: So assume that Vα is an open covering of V ⊆ X and that sα ∈ Γ(Vα, F )
is a bunch of sections matching on the intersections Vαβ. Since F |Ui∩V is a sheaf—
we just checked that F |Ui is isomorphic to Fi—the sections sα|Vα∩Ui patch together
to give sections si in Γ(Ui∩V, F ) matching on Uij ∩V . This last condition means
that τij(si) = sj . By definition (si)i∈I then this is a section in Γ(V, F ) restricting
to si, and we are done. The locality axiom is easy to verify and is left to reader
to verify (do it!).
Exercise 15. Show the uniqueness statement in the proposition.
3.2 Gluing maps of sheaves
This is the easiest gluing situation we encounter in this course. The setting is
as follows. We are given two sheaves F and G on the scheme X and an open
covering Ui i of X. On each open set Ui we are given a map φi : F |Ui → G|Uiof sheaves, and the gluing conditions
• φi|Uij = φj|Uij
are assumed to be satisfied for all i, j ∈ I. In this context we have
Proposition 3.2. There exists a unique map of sheaves φ : F → G such that
φ|Ui = φi.
Proof. The salient point is this: Take any s ∈ Γ(V, F ) where V ⊆ X is open,
and let Vi = Ui ∩ V . Then φi(s|Vi) is a well defined element in Γ(Vi, G), and it
holds true that φi(s|Vij) = φj(s|Vij) by the gluing condition. Hence the sections
φi(s|Vi)’s of G|Vij patch together to a section of G over V which we define to be
φ(s). The checking of all remaining details is hassle-free and left to the zealous
student.
3.3 The gluing of schemes
The possibility of gluing different schemes together along open subschemes, gives
rise to many new schemes. The most prominent one being the projective spaces.
The gluing process is also an important part in many general existence proofs,
like in the construction of the fiber-product, which as we are going to show,
exists without restrictions in the category of schemes.
67
3.3. The gluing of schemes
In the present context of scheme gluing we are given a family Xi i∈I of
schemes indexed by the set I. In each of the schemes Xi we are given a collection
of open subschemes Xij, where the second index j runs through I. They form
the glue lines in the process; i.e., the contacting surfaces that are to be glued
together. In the glued scheme they will be identified and will be equal to the
intersections of Xi and Xj. The identifications of the different pairs of the Xij’s
are encoded by a family of scheme-isomorphisms τji : Xij → Xji. Furthermore,
we let Xijk = Xik ∩Xij— this are the different parts of the triple intersections
before the gluing has been done—and we have to assume that τij(Xijk) = Xjik.
Notice that Xijk is an open subscheme of Xi.
The three following gluing condition, very much alike the ones we saw for
sheaves, must be satisfied for the gluing to be doable:
• τii = idXi .
• τij = τ−1ji .
• The isomorphism τij takes Xijk into Xjik and one has τki = τkj τji over
Xijk.
Proposition 3.3. Given gluing data as above. Then there exists a scheme X
with open immersions ψi : Xi → X such that ψi|Xij = ψj|Xji τji, and such
that the images ψi(Xi) form an open covering of X. Furthermore one has
ψi(Xij) = ψi(Xi) ∩ ψj(Xj). The scheme X is uniquely characterized by these
properties
Proof. To build the scheme X we first build the underlying topological space
X and subsequently equip it with a sheaf of rings. For the latter, we rely on
the patching technic for sheaves presented in proposition 3.1. And finally, we
need to show that X is locally affine, this follows however immediately once the
immersions ψi are in place—the Xi’s are schemes and therefore locally affine.
On the level of topological spaces, we start out with the disjoint union∐
iXi
and proceed by introducing an equivalence relation on it. We require that two
points x ∈ Xij and x′ ∈ Xji be equivalent if x′ = τji(x)—observe that if the
point x does not lie in any Xij with i 6= j, we leave it alone, and it will not be
equivalent to any other point.
The three gluing conditions imply readily that we obtain an equivalence
relation. The first requirement entails that the relation is reflexive, the second
that it is symmetric, and the third ensures it is transitive. The topological space
X is then defined to be the quotient of∐
iXi by this relation, and we declare
the topology on X to be the quotient topology. If π denotes the quotient map, a
subset U of X is open if and only if π−1(U) is open.
Topologically the maps ψi : Xi → X are just the maps induced by the open
inclusions of the Xi’s in the disjoint union∐
iXi. They are clearly injective
68
Chapter 3. Gluing
since a point x ∈ Xi never is equivalent to another point in Xi. Now, X has the
quotient topology so a subset U of X is open if and only if π−1(U) is open, and
this holds if and only if ψ−1i (U) = Xi ∩ π−1(U) is open for all i. In view of the
formula
π−1(ψi(U)) =⋃j
τji(U ∩Xij)
we conclude that each ψi is and open immersion.
To simplify notation we now identify Xi and ψi(Xi), which is in concordance
with our intuitive picture of X as being the union of the Xi’s with points in
the Xij’s identified according to the τij’s. Then Xij becomes Xi ∩Xj ant Xijk
becomes the triple intersection Xi ∩Xj ∩Xk.
On Xij we have the isomorphisms τ#ji : OXj |Xij → OXi |Xij ; the sheaf-parts of
the scheme-isomorphisms τji : Xij → Xji. In view of the third gluing condition
τki = τkj τji, valid on Xijk, we obviously have τ#ki = τ#
ji τ#kj. The two first
gluing conditions translate into τ#ii = id and τ#
ji = (τ#ji )−1. The end of the story
is that the gluing properties needed to apply proposition 3.1 are satisfied, and
we are enabled to glue the different OXi ’s together and thus to equip X with a
sheaf of rings. This sheaf of rings restricts to OXi on each of the open subsets
Xi, and therefore it is a sheaf local rings. So (X,OX) is a locally ringed space
that is locally affine, hence a scheme.
The uniqueness property is, as usual, left to the industrious student.
Before and after gluing
3.4 Global sections of glued schemes
The standard exact sequence for computing global sections from an open covering
is a valuable tool in the setting of glued schemes. If X is obtained by gluing the
open subschemes Xi along τji : Xij → Xji it reads:
0 // Γ(X,OX) α //⊕
i Γ(Xi,OXi)ρ //⊕
i,j Γ(Xij,OXij)
where ρ(si)i∈I = (si|Xij − τ#ij (sj|Xji)) and α(s) = (ψ#
i (s))i∈I .
69
3.5. The gluing of morphisms
3.5 The gluing of morphisms
Assume given schemes X and Y and an open covering Ui i∈I of X. Assume
further that there is given a family of morphisms φ?i : Ui → Y which match on
the intersections Uij = Ui ∩Uj . The aim of this paragraph is show that they can
be glued together to give a morphism X → Y :
Proposition 3.4. In the situation just described, there exists a unique map of
schemes φ : X → Y such that φ|Ui = φi.
Proof. Clearly the underlying topological map is well defined, so if U ⊆ Y is
an open set, we have to define φ# : Γ(U,OY )→ Γ(U, φ∗OX) = Γ(φ−1U,OX). So
take any section s of OY over U . This gives sections ti = φ#i (s) of OX |Ui , but
since φ#i and φ#
j restrict to the same map on Uij, one has ti|Uij = tj|Uij . The titherefore patch together to a section of OX over U , which is the section we are
aiming at: We define φ#(s) to be t. The checking of the remaining details is left
to student (as usual).
3.6 The category of affine schemes
The assignment A → SpecA and φ → Spec(φ) gives a contravariant functor
from the category Rings of rings to the category AffSch of affine schemes. There
is also a contravariant functor the other way around. Taking the global sections
of the structure sheaf OSpecA gives us the ring A back. Furthermore, a map of
affine schemes f : SpecB → SpecA, comes with a map of sheaves f# : OSpecA →f∗OSpecB. Taking global sections gives a ring homomorphism
A = Γ(SpecA,OSpecA)→ Γ(SpecA, φ∗OSpecB) = B.
which coincides with φ in the case f = Spec(φ).
Let X,T be two schemes and let HomAffSch(X, Y ) denote the set of morphisms
f : X → Y between them. We can define a canonical map
ρ : HomAffSch(X, Y )→ HomRings(OY (Y ),OX(X)) (3.6.1)
which sends (f, f#) to the map f#(Y ) : OY (Y )→ OX(X).
Lemma 3.5. If X and Y are affine, the map ρ is bijective.
Proof. Write X = SpecB and Y = SpecA. By definition, we have A = OY (Y )
and B = OX(X). Given a morphism f : X → Y , let φ = ρ(f) = f#(Y ) :
B → A be the corresponding ring homomorphism. As we noted above, any ring
homomorphism induces a morphism of the corresponding ring spectra, so we
70
Chapter 3. Gluing
obtain a morphism of affine scheme fφ : X → Y . To prove the lemma, we need
only prove that fφ = f .
Let x ∈ X be a point, corresponding to the prime ideal q ⊆ B, and let p ⊆ A
be the prime ideal corresponding to f(x) ∈ Y . We have a diagram
Aφ //
B
Ap
// Bq
where the vertical maps are the localization maps. We have φ(A− p) ⊆ B − q.
Hence φ−1(q) ⊆ p. Now f#x is a local homomorphism, and so in fact φ−1(q) = p.
Hence f and fφ induce the same map on the underlying topological spaces.
Moreover, we have f#x = f#
φ,x for every x ∈ X, and so also f# = f#φ .
We have established the all important theorem:
Theorem 3.6. The two functors Spec and Γ are mutually inverse and define
an equivalence between the categories Rings and AffSch.
Thus affine schemes X are characterized by their rings of global sections
Γ(X,OX), and morphisms between affine schemes X → Y are in bijective
correspondence with ring homomorphisms Γ(Y,OY )→ Γ(X,OX).
3.7 Universal properties of maps into affine schemes
For a general schemeX one can consider the associated affine scheme Spec Γ(X,OX),
but this is in general very different from X (for the projective line introduced in
the next section, Spec Γ(X,OX) is just a point). There is however still a canonical
morphism X → Spec Γ(X,OX) enjoying the following universal property:
Proposition 3.7. Let X be any scheme. Then there is a canonical (unique)
map of schemes X → Spec Γ(X,OX) inducing the identity on global sections of
the structure sheaves.
This will follow by applying the following theorem to A = Γ(X,OX):
Theorem 3.8. For any scheme X, the canonical map
ΦX : HomSch(X, SpecA)→ HomRings(A,Γ(X,OX))
given by f → f#, is bijective.
71
3.7. Universal properties of maps into affine schemes
Proof. Let Ui be an affine covering of X, and consider the commutative
diagram
Hom(X, SpecA)ΦX //
_
Hom(A,Γ(X,OX)) _
∏i Hom(Ui, SpecA)
ΠΦUi//∏
i Hom(A,Γ(Ui,OX))
where the vertical maps are induced by the inclusion maps Ui → X. By the
affine schemes case (Theorem 3.6), we know that ΠΦUi is bijective. In particular,
this shows that ΦX is injective.
To show that ΦX is surjective, let β : A→ Γ(X,OX) be a ring homomorphism.
This induces maps βi : A → Γ(X,OX) → Γ(Ui,OX), and hence, morphisms
fi : Ui → SpecA. We check that the fi glue to a map f : X → SpecA. This is a
consequence of the fact that the following diagram commutes:
Γ(Ui,OUi)
**A
βi::
βj $$
β // Γ(X,OX) //
OO
Γ(Ui ∩ Uj,OX)→ Γ(V,OX)
Γ(Uj,OUj)
44
Indeed, note that for V ⊆ Ui∩Uj affine, the diagram implies that the restrictions
fi|V and fj|V induce the same element in Hom(A,Γ(V,OX)), and so they are
equal on V (using Theorem 3.6). Since this is true for any V , they are equal on
all of Ui ∩ Uj. So by gluing the fi, we obtain a morphism f : X → SpecA. We
must have ΦX(f) = β, by injectivity of ΦX and since f maps to∏
i βi via ΠΦUi .
This completes the proof.
Corollary 3.9. The canonical map ψ : X → Spec Γ(X,OX) is universal among
the maps from X to affine schemes; i.e., any map α : X → SpecA factors as
α = α′ ψ for a unique map α′ : Spec Γ(X,OX)→ SpecA.
Proof. In the theorem above, ψ corresponds to the identity map idΓ(X,OX) on
the right hand side. The morphism α′ is the map of Spec’s induced by the
ring map α# : A → Γ(X,OX). We check that it factors α: The morphism
(α′ ψ) : X → SpecA satisfies (α′ ψ)# = ψ# α# = α# and hence it coincides
with α by the above theorem.
Remark 3.10. Let X be any scheme. By the above we have a bijection
HomSch(X, SpecZ) = HomRings(Z,Γ(X,OX)).
72
Chapter 3. Gluing
The set on the right is clearly a one point set and thus we find that SpecZ is
final object in the category Sch.
Sch also has a initial object: The spectrum of the zero ring, Spec 0, which
has the empty set as underlying topological space. Given any scheme X there
is clearly a unique morphism Spec 0→ X, which on the level of sheaves sends
every section to zero.
Example 3.11 (The morphism Spec OX,x → X). Let X be a scheme and let
x ∈ X be a point. There is a canonical morphism
f : Spec(OX,x)→ X
which is induced from all the morphisms SpecAp → SpecA. The image I =
im f ⊆ X of f is exactly the intersection of all neighbourhoods of x in X: it is a
kind of microgerm of X at x.
More geometrically consider the irreducible closed subset V = x ⊆ X
whose generic point is x. Let Y ⊆ X be a closed irreducible subset on which
V lies: V ⊆ Y and let ηY be the generic point of Y . Then our subset im f is
exactly the set of all those generic points ηY . We say that im f is the set of
specializations of x.
For example, if X = A2k = Spec(k[x, y]) and x is a closed point corresponding
to the maximal ideal m = (x − a, x − b), then I is the set consisting of x, the
generic point of A2k and the generic points of all irreducible curves passing through
x, like for example the curve (y − b)2 − (x− a)3 = 0.
3.8 The functor from varieties to schemes
We have already stated that schemes form a generalization of algebraic varieties.
On the other hand, we have also seen that even the simplest schemes (e.g.,
A2k = Spec k[x, y]) behave differently than varieties in the sense that they usually
have many non-closed points. Thus for this generalization to make sense, we
should expect that there to be a canonical way to ‘add non closed points’ to an
algebraic variety V , so that the resulting topological space has the structure of a
scheme. Let us explain what this means more precisely.
Let k be an algebraically closed field, and let V be an algebraic variety over
k. Let us first consider the case where V is an affine variety, i.e., V ⊆ Ank is
the zero-set of some ideal I ⊆ k[x1, . . . , xn]. The coordinate ring A = A(V ) =
k[x1, . . . , xn]/I is the ring of regular functions f : V → k. The scheme V sh =
SpecA is an affine scheme whose closed points is in bijection with the points of
V . So any affine variety determines a canonical affine scheme.
In the general case, a variety has an open cover by affine varieties, and gluing
can be performed in both the category of varieties, as well as the categories of
73
3.8. The functor from varieties to schemes
schemes, and it is a matter of checking that this gives a well-defined scheme (see
Chapter 3).
This assignment works also for morphisms; a morphism of varieties φ : V → W
is determined by a ring homomorphism A(W ) → A(V ), and so taking Spec
we obtain a morphism φsh : V sh → W sh, which extends φ. It follows that the
assignment V 7→ V sh in fact determines a functor t : Var/k → Sch/k. It is a
theorem (Hartshorne Chapter II), that this functor is fully faithful, in the sense
that
HomVar/k(V,W ) = HomSch/k(Vsh,W sh)
is bijective. So two varieties give rise to the same scheme if and only if they
are isomorphic by a unique isomorphism. In particular, this tells us that the
category of varieties Var/k is equivalent to a full subcategory of Sch/k.
74
Chapter 4
More examples
4.1 A scheme that is not affine
Consider A = k[u, v] and let X = SpecA. Let U = X − V (u, v); this is the
affine plane A2k minus the origin. We claim that U is not an affine scheme. The
point of this example is that the restriction map Γ(X,OX) → Γ(U,OU) is an
isomorphism. This shows that U can not be an affine scheme, since in that
case, by Theorem 3.8 above, the inclusion map would be an isomorphism which
obviously is not true, since it is not surjective.
Let us check that that restriction map really is an isomorphism. The two
distinguished open sets D(u) and D(v) form an open covering of U . The proof
is based on the sequence from before associated to that covering:
0 // Γ(U,OU) // Av × Auρ // Auv
A
i#
OO
where ρ is the difference between the two localization maps; that is; it maps
a pair (av−m, bu−n) to av−m − bu−n. We have included the restriction map i#
in the diagram, which is the just the action of the inclusion map i : U → X on
global sections of the structure sheaves. It sends an element a ∈ A to the pair
(a, a).
If the pair (av−m, bu−n) lies in kernel of ρ, we have the relation
aun = bvm,
75
4.2. The projective line
which (since A is a UFD) implies that there is an element c ∈ A with a = cvm
and b = cun; that is, av−m = bu−n. This shows that i# is surjective, and hence
it is an isomorphism since it obviously is injective.
Note that the same example works for any is an integral domain A with an
ideal a generated by two elements u and v having the following property: If
there is a relation aun = bvm with a, b ∈ A and n,m ∈ N , one may find a c ∈ Asuch that a = cvm and b = cun.
4.2 The projective line
In elementary courses on complex function theory one learns about the Riemann
sphere. That is the complex plane with one point added, the point at infinity.
If z is the complex coordinate centered at the origin, the inverse 1/z is the
coordinate centered at infinity. Another name for the Riemann sphere is the
complex projective line, denoted P1C.
The construction of P1C can be vastly generalized, and works in fact over any
ring A. Let u be a variable (“the coordinate function at the origin”). and let
U1 = SpecA[u]. The inverse u−1 is a variable as good as u (“the coordinate at
infinity”), and we let U2 = SpecA[u−1]. Both are copies of the affine line A1A
over A.
Inside U1 we have the open set U12 = D(u) which is canonically equal to
the prime spectrum SpecA[u, u−1], the isomorphism coming from the inclusion
A[u] ⊆ A[u, u−1]. In the same way, inside U2 there is the open set U21 = D(u−1).
This is also canonical isomorphic to the spectrum SpecA[u−1, u], the isomorphism
being induced by the inclusion A[u−1] ⊆ A[u−1, u]. Hence U12 and U21 are
isomorphic schemes (even canonically), and we may glue U1 to U2 along U12.
The result is called the projective line over A and is denoted by P1A.
Gluing two affine lines to get P1
In this example, we will carry out our first computations with sheaf cohomology
using the Cech complex:
Proposition 4.1. We have Γ(P1,OP1A
) = A.
76
Chapter 4. More examples
Proof. The sheaf axiom exact sequence gives us
Γ(P1,OP1A
) // Γ(U1,OP1A
)⊕ Γ(U2,OP1A
) //
'
Γ(U12,OP1A
)
'
A[u]⊕ A[u−1] ρ// A[u, u−1]
,
where the map ρ sends a pair (f(u), g(u−1)) of polynomials with coefficients in
A, one in the variable u and one in u−1, to their difference. We claim that the
kernel of ρ equals A; i.e., the polynomials f and g must both be constants.
So assume that f(u) = g(u−1), and let f(u) = aun + lower terms in u , and
in a similar way, let g(u−1) = bu−m+ lower terms in u−1 , where both a 6= 0 and
b 6= 0, and without loss of generality we may assume that m ≥ n. Now, assume
that m ≥ 1. Upon multiplication by um we obtain b + uh(u) = umf(u) for
some polynomial h(u), and putting u = 0 we get b = 0, which is a contradiction.
Hence m = n = 0 and we are done.
In particular, the global sections of OX of X = P1C is just C. In particular,
knowing Γ(X,OX) does not recover X, in constrast to the case when X = SpecA
is affine.
4.2.1 Affine line a doubled origin
Keeping the notation from the previous example, if we now glue U1 = U2 =
SpecA[v] along U12 = SpecA[u, u−1] using the identity map on the overlap, we
obtain the ‘affine line with two origins’.
This scheme is not affine: Using the sheaf axiom sequence, we find that
Γ(X,OX) = A[u]. Note that both of the points above the origin would correspond
to the same ideal, which is not possible on an affine scheme.
More strikingly, the cokernel of the map A[u] ⊗ A[u−1] → A[u, u−1] is
A[u, u−1]/A[u], and so H1(X,OX) is not finitely generated as an A-module.
We will see that this cannot happen on affine schemes. In fact, for X affine, all
higher cohomology groups vanish: H i(X,OX) = 0 i > 0.
4.3 The blow-up of the affine plane
4.3.1 Blow-up as a variety
Recall the classical construction of a blow-up. There is a rational map f :
A2k 99K P1
k sending a point (x, y) to (x : y) (in homogeneous coordinates on P1k).
77
4.3. The blow-up of the affine plane
This map is not defined at the origin, but we can still associate to it its graph
X ⊆ A2k×P1
k, consisting of all pairs (x, y)× (s : t) so that f(x, y) = (s : t). From
the definition, we see that X is defined by a single equation in A2k × P1
k, namely
X = Z(xt− ys) ⊆ A2k × P1
k
We also have two projection maps p : X → A2 and q : X → P1. The situation is
depicted in Figure 4.1
Figure 4.1: The blow-up of the plane at a point
Let us analyze the fibers of these maps.
The fibers of p are easy to describe. If (x, y) ⊆ A2k is not the origin, then
p−1(x, y) consists of a single point; the equation xt = ys allows us to determine
the point (s : t) uniquely. However, when (x, y) = (0, 0), any s and t satisfy
the equation, so p−1(0, 0) = (0, 0)× P1. In particular, that this inverse image is
1-dimensional - it is called the exceptional divisor of X, and is usually denoted
by E.
Similarly, if (s : t) ∈ P1k is a point, the the fiber q−1(s : t) = (x, y) × (s :
t)|xt = ys ⊆ A2k × (s : t) is a line in A2. The map q is an example of a line
bundle; all of its fibers are A1k’s. We will see this later on in the book.
The standard covering of P1k as a union of two A1
k’s gives an affine cover of X:
If U ⊆ P1k is the open set where s 6= 0, we can normalize so that s = 1, and solve
the equation xt = sy for y. Hence x and t give coordinates on q−1(U) ' A2x,t.
In these coordinates, the morphism p : X → A2k, restricts to a map A2
x,t → A2x,y
given by (x, t) 7→ (x, xt). Similarly, q−1(V ) = A2y,s and the map p is given here
as (y, s) 7→ (sy, y).
4.3.2 The blow-up as a scheme
From the above discussion, we can define the scheme-analogue of the blow-up
of A2k at a point. We will define this as a scheme over Z, rather than over a
78
Chapter 4. More examples
field k. Also, in addition to the scheme X, we also want a morphism of schemes
p : X → A2Z having similar properties to that above.
Consider the affine plane A2 = SpecZ[x, y]. The prime ideal p = (x, y) ⊆Z[x, y] corresponds to the origin in A2. Consider the diagram
Z[x, y]
Z[x, t] Z[y, s]
R = Z[x, y, s, t]/(xt− y, st− 1)
Here the diagonal maps on the top are given by (x, y) 7→ (x, xt) and (x, y) 7→(ys, y) respectively.
Note that the ring R is isomorphic to Z[x, s, t]/(st− 1) = Z[x, s, s−1], as well
as Z[y, s, t]/(st− 1) = Z[y, s, s−1]. Since this is a localization of both Z[x, t] and
Z[y, s] we can identity its spectrum as an open set of SpecZ[x, t] and SpecZ[y, s]
respectively. This gives a digram
SpecZ[x, y]
U = SpecZ[x, t] V = SpecZ[y, s]
SpecR
Hence we can glue these two affine spaces along SpecR to a new scheme
X. By construction, the maps SpecZ[x, t] → SpecZ[x, y] and SpecZ[y, s] →SpecZ[x, y] coincide with the map SpecR → SpecZ[x, y] which is induced by
Z[x, y]→ R. Hence they glue to a morphism
p : X → A2Z.
To complete the discussion, we should define the corresponding morphism
q : X → P1. Again we work locally. On the affine open U = SpecZ[x, t], we have
a map U 7→ A1 = SpecZ[t] induced by the inclusion Z[t] ⊆ Z[x, t]. Similarly, on
V = SpecZ[y, s], we have a map V 7→ A1 = SpecZ[s]. To see if we can glue, we
have to see what happens on the overlap U ∩ V = SpecR. However, on SpecR,
t = s−1, so using the standard gluing on P1, we see that the maps Z[t]→ R and
Z[s]→ R induce the desired morphism q : X → P1.
79
4.4. Hyperelliptic curves
4.4 Hyperelliptic curves
Let k be a field and consider the scheme X glued together by the affine schemes
U = SpecA and V = SpecB, where
A =k[x, y]
(−y2 + a2g+1x2g+1 + · · ·+ a1x)and B =
k[u, v]
(−v2 + a2g+1u+ · · ·+ a1u2g+1)
Here we glue D(x) = SpecA(x) to D(u) = SpecB(u) using the identifications
u = x−1 and v = x−g−1y. These curves are so called hyperelliptic curves or
double covers’ of P1. In the case g = 1 they are called elliptic curves. Here is an
illustration of one of the affine charts:
The term ‘double cover’ comes from the fact that it admits a map X → P1
coming from the inclusions k[x] ⊆ A and k[u] ⊆ B; note that u = x−1 gives the
standard gluing of P1 by the two affine schemes Spec k[x] and Spec k[u]. The
corresponding morphism X → P1 is finite of degree 2, as we are adjoining a
square root of an element of k[x].
80
Chapter 5
Geometric properties of schemes
5.1 Some topological properties
Many basic geometric concepts of schemes are formulated just in terms of their
underlying topological space. For instance, a scheme (X,OX) is said to be
connected if the topological space of X connected (ie it cannot be written as a
union of two proper open sets.)
Example 5.1. If R = Z×Z. Then SpecR is not connected. Indeed, the proper
prime ideals of R are of the form p× Z or Z× p, for p ⊆ Z a prime ideal. And
so SpecR = SpecZ ∪ Spec(Z).
More generally, SpecA is disconnected if and only if A ' B × C for some
non-zero rings B,C.
Likewise, we say that X is irreducible (irredusibelt) if its space is not the
union of two proper, closed subsets. That is, if X = Z ∪ Z ′ with Z and Z ′ both
closed one has either Z = X or Z ′ = X. An equivalent formulation is that any
two nonempty open subsets of X has a non empty intersection; indeed if U and
U ′ are open, it holds that U ∩ U ′ = ∅ is equivalent to U c ∪ (U ′)c = X.
From the the theory of varieties we know that the coordinate ring of a variety
(which is assumed to be irreducible), is an integral domain, and very simple
examples illustrate that reducibility is strongly connected to zero divisors in ring
of functions:
Example 5.2. The scheme X = Spec k[x, y]/(xy) is the prime example of
something that is not irreducible. The coordinate functions x and y are zero-
divisors in the ring k[x, y]/(xy), and their zero-sets V (x) and V (y) show that X
has two components.
81
5.2. Properties of the scheme structure
We have seen that the irreducible subsets of an affine scheme SpecA cor-
respond exactly to the prime ideals p in A. Specializing to the irreducible
components of X, we find:
Proposition 5.3. Let X = SpecA be an affine scheme.
1. The irreducible components of X are precisely the closed subsets of the
form V (p) where p ⊆ A is a minimal prime.
2. X is irreducible if and only if A has a unique minimal prime, if and only
if√
0 is prime.
Note that the ring R = k[t]/(t2) has only one prime ideal (t), so X = SpecR
is just point, hence irreducible. However k[t]/(t2) is not an integral domain.
5.2 Properties of the scheme structure
A ring A is said to be reduced if it has no nilpotent elements. A scheme (X,OX)
is said to be reduced if for every x ∈ X, the local ring OX,x is reduced. This
condition holds if and only if for every open U ⊆ X, the ring OX(U) has no
nilpotents. Indeed, suppose (X,OX) is reduced. Clearly, then, for every x ∈ X,
the local ring OX,x has no nilpotents via the description of its elements as germs
of OX around x. The converse follows similarly: a nilpotent section of OX(U)
would pass to a nilpotent element in the local ring at any point x ∈ U .
Given (X,OX), we can define a new ringed space (X, (OX)red) whose under-
lying topological space is the same as that of X, but the structure sheaf of OXred
is defined as the sheafification of the presheaf
F(U) = OX(U)red
where OX(U)red = OX(U)/N (U) is the quotient of OX(U) by its nilradical N (U)
of nilpotent elements. The collection N (U) forms the so-called nilradical sheaf
N . You should convince yourself that N is indeed a sheaf.
Lemma 5.4. (X,OXred) is a scheme.
Proof. We find that OXred,x = (OX,x)red = OX,x/√
0 is the quotient of a local ring
and hence local. It follows that (X, (OX)red) is a locally ringed space. Further-
more, if (X,OX) is locally isomorphic to affine schemes (SpecAi,OSpecAi), and
then (X, (OX)red) is locally isomorphic to affine schemes (SpecAi/Ni,OSpecAi/Ni),
where Ni is the nilradical of Ai. Hence (X, (OX)red) is a scheme.
We refer to (X,OXred) as simply Xred, the reduced scheme associated to X.
Xred comes with a canonical morphism of schemes r : Xred → X defined as
82
Chapter 5. Geometric properties of schemes
follows. The underlying topological space of Xred is that of X, and hence there is a
homeomorphism r : Xred → X. There is also a map of sheaves r# : OX → r∗OXred
that on each open U ⊆ X is the quotient map OX(U) → (OX)red(U). As the
induced map on stalks is a quotient map as well, it is a local homomorphism,
and we obtain a morphism (r, r#) : Xred → X of schemes.
The scheme Xred and the morphism r satisfy the following universal property:
Proposition 5.5. Let f : Y → X be a morphism of schemes, with Y reduced.
Then f factors uniquely through the natural map r : Xred → X, i.e. there exists
a unique g : Y → Xred such that f = r g.
Proof. The only difference between X and Xred is the structure sheaf, so define
g on the level of topological spaces by f . On the level of sheaves we find that
(over any open U ⊆ X) the map f# : OX(U)→ f∗OY (U) takes all nilpotents to
zero as Y is reduced. By the universal property of quotients there must exist a
unique morphism of rings g#(U) : OXred(U)→ f∗OX(U) such that f# = g# r#.
This gives the required morphism (g, g#) of schemes.
5.2.1 Integral schemes
Definition 5.6. A scheme is integral if it is both irreducible and reduced.
In particular, an affine scheme X = SpecA is integral if and only if A is an
integral domain. Indeed, X is reduced if and only if A has no nilpotents. X
is irreducible if and only if the nilradical of A N = ∩pp is prime. These two
statements imply that the zero-ideal (=∩p) is prime, and so A is an integral
domain.
Moreover, it is not so hard to prove the following:
Proposition 5.7. A scheme X is integral if and only if OX(U) is an integral
domain for each open U ⊆ X.
If X is an integral scheme, there is a unique generic point η of X which
is dense in X. Indeed, take any U = SpecA ⊆ X and let η be the point
corresponding to the zero ideal (which is prime). The local ring OX,η is equal to
the field of fractions of A. We define the function field K(X) of X as the field of
fractions of OX,η at the generic point.
Example 5.8. The function field of Ank = Spec k[x1, . . . , xn] is k(x1, . . . , xn).
Example 5.9. The function field of SpecZ is Q.
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5.2. Properties of the scheme structure
5.2.2 Noetherian spaces and quasi-compactness
Definition 5.10. A topological space is called noetherian (noetersk) if every
descending chain of closed subsets is eventually constant; that is, if Zi i∈N is a
family of closed subsets Zi with Zi+1 ⊆ Zi, there is an i0 such that Zi = Zi+1 for
i ≥ i0.
An equivalent formulation is any any non-empty family of closed subsets has
a minimal element. This happens if and only if every non-empty family of open
subsets has a maximal element.
Clearly if A is a noetherian ring, the prime spectrum SpecA is a noetherian
topological space. The converse however, is not true. A simple example is the
following: Take the polynomial ring k[t1, t2, . . . ] in countably many variables tiand mod out by the square m2 of the maximal ideal generated by the variables,
i.e., with m = (t1, t2, . . . ). The resulting ring A has just one prime ideal, the
one generated by the ti’s, so SpecA has just one point. The ring A, however, is
clearly not noetherian; the sole prime ideal requiring infinitely many generators,
namely all the ti’s.
Given this example, we take a different route to define noetherianness for
schemes:
Definition 5.11. (i) A scheme is quasi-compact if every open cover of X has
a finite subcover.
(ii) A scheme is locally noetherian if it can be covered by open affine subsets
SpecAi where each Ai is a noetherian ring
(iii) A scheme is noetherian if it is both locally noetherian and quasi-compact.
An affine scheme X = SpecA is always quasi-compact: If X =⋃
SpecAi,
each SpecAi can be covered by the distinguished opens D(fij), and X is covered
by such subsets if and only if 1 belongs to the ideal generated by the fij; and in
this case finitely many of them will do. Hence a general scheme is noetherian if
and only if it can be covered by finitely many SpecAi where each Ai is noetherian.
In fact, now we have
Proposition 5.12. SpecA is noetherian (as a scheme) if and only if A is
noetherian.
This is really a purely algebraic fact: Refining the cover, we may assume that
each Ai = Afi . By a theorem in commutative algebra, A is noetherian provided
that each Afi is noetherian and 1 ∈ (f1, . . . , fr).
Proposition 5.13. If X is a noetherian scheme, then its underlying topological
space is noetherian.
84
Chapter 5. Geometric properties of schemes
By covering X with open affine subsets, it suffices to show the proposition for
X = SpecA, for A a noetherian ring. In ths case a descending chain of closed
subsets is of the form V (I1) ⊇ V (I2) ⊇ · · · , where we may assume that the
ideals In are radical. Then the condition that V (In) is decreasing, corresponds
the sequence (In) being increasing, and so it has to be stationary, because A is
noetherian.
Proposition 5.14. Let X be a (locally) noetherian scheme. Then any open
subscheme of X is also (locally) noetherian.
This is clear for the locally noetherian case. For the noetherianness, we need
to show that U is also quasi-compact. Since X is noetherian, U is noetherian as
a topological space. Now if (Vi)i∈I is a cover of U , then the collection of unions
of the form ∪i∈FVi for F finite, has a maximal element, which gives a finite
subcover of U .
We also see that if X is locally noetherian, then all the local rings OX,x are
noetherian. The converse however, does not hold:
Example 5.15. Let A = C[y, x1, x2, . . . ]/I where I is the ideal generated by
xi−(y−i)xi+1, x2i for i = 1, 2, . . . . Then A is not noetherian, but Ap is noetherian
for every prime ideal p. The nilradical N is (x1, x2, . . . ) – the xi are nilpotent,
and A/(x1, . . . ) is a domain. In particular, A is not noetherian. We have
A/N ' C[y], so the maximal ideals of A are of the form m = (y − a, x1, x2, . . . )
for some a ∈ C. If a is not a positive integer, then Am is the localization of a ring
generated by y and x1, and hence is noetherian. If a = n is a positive integer,
we have xi+1 = xi/(y − i) for i 6= n in m. Hence Am is the localization of a ring
generated by y, xn, and hence is Noetherian.
Proposition 5.16. Let X be a noetherian scheme. Then any closed subset
Y ⊆ X can be written as a finite union Y =⋃Yi where each Yi is irreducible.
Proof. If W is the set of closed subsets that cannot be written as a finite union
of closed irreducible subsets, and W is non-empty, it has a minimal element
Z ∈ W . Z is not irreducible, so it can be written as a union of two proper closed
subsets which do not lie in W , a contradiction.
The Yi are called the irreducible components of Y . They are a priori defined
only at the level of topological spaces, but we will see in Chapter 7 (Lemma
8.37) that one can define a scheme structure on them in a natural way, so that
they become ‘closed subschemes’ of X.
85
5.3. The dimension of a scheme
5.3 The dimension of a scheme
Recall that the Krull dimension of a ring A is defined as the suprenum of the
length of all chains of prime ideals in A. For a scheme, we make the following
similar definition:
Definition 5.17. Let X be a scheme. The dimension of X is the suprenum of
all integers n such that there exists a chain
Y0 ⊆ Y1 ⊆ · · · ⊆ Yn
of distinct closed irreducible closed subsets of X.
Note that this suprenum might not be a finite number, in which case we
say that dimX =∞. Note also that the dimension of X only depends on the
underlying topological space.
In the case where X = SpecA is affine, we know that the closed irreducible
subsets are of the form V (p) where p is a prime ideal. Using this observation we
find
Proposition 5.18. The dimension of X = SpecA equals the Krull dimension
of A.
Example 5.19. 1. The dimension of AnA = SpecA[x1, . . . , xn] is n + dimA.
In particular, when A = k is a field, Ank has dimension n.
2. dim SpecZ is 1.
3. dim Spec k[ε]/ε2 = 0.
Remark 5.20. Having finite dimension does not guarantee that the scheme is
noetherian. For an explicit counterexample, take SpecR where R is the ring of
integers in Qp.
More surprisingly, there might may be noetherian rings having infinite Krull
dimension! First counterexamples were constructed by Nagata. In short the
counterexample involves taking the polynomial ring in infinitely many variables
k[x1, x2, . . . ] and dividing out by a certain ideal, ensuring that all prime chains
are finite, but there is a sequence of prime chains whose lengths tend to infinity!
Definition 5.21. Let Y ⊆ X be a closed subset of X. We define the codimension
of Y as the suprenum of all integers n such that there exists a chain
Y = Y0 ⊆ Y1 ⊆ · · ·Yn
of distinct irreducible closed subsets of X.
86
Chapter 5. Geometric properties of schemes
The codimension of V (p) in SpecA is the height of the prime p in A.
Note that the formula dim Y + codimY = dimX does not always hold, even
if X is the spectrum of a ring. Indeed, let X = SpecR[t] where R = k[[t]]. The
prime p = (tu− 1) has height 1, but A/p ' R[1/t] is a field, hence of dimension
0. However, dimA = dimR + 1 = 2.
On the other hand, for integral schemes, we can say something:
Theorem 5.22. Let X be an integral scheme of finite type over a field k, with
function field K. Then
(i) dimX equals the trancendence degree of K over k (in particular, dimX <
∞).
(ii) For each U ⊆ X open, dim U = dimX.
(iii) For p ∈ X closed, dimX = dim OX,p.
Proof. Note that (ii) follows from (i) since X and an open subset U have the
same function field. To prove (i) we may assume that X = SpecA is affine. The
hypothesis on X gives that A is a finitely generated k-algebra with quotient field
K. In this case, (i) is a consequence of Noether normalization (see Chapter 12
of Atiyah–MacDonald). To prove (iii), we may again assume that X = SpecA,
and use the formula
dimA/p + ht p = dimA
which folds for prime ideals in finitely generated k-algebras.
For schemes which aren’t integral but still of finite type, we still have a good
control over its dimension. First of all, the dimension of X is the same as Xred,
so we may assume that X is reduced. Then if X =⋃Xi is the decomposition
into irreducible components, we have Xi integral, and dimX is the suprenum of
all dimXi.
5.4 More examples
The rings Z(2) and Z(3) are both discrete valuation rings with maximal ideal
being (2) and (3) respectively. Their fraction field are both equal to Q. Let
X1 = SpecZ(2) and A2 = SpecZ(3). Both have a generic point that is open, so
there is a canonical open immersion SpecQ → Xi for i = 1, 2. Hence we can
glue. We obtain a scheme X with one open point η and two closed points. Let
us compute the global sections of OX using the now classical restriction sequence
87
5.4. More examples
for the open covering X1, X2 :
Γ(X,OX) // Γ(X1,OX)⊕ Γ(X2,OX) //
=
Γ(X1 ∩X2,OX)
=
Z(2) ⊕ Z(3) ρ
// Q
.
The map ρ sends a pair (an−1, bm−1) to the difference an−1 − bm−1, hence the
kernel consists of the diagonal, so to speak, in Z(2) ⊕ Z(3), which is isomorphic
to the intersection Z(2) ∩ Z(3). This is a semi local ring with the two maximal
ideals (2) and (3). Hence there is a map X → SpecZ(2) ∩ Z(3) and it is left as
an exercise to show this is an isomorphism.
More generally, if P = p1, . . . , pr is a finite set different prime numbers,
one may let Xp = SpecZ(p) for p ∈ P . There is, as in the previous case, canonical
open embedding SpecQ → Xp. Let the image be ηp . Obviously the gluing
conditions are all satisfied (the transition maps are all equal to idSpecQ and
Xpq = ηp for all p). We do the gluing and obtain a scheme X. Again, to
compute the global sections, we use the sequence
Γ(X,OX) //⊕
p∈P Γ(Xp,OX) //
=
⊕p,q∈P Γ(X1 ∩X2,OX)
=
⊕p∈P Z(p) ρ
//⊕
p,q∈P Q.
The map ρ sends the sequence (ap)p∈P to the sequence (ap − aq)p,q∈P and the
kernel of ρ is the intersection AP =⋂p∈P Z(p), which is a semilocal ring whose
maximal ideals are the (p)Ap’s for p ∈ P . There is a canonical morphism
X → SpecAp, and again we leave it to the industrious student to verify that
this is an isomorphism.
5.4.1 A more fancy example
In the previous example we worked with a finite set of primes, but the hypothesises
of the gluing theorem impose no restrictions on the number of schemes to be
glued together, and we are free to take P infinite, for example we can use the set
P of all primes! The glued scheme XP is a peculiar animal: It is neither affine
nor noetherian, but it is locally noetherian. There is a map φ : XP → SpecZwhich is bijective and continuous, but not a homeomorphism, and it has the
property that for all open subsets U ⊆ SpecZ the map induced on sections
φ# : Γ(U,OSpecZ)→ Γ(φ−1U,OXP ) is an isomorphism!
As before we construct the scheme XP by gluing the different SpecZ(p)’s
together along the generic points. However, when computing the global sections,
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Chapter 5. Geometric properties of schemes
we see things change. The kernel of ρ is still⋂p∈P Z(p), but now this intersection
equals Z! Indeed, a rational number α = a/b lies in Z(p) precisely when the
denominator b does not have p as factor, so lying in all Z(p), means that b has
no non-trivial prime-factor. That is, b = ±1, and α ∈ Z.
There is a morphism XP → SpecZ which one may think about as follows.
Each of the schemes SpecZ(p) maps in a natural way into SpecZ, the mapping
being induced by the inclusions Z ⊆ Z(p). The generic points of the SpecZp’sare all being mapped to the generic point of SpecZ. Hence they patch together
to give a map XP → SpecZ. This is a continuous bijection by construction, but
it is not a homeomorphism! Indeed, the subsets SpecZ(p) are open in XP by the
gluing construction, but they are not open in SpecZ, since their complements
are infinite, and the closed sets in SpecZ are just the finite sets of maximal
ideals.
The topology of the scheme XP is not noetherian since the subschemes
SpecZ(p) form an open cover that obviously can not be reduced to a finite
cover. However, it is locally noetherian the open subschemes SpecZ(p) being
noetherian. The sets Up = XP − (p) map bijectively to D(p) ⊆ SpecZ and
Γ(Up,OXP ) = Zp, but Up and D(p) are not isomorphic.
Exercise 16. Glue X2 to itself along the generic point to obtain a scheme X.
Show that X is not affine. Hint: Show that Γ(X,OX) = Z(2).
89
Chapter 6
Projective schemes
6.1 Motivation
Let us consider the usual construction of complex projective space: As a topo-
logical space, CPn is the quotient space
CPn =(Cn+1 − 0
)/C∗
where C∗ acts on Cn+1 by scaling the coordinates. We can translate this into
algebra as follows: If f is a function on Cn+1 and λ ∈ C∗ then we get a (right)
action of C∗ on f by defining f.λ(x) = f(λx). In other words, C∗ acts on the
ring C[x0, . . . , xn]. Now, the actions C∗ are well-understood: Any time C∗ acts
on a complex vector space V , we can decompose that vector space as a direct
sum
V =∑d∈Z
Vd
where λ ∈ C∗ acts on v ∈ Vd by λ.v = λdv. Thus we obtain a grading on
C[x0, . . . , xn]. More generally, affine schemes over C with an action of C∗correspond to graded C-algebras.
We want to take the quotient of An+1C −0 = SpecC[x0, . . . , xn]−V (x0, . . . , xn)
by this action. We write PnC for the corresponding quotient space. The notation
PnC, rather than CPn, is used to emphasise that the quotient is taken with the
Zariski topology, and not the usual topology.
We can try to put a scheme structure on PnC by looking for reasonable open
covers. Note that the open subsets of PnC correspond to C∗-invariant open subsets
of An+1C − 0. If is not too hard to see that D(f) ⊆ An+1
C is C∗-invariant if and
91
6.2. Basic remarks on graded rings
only if f is a homogeneous polynomial. We write D+(f) ⊆ PnC for the open
subset corresponding to D(f) ⊆ An+1C − 0.
To define a structure sheaf we should figure out what OPnC(D+(f)) should
be. While it is true that D(f), being an affine scheme, has a structure sheaf,
we have to take more care in deciding which sections to take, to make things
compatible with the C∗-action: A function on D+(f) should be a function on
D(f) that is invariant under the action of C∗. That is, we should have g.λ = g,
which means precisely that g has graded degree zero. The functions of with this
property are exactly the ones of graded degree zero, thus we define
OP (D+(f)) = C[x0, . . . , xn, f−1]0.
where the degree means that we take the degree 0 part.
We can generalize the above for any C-scheme with an action of C∗. Such a
scheme corresponds to a graded C-algebra R. To make a reasonable quotient,
we must remove the locus in SpecR that is fixed by C∗. It is not too hard to
prove the following:
Lemma 6.1. The fixed locus of C∗ acting on SpecR is V (R+), where R+ denotes
the ideal generated by element of positive degree.
So we then attempt to take a quotient of SpecR− V (R+) by C∗. Again, the
C∗-invariant open subsets are of the form D(f) where f is homogeneous, and
these correspond to open subsets D+(f) ⊆ P = (SpecR− V (R+)) /C∗. We can
then define a B-sheaf on P by setting OP (D+(f)) = OSpecS(D(f))0, and check
that we get a scheme P .
Beside of inducing a grading on S, the action of C∗ plays very little role here.
Realizing this, we can in fact build a scheme P from any graded ring R: We
construct the topological space of P from the set of homogeneous prime ideals of
R (with the induced Zariski topology), and define a structure sheaf on it by the
formula like to the one above. This is essentially the ‘Proj’-construction.
6.2 Basic remarks on graded rings
A graded ring R is a ring with a decomposition
R = R0 ⊕R1 ⊕ · · ·
such that Rm · Rn ⊆ Rm+n for each m,n ≥ 0, and such that R0 is a subring
of R. We say that an R-module M is graded if it has a similar decomposition
M =⊕
n≥0Mn such that Rm ·Mn ⊆Mm+n for each m,n ≥ 0. As usual we say
that an element f ∈ R (resp. m ∈M) is homogeneous of degree d if it lies in Rd
92
Chapter 6. Projective schemes
(resp. m ∈Md). An ideal I ⊆ R is homogeneous if the homogeneous components
of any element in I belongs to I.
We will write R+ for the sum⊕
n>0Rn; this is naturally a homogeneous ideal
of R, which we call the irrelevant ideal.
We let R(d) denote the subring of R given by⊕
n≥0Rnd
6.2.1 Localization
Given a homogeneous prime ideal p let T (p) be the set of homogeneous elements
f ∈ R−p. We obtain a grading on the localization T (p)−1R by setting deg(f/g) =
deg f − deg g.
Definition 6.2. For a homogeneous prime ideal p ⊆ R and f ∈ R homogeneous,
define for an R-module M
• M(p) = (T (p)−1M)0
• M(f) = (1, f, f 2, . . . , −1M)0
where the subscript means the degree 0 part.
Example 6.3. For R = A[x0, . . . , xn], we have R(xi) = A[x−1i xj]0≤j≤n.
6.3 The Proj construction
Definition 6.4. Let ProjR denote the set of homogeneous prime ideals of R
that do not contain the irrelevant ideal R+.
We can put a topology on ProjR by setting, for a homogeneous ideal b,
V (b) = p ∈ ProjR : p ⊇ b.
These sets satisfy the following identities
1. V (∑
bi) =⋂V (bi).
2. V (ab) = V (a) ∪ V (b).
3. V (√a) = V (a).
In particular, the V ’s do in fact yield a topology on ProjR (setting the open
sets to be complements of the V ’s). Notice that this is nothing by the induced
topology from the inclusion ProjR ⊆ SpecR.
As with the affine case, we can define the distinguished open sets. For f
homogeneous of positive degree, define D+(f) to be the collection of homogeneous
93
6.3. The Proj construction
ideals (not containing R+) that do not contain f . These are open sets with
respect to the Zariski topology.
Let a be a homogeneous ideal. Then we claim that:
Lemma 6.5. V (a) = V (a ∩R+).
Proof. Indeed, suppose p is a homogeneous prime not containing R+ such that
all homogeneous elements of positive degree in a (i.e., anything in a∩R+) belongs
to p. We will show that a ⊆ p.
Choose a ∈ a ∩R0. It is sufficient to show that any such a belongs to p since
we are working with homogeneous ideals. Let f be a homogeneous element of
positive degree that is not in p. Then af ∈ a ∩ R+, so af ∈ p. But f /∈ p, so
a ∈ p.
Thus, when constructing these closed sets V (a), it suffices to work with ideals
contained in the irrelevant ideal. In fact, we could take a in any prescribed power
of the irrelevant ideal, since taking radicals does not affect V . Note incidentally
that we would not get any more closed sets if we allowed all ideals b, since to
any b we can consider its “homogenization.”
Proposition 6.6. We have D+(f)∩D+(g) = D+(fg). Also, the D+(f) form a
basis for the topology on ProjR.
Proof. The first part is evident, by the definition of a prime ideal. We prove the
second. Note that V (a) is the intersection of the V ((f)) for the homogeneous
f ∈ a ∩R+. Thus ProjR− V (a) is the union of these D+(f). So every open set
is a union of sets of the form D+(f).
Proposition 6.7. Let f ∈ R be homogeneous of degree d. There is a canonical
homeomorphism φf : D+(f)→ SpecR(f) given by
φf : p→ pRf ∩R(f)
sending homogeneous prime ideals of R not containing f into primes of R(f).
Moreover,
(i) For g ∈ R such that D+(g) ⊆ D+(f). If u = gdf− deg g ∈ R(f), then
φ(D+(g)) = D(u).
(ii) For any graded R-module M , there is a canonical homomorphism M(f) →M(g), which induces an isomorphism (M(f))u 'M(g).
(iii) If I ⊆ R is a homogeneous ideal, then φ(V (I)∩D+(f)) = V (IRf ∩R(f)).
94
Chapter 6. Projective schemes
Proof. Note that φ is just the restriction of the canonical map SpecRf →SpecR(f) induced by the inclusion map R(f) ⊆ Rf . Therefore it is continuous.
We now check that it is a bijection.
We construct the inverse map ψ : Spec(R(f))→ D+(f) as follows. Given a
prime ideal p ∈ Spec(R(f)), define ψ(p) =⊕
n≥0 ψ(p)n where
ψ(p)n = x ∈ Rn|xd/fn ∈ p
(Recall that d = deg f). We need to check that this is a homogeneous prime ideal.
Given x, y ∈ ψ(p)n, we have, by definition, that xd/fn ∈ p and yd/fn ∈ p, and
so by the Binomial theorem, (x+ y)2d/f 2n ∈ p. Therefore, (x+ y)d/fn ∈ p, and
so ψ(p) is closed under addition. The other conditions for being a homogeneous
ideal are proved similarly. To show that ψ(p) is prime, let x ∈ Rm and y ∈ Rn
so that (xy)d/fm+n ∈ p. Then (xy)d ∈ p, and so either x ∈ p or y ∈ p since p is
prime. Hence either xd/fm ∈ ψ(p)m or yd/fn ∈ ψ(p)n, and so ψ(p) is prime.
The fact that φ and ψ are mutually inverse follows almost by construction.
Indeed, to verify ψ(pRf ∩ R(f)) = p, note that x ∈ pn ⊆ Rn satisfies xd/fn ∈pRf ∩R(f). Conversely, if x ∈ ψ(pRf ∩R(f))n, then xd/fn ∈ pRf ∩R(f), and so
there is an N > 0 so that fNxd ∈ p. As p is prime, and f 6∈ p, we have xd ∈ p,
and hence x ∈ p. Proving φ ψ = id is similar. This shows that φ is bijective.
The fact that φ is a homeomorphism follows from the fact that it is continuous,
and the point (i) showing that it is open.
(i) Let g ∈ R be an element with D+(g) ⊆ D(f). Then for p ∈ D+(f), we
have p ∈ D+(g) ⇔ gr/f s 6∈ pRf for some r, s > 0 ⇔ gr/f s 6∈ pRf for all r, s > 0
⇔ gdeg f/fdeg g 6∈ p ∩R(f) = φ(p). Hence φ(D+(g)) = D(u).
(ii) Write gk = af . The localization map Mf → Mg is given by xfm7→ amx
gmk,
where x ∈ M . This induces a map M(f) → M(g) (since deg x + m(deg a −k deg g) = deg x−m deg f). The element u acts as an invertible element on R(g),
so the map M(f) →M(g) factors via a map
ρ : (M(f))u →M(g)
We claim that this is an isomorphism.
ρ surjective: Explicitly, we have
ρ
(xf−n
um
)=f tm−nx
gdm
where t = deg g. Take an element y/gl ∈M(g) where deg y = tl. Choose m large
so that dm ≥ l. Define x = gdm−lyf tm−n
∈ M(f). We have deg x = nd, and hencexf−n
um∈ (M(f))u is an element that maps to the given y
gl.
ρ is injective: If x/fn ∈ M(f) maps to 0 in M(g), then there is an l > 0
95
6.3. The Proj construction
so that gldanx = 0 ∈ M . Multiplying up by powers of a and f , we get a
relation of the form g(l+n)dx = 0 ∈M , and hence u(l+n)dx = 0 ∈M(f). But then
x/fn = 0 ∈ (M(f))u.
(iii) Follows by the definition of φ. Let p ∈ V (I)∩D+(f), so p ⊇ I and p 63 f .
Hence pRf∩R(f) ⊇ IRf∩R(f) which gives the ‘⊆′ containment. Conversely, given
a prime ideal p ⊆ R(f) such that p ⊇ IRf ∩R(f), then its preimage p′ in R is a
homogeneous prime ideal not containing f , and φ(p′) = p′Rf ∩R(f) ⊇ IRf ∩R(f).
6.3.1 ProjR as a scheme
We shall now make X = ProjR into a locally ringed space. Let B be the base
of ProjR made up by the distinguished open subsets. For each D+(f) we let
OX(D+(f)) = R(f)
The previous proposition shows that this gives a well-defined B-presheaf of rings,
and using the homeomorphism φ to SpecR(r), we see that it is a actually a
B-sheaf. (Alternatively, we could modify the proof for the case of Spec to see
this directly). We will denote the unique sheaf extension by OX .
It follows that X is has the structure of a ringed space. This is in fact a locally
ringed space, because the stalk Op is just R(p), which is a local ring. Indeed,
the unique maximal ideal is generated by p. Moreover, the previous discussion
has shown that the basic open sets D+(f) are each isomorphic as locally ringed
spaces to SpecR(f), which are affine schemes, and so ProjR is a scheme.
Definition 6.8. For a graded ring R, we call the scheme (ProjR,O) the projec-
tive spectrum of R.
In fact, ProjR is naturally a scheme over R0: The homomorphisms R0 → R(f)
induce a morphism
ProjR→ Spec(R0)
Moreover, if R is a finitely generated over R0, this is of finite type over SpecR0.
This follows by looking at the distinguished opens, D+(f) - each R(f) is finitely
generated as an R0-algebra if R is.
Example 6.9. Let A be a ring and let R = A[t] with the grading given by
deg t = 1 and deg a = 0 for all a ∈ A. Then ProjR ' SpecA.
Example 6.10. Let us study the case of a polynomial ring in two variables.
R = Z[s, t] where s and t have degree 1, and k is any ring. To see that
X = ProjR coincides with P1 as defined in Chapter 3, we see how it is glued
from two affine schemes. Note that X is covered by D+(s) and D+(t) (since
96
Chapter 6. Projective schemes
these generate the irrelevant ideal). Write for simplicitly U = D+(s) ' R(s)
and V = D+(t) ' R(t). Let us check how X is glued together from U, V along
U ∩ V = D+(st) ' SpecR(st).
Note first that Rs ' Z[s, s−1, t] has degree 0 part equal to Z[s−1t] ⊆ Rs.
Similarly, R(t) = Z[st−1]. The intersection D+(st) is the degree 0 part of Rst, i.e.,
R(st) = Z[s−1t, st−1]. In other words, if we write u = s−1t, we have R(s) = Z[u],
R(t) = Z[u−1] and R(st) = Z[u, u−1] = Z[u]u. It follows that U ' SpecZ[u] = A1Z
and V ' SpecZ[u−1] ' A1Z are glued along SpecR(st) which is an open set in
both. From this we see that ProjR coincides with out previous defintion of P1.
Example 6.11. Let R = k[x, y]/(xy). SpecR is the union of the x- and y-axes.
So SpecR− V (x, y) is the axes excluding the origin. ProjR consists of just two
points: We obtain ProjR by gluing Spec(R(x)) and Spec(R(y)). Now,
R(x) = k[x, y](x)/xy = k[x, y](x)/y = k[x](x) = k
and the corresponding chart of ProjR is Spec k. Similarly, the other chart is
Spec k. We have R(xy) = 0, so the overlap is empty and ProjR is two points.
Definition 6.12. We define the projective n-space to be the scheme
Pn = ProjZ[x0, . . . , xn].
More generally, for a ring A, the projective space over A
PnA = ProjA[x0, . . . , xn]
In this notation, Pnk is a scheme whose closed k-points Pn(k) coincides with
the variety of projective n-space.
Proposition 6.13. Let R be a graded ring.
(i) If R is noetherian, then ProjR is noetherian.
(ii) If R is finitely generated over R0, then ProjR is of finite type over SpecR0.
(iii) If R is an integral domain, then ProjR is integral.
Proof. The properties listed are properties one can verify on an affine covering.
In out case ProjR is covered by the affines SpecR(f) which are noetherian (resp.
of finite type, integral) provided R is noetherian (resp. finitely generated, an
integral domain).
97
6.4. Functoriality
6.4 Functoriality
Unlike the case of affine schemes, a graded ring homomorphism φ : R→ S does
not induce a morphism between the projective spectra ProjS and ProjR. The
reason is that some primes in S may pullback to R to contain the irrelevant ideal
R+. However, as we will see shortly, this is the only obstruction to defining a
morphism.
Given a homomorphism φ : R→ S, we define the set G(φ) ⊆ ProjS to be the
set of prime ideals p such that p ⊆ S that do not contain φ(R+), in particular,
such that φ−1(p) is in ProjR. This sets up a map
ψ : G(φ) → ProjR
p 7→ φ−1(p)
Note that G(φ) is an open set of ProjS, so in particular it has an induced scheme
structure. We claim that the map G(φ)→ ProjR is a morphism of schemes.
First of all, the map ψ is continuous, as the Zariski topologies are induced
from those of SpecS and SpecR. More explicitly, the inverse image φ−1a (D+(f))
is equal to G(φ) ∩D+(φ(f)), which is open.
Write X = G(φ) and Y = ProjR. We now define the map ψ on the level of
sheaves, i.e., a map
ψ# : OY → ψ∗OX
On the distinguished open set D+(f) ⊆ ProjR, we can define this map using the
isomorphism to SpecR(f). We have a map R(f) → Sφ(f) induced by φ. Moreover,
since ψ−1(D+(f)) is open in SpecS(φf), we get the map
OY (D+(f)) = R(f) → OX(ψ−1(D+(f))
by restriction. We have shown the following
Proposition 6.14. Let φ : R → S be a morphism of graded rings. The map
ψ : G(φ)→ ProjR is a morphism of schemes.
Example 6.15. To see why restriction to the open set G(φ) is necessary, we
consider the case where R = k[x0, x1], S = k[x0, x1, x2] and φ is the inclusion
map. Note that the prime ideal a = (x0, x1) defines an element in ProjS, but its
restriction to R is the whole irrelevant ideal of R. In fact, G(φ) = ProjS− V (a),
and the map
ψ : P2k − V (a)→ P1
k
is nothing but the projection from the point [0, 0, 1]. It is a good exercise to
prove that there can be no morphisms Pmk → Pnk for m > n in general.
98
Chapter 6. Projective schemes
6.4.1 Closed immersions
The primary example of the above construction is considering the graded quotient
homomorphism φ : R → R/I, where I ⊆ R is a homogeneous ideal. Here we
have G(φ) = Proj(R/I), so we get a map
ψ : Proj(R/I)→ ProjR.
We claim that this is a closed immersion. As usual, homogeneous primes
in R/I not containing R+ correspond to homogeneous primes in R containing
I but not R+. It follows that ψ is injective, mapping onto the closed subset
V (I) in ProjR. Finally, ψ# is surjective on stalks (where it is just the map
R(p) → (R/I)(p)), and so ψ is a closed immersion. We will see later that there is
a converse to this statement, under some mild assumptions on R.
6.5 The Veronese embedding
Let R be a graded ring and let d > be an integer. The inclusion φ : R(d) → R
induces a morphism
νd : ProjR→ ProjR(d)
Indeed, in this case G(φ) = ProjR, since any prime p such that p ⊇ R+ ∩R(d)
must also contain all of R+ – if r ∈ R+, note that rd ∈ R+ ∩R(d) and so r ∈ p
as well! This map is called the Veronese embedding, or d-uple embedding of X.
Proposition 6.16. The Veronese embedding νd is an isomorphism.
Proof. νd is injective: If p, q ∈ ProjR such that p ∩R(d) = q ∩R(d). Then for a
homogeneous element x ∈ R we have
x ∈ p⇔ xd ∈ p⇔ xd ∈ q⇔ x ∈ q
And so p = q. To show that νd is surjective, let q ∈ ProjR(d), and define the
homogeneous ideal in R by
p =∞⊕n=0
x ∈ Rn|xd ∈ q
.
It is not too hard to check that p is prime, and that p ∩ R(d) = q. Hence
νd is bijective, and even a homeomorphism. νd induces an isomorphism when
restricted to the open affines D+(f) as well, and so we are done.
99
6.5. The Veronese embedding
6.5.1 Remark on rings generated in degree 1
We will frequently assume that the ring R is generated in degree 1, that is, R
is generated as an R0-algebra by R1. The reason for this will become clear in
the next section. Intuitively, it is because we want ProjR to be covered by the
‘affine coordinate charts’ D+(x) where x should have degree 1.
We remark that this assumption is in fact not not too restrictive: Any
projective spectrum of a finitely generated ring is isomorphic to the Proj of a
ring generated in degree 1. This is because of the basic algebraic fact that if R
is finitely generated, then some subring R(d) will have all of its generators in one
degree, and since ProjR(d) ' ProjR, we don’t change the Proj by replacing R
with R(d).
6.5.2 The Blow-up as a Proj
Consider the ring A = k[x, y] and the ideal I = (x, y). We can form a new
graded ring by introducing a formal variable t and setting
R =⊕m≥0
Iktk
where I0 = A. In R, t has degree 1, and all other variables have degree 0. This
gives a morphism
π : ProjR→ SpecA = A2k
This ProjR is glued together by two affine schemes SpecR(xt) and SpecR(yt)
which are both isomorphic to A2k. It is a nice exercise to check that this coincides
with the gluing of the blow-up of A2k, and that π is the blow-up morphism.
Claim: We have R ' A[u, v]/(xv − yu).
Consider the map φ : A[u, v] → R given by φ(u) = xt, φ(v) = yt. This is
clearly surjective. It suffices to show that kerφ = I where I = (xv − yu). The
‘ ⊇′-direction is clear. Conversely, we can write, modulo I, any element p of
k[x, u, y, v] as ∑ai,j,kx
iujvk +∑
bi′,j′,k′xi′yj
′vk′.
If p ∈ kerφ, we have
0 = φ(p) =∑
ai,j,kxi+jyktj+k +
∑bi′,j′,k′x
j′yi′+j′tk
′.
Hence the coefficients ai,j,k, bi′,j′,k′ vanish except possibly when we have the same
monomials appearing in each sum i.e. when i+ j = i′, k = j′ + k′, i = i′ + j′ and
j + k = k′, in which case we have ai,j,k = −bi′,j′,k. These conditions imply that
j = −j′ which must then both be 0, and so i = i′, k = k′. Hence p = 0 mod I
and so kerφ ⊆ I.
100
Chapter 6. Projective schemes
Now, writing R = A[u, v]/(xv − yu), we see that ProjR is covered by the
two open sets D+(u) = SpecR(v) and D+(v) = SpecR(v). Here
R(u) ' (A[u, v]u/(xv − yu))0 = k[x, v/u]
and
R(v) ' (A[u, v]v/(xv − yu))0 = k[y, u/v].
These are glued along
R(uv) ' (A[u, v]uv/(xv − yu))0 = k[x, y, u/v, v/u]/(x · v/u− y) ' k[x, u/v, v/u]
so we see that ProjR coincides with the previous blow-up description.
101
Chapter 7
Fiber products
7.1 Introduction
One of the most fundamental properties of the category of schemes is that all
fiber products exist. The fiber product is extremely useful in many situations
and takes on astonishingly versatile roles.
The aim of this chapter is to prove the existence theorem for fiber products
and explain the various contexts where fiber products appear. We begin with
recalling the definition of the fibre product of sets, then transition into a very
general situation to discuss fibre product in general categories, for then to return
to the present context of schemes. At the end of the chapter we will go through
a series of examples.
7.1.1 Fiber products of sets.
As a warming up we use some lines on recalling the fiber product in the category
Sets of sets. The points of departure is two sets X1 and X2 both equipped with
a map to a third set S; i.e., we are given a diagram
X1
ψ1
X2
ψ2~~S.
103
7.1. Introduction
The fibre product X1×SX2 is the subset of the cartesian product X×Y consisting
of the pairs whose two components have the same image in S; that is, we have
X1 ×S X2 = (x1, x2) | ψ1(x1) = ψ2(x2) .
Clearly the diagram below where π1 and π2 denote the restrictions of the two
projections to the fiber product—that is, πi(x1, x2) = xi—is commutative,
X1 ×S X2
π2
%%
π1
yyX1
ψ1 %%
X2
ψ2yyS.
(7.1.1)
And more is true; the fibre product enjoys a universal property: Given any two
maps φ1 : Z → X1 and φ2 : Z → X2 such that ψ1 φ1 = ψ2 φ2 there is a unique
map φ : Z → X1 ×S X2 satisfying π1 φ = φ1 and π2 φ = φ2. To lay your
hands on such a φ, just use the map whose two components are φ1 and φ2 and
observe that it takes values in X1×SX2 since the relation ψ1 φ1 = ψ2 φ2 holds.
Giving the two φi’s is to give a commutative diagram like 7.1.1 above with Z
replacing the product X ×S Y , and the universal property is to say that 7.1.1 is
universal—a more precise usage would be to say it is final among such diagrams.
7.1.2 The fiber product in general categories
The notion of a fiber product—formulated as the solution to a universal problem
as above—is mutatis mutandis meaningful in any category C. Given any two
arrows ψi : Xi → S in the category C, an object—that we shall denote by
X1 ×S X2—is said to be the fiber product (fiberproduktet) of the objects Xi, or
more precisely of the two arrows ψi : Xi → S, if the following two conditions are
fulfilled:
• There are two arrows πi : X1 ×S X2 → Xi in C such that ψ1 π1 = ψ2 π2
(called the projections).
• For any two arrows φi : Z → Xi in C such that φ1 ψ1 = φ2 ψ2, there is
a unique arrow φ : Z → X1 ×S X2 satisfying πi φ = φi for i = 1, 2.
The two arrows π1 φ and π2 φ that determine the arrow φ : Z → X1×SX2,
are called the components (komponentene) of φ, and the notation φ = (φ1, φ2) is
sometimes used. If φ1 : Y1 → X1 and φ2 : Y2 → X2 are two arrows over S, there
is a unique arrow denoted φ1× φ2 from Y1×S Y2 to X1×S X2 whose components
are φ1 πY1 and φ2 πY2 .
104
Chapter 7. Fiber products
If the fiber product exists, it is unique up to a unique isomorphism, as is true
for solutions to any universal problem. However, it is a good exercise to check
this in detail in this specific situation.
Exercise 17. Show that if the fiber product exists in the category C, it is unique
up to a unique isomorphism.
It is not so hard to come up with examples of categories where fiber products
do not exist. For instance, consider the category C of subsets X of the integers
with an even number of elements, and morphisms given by inclusions Y ⊆ X.
The fiber product of Y ⊆ X and Z ⊆ X over X would be Y ∩Z ⊆ X – however
Y ∩ Z does not necessarily have an even number of elements!
What is of course much more disappointing is that fiber products fail to
exist in our good old categories like manifolds or affine varieties. This shows yet
another reason why we need to make the transition from varieties to schemes.
Exercise 18. (i) Give an example showing that the fiber product does not
always exist in the category of manifolds.
(ii) Give an example showing that the fiber product does not always exist in
the category of affine varieties.
7.2 Products of affine schemes
The category Aff of affine schemes is, more or less by definition, equivalent to the
category of rings, and in the category of rings we have the tensor product. The
tensor product enjoys a universal property dual to the one of the fiber product.
To be precise, assume A1 and A2 are B-algebras, i.e., we have two maps of rings
αiA1 A2
B.
α1
==
α2
aa
There are two maps βi : Ai → A1⊗B A2 sending a1 ∈ A1 to a1⊗ 1 and a2
to 1⊗ a2, respectively. These are both ring homomorphisms since aa′⊗ 1 =
(a⊗ 1)(a′⊗ 1) respectively 1⊗ aa′ = (1⊗ a)(1⊗ a′), and they fit into the follow-
ing commutative diagram as α1(b)⊗ 1 = 1⊗ α2(b) by the definition of the tensor
product A1⊗B A2 (this is the significance of the tensor product being taken over
105
7.2. Products of affine schemes
B; one can move elements in B from one side of the ⊗-glyph to the other):
A1⊗B A2
A1
β1::
A2
β2dd
B.
α1
99
α2
ee
(7.2.1)
Moreover, the tensor product is universal in this respect. Indeed, assume that
γi : Ai → C are B-algebra homomorphisms, i.e., γ1 α1 = γ2 α2; or said
differently, they fit into the commutative diagram analogous to (7.2.1) with the
βi’s replaced by the γi’s. The association a1⊗ a2 → γ1(a1)γ(a2) is B bilinear,
and hence it extends to a B-algebra homomorphism γ : A1⊗B A2 → C, that
obviously have the property that γ βi = γi.
Applying the Spec-functor to all this, we get the diagram
Spec(A1⊗B A2)
π1vv
π2
((SpecA1
((
SpecA2
vvSpecB
(7.2.2)
and the affine scheme Spec(A1⊗B A2) enjoys the property of being universal
among affine schemes sitting in a diagram like 7.2.2. Hence Spec(A1⊗B A2)
equipped with the two projections π1 and π2 is the fiber product in the category
Aff of affine schemes. One even has the stronger statement; it is the fiber product
in the bigger category Sch of schemes.
Proposition 7.1. Given φi : SpecAi → SpecB. Then Spec(A1⊗B A2) with the
two projection π1 and π2 defined as above is the fiber product of the SpecAi’s
in the category of schemes. That is, if Z is a scheme and ψi : Z → SpecAiare morphisms with ψ1 π1 = ψ2 φ2, there exists a unique morphism ψ : Z →Spec(A1⊗B A2) such that πi ψ = ψi for i = 1, 2.
Proof. We know that the proposition is true whenever Z is an affine scheme;
so the salient point is that Z is not necessarily affine. For short, we let X =
Spec(A1⊗B A2). The proof is just an application of the gluing lemma for
morphisms. One covers Z by open affines Uα and covers the intersections
Uαβ = Uα ∩ Uβ by open affine subsets Uαβγ as well. By the affine case of the
proposition, for each Uα we get a map ψα : Uα → X, such that ψα πi = ψi|Uα .
106
Chapter 7. Fiber products
By the uniqueness part of the affine case, these maps coincide on the open affines
Uαβγ , and therefore on the intersections Uαβ. They can thus be patched together
to a map ψ : Z → X, which is is unique since the ψα’s are unique.
Example 7.2. Let k be a field and x and y two variables. Consider the tensor
product A = k(x)⊗k k(y). We can regard this as a localization of k[x, y] where
we invert everything in the multiplicative set S = p(x)q(x)|p(x), q(y) 6= 0.Suppose that m ⊆ A is a maximal ideal; it has the form S−1p for some prime
ideal p ⊆ k[x, y] which is maximal among the primes that do not intersect S.
In this case we must have pk[x] = 0, since otherwise there will be a non-zero
p(x) ⊆ p ∩ S. Similarly p ∩ k[y] = 0, which implies that p has height at most 1.
Hence either p = (0), of p = (f) for some irreducible polynomial f ∈ k[x, y] not
contained in k[x] ∪ k[y]. If follows that A has dimension 1, and A has infinitely
many maximal ideals.
This example shows how strange the fiber product really is – SpecA is an
infinite set, even though it is the fiber product of two schemes with one-point
sets. We will see more examples like this in the end of this chapter.
7.3 A useful lemma
Lemma 7.3. If X ×S Y exists and U ⊆ X is an open subscheme, then U ×S Yexists and is (canonically isomorphic to) an open subset of X×SY and projections
restrict to projections. Indeed π−1X (U) with the two restrictions πY |π−1
X (U) and
πX |π−1X (U) as projections is a fiber product.
Proof. Displayed the situation appears like
Y
π−1X (U)
99
// X ×s YπX
πY
OO
U // X,
and we are to verify that π−1X (U) together with the restriction of the two projec-
tions to π−1X (U) satisfy the universal property. If Z is a scheme and φX : Z → U
and φY : Z → Y are two morphisms over S we may consider φU as a map into X,
and therefore they induce a map of schemes φ : Z → X ×S Y whist φX = πX φand φY = πY φ. Clearly πX φ = φU takes values in U and therefore φ takes
values in π−1X (U). It follows immediately that φ is unique (see the exercise below),
and we are through.
107
7.4. The gluing process
Exercise 19. Assume that U ⊆ X is an open subscheme and let ι : U → X be
the inclusion map. Let φ1 and φ2 be two maps of a schemes from a scheme Z to
U and assume that ι φ1 = ι φ2. Then φ1 = φ2.
When identifying π−1X (U) with U ×S Y , the inclusion map π−1
X (U) ⊆ X ×S Ywill correspond to the map ι × idY where ι : U → X is the inclusion, so a
reformulation of the lemma is that open immersions stay open immersion under
change of basis.
7.4 The gluing process
The following proposition will be basis for all gluing necessary for the construction:
Proposition 7.4. Let ψX : X → S and ψY : Y → S be two maps of schemes,
and assume that there is an open covering Ui i∈I of X such that Ui ×S Y exist
for all i ∈ I. Then X ×S Y exists. The products Ui ×S Y form an open covering
of X ×S Y and projections restrict to projections.
Proof. We need some notation. Let Uij = Ui ∩ Uj be the intersections of the
Ui’s, and let πi : Ui ×S Y → Ui denote the projections. By lemma 7.3 there
are isomorphisms θji : π−1i (Uij) → Uij ×S Y , and gluing functions we shall use
τji = θ−1ij θji that identifies π−1
i (Uij) with π−1j (Uij). The picture is like this
Ui ×S Y ⊇ π−1i (Uij)
θji
'//Uij ×S Y
θ−1ij
'//π−1j (Uji) ⊆ Uj ×S Y.
The gluing maps τij clearly satisfy the gluing conditions being compositions
of that particular form, and the scheme emerging from gluing process is X ×S Y .
The two projections are essential parts of the product and mut not be
forgotten: The projection onto Y is there all the time since Y is nevener touched
during the construction. The projection onto X is obtained by gluing the
projections πi along the π−1i (Uij). By lemma 7.3 we know that when identifying
π−1i (Uij) with the product Uij×SY , the projection πij onto Uij corresponds to the
restriction πi|π−1i (Uij)
. This means that πi|π−1i (Uij)
= πijθji. Saying that πi|π−1i (Uij)
and πj|π−1j (Uij)
become equal after gluing, is to say that πi|π−1i (Uij)
= πj|π−1j (Uij)
τji(remember that in the gluing process points x and τji(x) are identified), but this
holds true since
πj|π−1j (Uij)
τji = πij θij τji = πij θij θ−1ij θji = πij θji = πi|π−1
i (Uij).
Hence we can glue the πi’s together to obtain πX .
It is a matter of easy verification that the the glued scheme with the two
projections has the universal property.
108
Chapter 7. Fiber products
It is worth while commenting that the product X ×S Y is not defined
as a particular scheme, it is just an isomorphism class of schemes (having
the fundamental property that there is a unique isomorphism respecting the
projections between any two). In the proof above both π−1i (Uij) and π−1
j (Uij) are
products Uij ×S Y , they are however not equal, merely canonically isomorphic.
In the construction we could have use any of them, or as we in fact did, any
non-specified representative of the isomorphism class, as this makes the situation
more symmetric in i and j.
An immediate consequence of the gluing proposition 7.4 is that fibre products
exist over an affine base S.
Lemma 7.5. Assume that S is affine, then X ×S Y exists.
Proof. First if Y as well is affine, we are done. Indeed, cover X by open affine
sets Ui. Then Ui ×S Y exists by the affine case, and we are in the position to
apply proposition 7.4 above. We then cover Y by affine open sets Vi. As we just
verified, the products X ×S Vi all exist, and applying proposition 7.4 once more,
we can conclude that X ×S Y exists.
7.5 The final reduction
Let Si be an open affine covering of S and let Ui = ψ−1(Si) and Vi = ψ−1Y (Si).
By lemma 7.5 the products Ui ×Si Vi all exist. Using the following lemma and,
for the third time, the gluing proposition 7.4 we are trough:
Lemma 7.6. With current notation, we have the equality Ui ×Si Vi = Ui ×S Y .
Proof. The key diagram is
Zf
g
Ui
ψX |Ui
Y
ψY
⊇ Vi
S
where f and g are given maps. If one follows the left path in the diagram, one
ends up in Si, and hence the same must hold following the right path. But then,
Vi being equal the inverse image ψ−1Y (Si), it follows that g necessarily factors
through Vi, and we are done.
109
7.6. Notation.
7.6 Notation.
If S = SpecA on often writes X×AY in short for X×SpecAY . If S = SpecZ, one
writes X × Y . In case Y = SpecB the shorthand notation X ⊗AB is frequently
seen as well—it avoids writing Spec twice.
Diagrams arising from fiber products are frequently called cartesian diagram
(kartesiske diagrammer); that is, the diagram
Zπ1 //
π2
X
ψX
YψY// S
is said to be a cartesian diagram if there is an isomorphism Z ' X ×S Y with
π1 and π2 corresponding to the two projections.
Exercise 20. Let X, Y and Z be three schemes over S. Show that X×S S = X,
that X ×S Y ' Y ×S X and that (X ×S Y )×S Z ' X ×S (Y ×S Z). If T is a
scheme over S, show that X ×S T ×T Y ' X ×S Y .
Exercise 21. Show by using the universal property that if φ : X ′ → X and
g : Y ′ → Y are morphisms over S, then there is morphism f × g : X ′ ×S Y ′ →X ×S Y such that
X ′ ×S Y ′f×g //
πX′
X ×S YπX
X ′ // X
and a corresponding diagram involving Y and Y ′ commute
7.7 Examples
7.7.1 Varieties versus Schemes
In the important case that X and Y are integral schemes of finite type over the
algebraically closed field k the product of the two as varieties coincides with
their product as schemes over k, with the usual interpretation that the variety
associated to the scheme X is the set closed points X(k) with induced topology.
The product X ×k Y will be a variety (i.e., an integral scheme of finite type
over k) and the closed points of the product X ×k Y will be the direct product
of the closed points in X and Y .
It is not obvious that A⊗k B is an integral domain when A and B are, and
in fact, in general, even if k is a field, it is by no means true. But it holds true
110
Chapter 7. Fiber products
whenever A and B are of finite type over k and k is an algebraically closed field.
The standard reference for this is Zariski and Samuel’s book Commutative algebra
I which is the Old Covenant for algebraists. It is also implicit in Hartshorne’s
book, exercise 3.15 b) on page 22.
However that the tensor product A⊗k B is of finite type over when A and B
are, is straightforward. If u1, . . . , um generate A over k and v1, . . . , vm generate
B over k the products ui⊗ 1 and 1⊗ vj generate A⊗k B.
7.7.2 Non-algebraically closed fields
This case the situation is more subtle when one works over fields that are not
algebraically closed. To illustrate some of the phenomena that can occur, we
study a few basic examples.
Example 7.7. A simple but illustrative example is the product SpecC×SpecRSpecC. This scheme has two distinct closed points, and it is not integral—it is
not even connected!
The example also shows that the underlying set of the fiber product is not
necessarily equal to the fiber product of the underlying sets, although this was
true for varieties over an algebraically closed field. In the present case the three
schemes involved all have just one element and the their fibre product has just
one point. So we issue warnings: The product of integral schemes is in general
not necessarily integral! The underlying set of the fiber product is not always
the fiber product of the underlying sets.
The tensor product C⊗R C is in fact isomorphic to the direct product C×C of
two copies of the complex field C; indeed, we compute using that C = R[t]/(t2+1)
and find
C⊗R C = R[t]/(t2 + 1)⊗RC = C[t]/(t2 + 1) = C[t]/(t− i)(t+ i) = C× C
where for the last equation we use the Chinese remainder theorem and that the
rings C[t]/(t± i) both are isomorphic to C.
Example 7.8. This little example can easily be generalized: Assume that L is
a simple, separable field extension of K of degree d; that is L = K(α) where
the minimal polynomial f(t) of α over K is separable and of degree d. Let
Ω be a field extension of K in which the polynomial f(t) splits completely—
e.g., a normal extension of L or any algebraically closed field containing K—
then by an argument completely analogous to the one above one finds that
L⊗K Ω=Ω× . . .×Ω where the product has d factors. Consequently the product
scheme SpecL ×Spec k Spec Ω has an underlying set with d points, even if the
three sets of departure all are prime spectra of fields and thus singletons.
111
7.7. Examples
One may push this further and construct examples where SpecK ⊗Spec k Spec Ω
is not even noetherian and has infinity many points!
Exercise 22. With the assumptions of the example above, check the statement
that L⊗K Ω ' Ω× . . .× Ω, the product having d factors.
Exercise 23. Assume that A is an algebra over the field k having a countable
set e1, e2, . . . , ei, . . . of mutually orthogonal idempotents, i.e., eiej = 0 if i 6= i
and eiei = 1, and assume that eiA ' k. Assume also that every element is a
finite linear combination of the ei’s.
• Show that the ideal Ij generated by the ei’s with i 6= j is a maximal ideal.
Example 7.9. In this example we let L ∈ C[x, y] be a linear form that is not real,
for example L = x+ iy + 1, and we introduce the real algebra A = R[x, y]/(LL).
The product LL of L and its complex conjugate is a real irreducible quadric;
which in the concrete example is (x+ 1)2 + y2. The prime spectrum SpecA is
therefore an integral scheme. However, the fiber product SpecA ×R SpecC is
not irreducible being the union of the two conjugate lines L = 0 and L = 0 in
SpecC[x, y].
The scheme SpecA has just one real point, namely the point (−1, 0) (i.e.,
corresponding to the maximal ideal (x + 1, y)). The C-points however, are
plentiful.
They are contained in the C-points A2R(C), which are of orbits (a, b), (a, b)
of the complex conjugation with (a, b) non-real, and form the subset of those
(a, b) such that L(a, b) = 0.
Example 7.10. Another example along same lines as example 7.8 shows that
the fiber product X ×S Y is not necessarily reduced even if both X and Y are;
the point being to use an inseparable polynomial f(t) in stead of a the separable
one in 7.8. Let k be a non-perfect field in characteristic p which means that
there is an element a ∈ k not being a p-th power of any element in k. Let L be
the field extension L = k(b) where bp = a. That is, L = k[t]/(tp − a). which is
a field since tp − a is an irreducible polynomial over k. However, upon being
tensored by itself over k, it takes the shape
L⊗k L = L[t]/(tp − a) = L[t]/(tp − bp) = L[t]/((t− b)p)
which is not reduced, the non-zero element t− b being nilpotent. So we issue a
third warning: the fiber product of integral schemes is not in general reduced!
One can elaborate these example and construct an example of two noetherian
schemes X and Y such that X ×S Y is not noetherian, even if S is the spectrum
of a field. The next examples shows that if X, Y and S are fields, the product
X ×S Y can even have an uncountable number of elements!
112
Chapter 7. Fiber products
Example 7.11. In this example we take L to the subfield of the field Q of
algebraic numbers that is generated by all the elements a such that a2n = 2 for
some n. The field L is the union of the ascending chain of fields
Q ⊆ Q(√
2) ⊆ Q(4√
2) ⊆ Q(8√
2) ⊆ . . . ⊆ Q(2r√
2) ⊆ . . . ⊆ L ⊆ Q
We denote r-th field in the chain Q( 2r√
2) by Lr. The next field Lr+1 is the
quadratic extension of Lr obtained simply by adjoining 2r+1√2, or, in other words,
the square root of 2r√
2.
We let Ar = Lr⊗QQ. This is a finite algebra of rank 2r over Q. It has a
refined algebraic structure given by the smaller algebras As for s ≤ r which are
all subalgebras of Ar. the algebra Ar splits as the direct product of two copies of
Ar−1; indeed, one has Lr⊗QQ = Lr⊗Lr−1 Q⊗Q Lr−1⊗Q Q = Ar−1 ⊕ Ar−1 since
Lr⊗Lr−1 Q ' Q⊕Q.
The two idempotents in Ar that induce this splitting are denoted by er,0and er,1. They are orthogonal and their sum equals one. Each of the two
subalgebras er,εAr are isomorphic to Ar−1 with 1 ∈ Ar−1 corresponding to er,ε,
they contain the idempotens er−1,ε which will correspond to the product er,εer−1,ε′
in Ar. Working our way down in Ar, this yields a sequence of idempotents
eI = er,εrer−1,εr−1 . . . e1,ε1 where I = (ε1, . . . , εr) is a sequence of 1’s and 0’s.
Take any sequence σ = (σi)i ∈ N . Let I ⊆ A be generated by the ei,εi with
εi /∈ σ. Then I is maximal. It contains all product except∏
i ei,σi . And these all
generate the same copy of Q ⊆ A!
Then of course L being a field is noetherian as is Q, but L⊗QQ is not!
Indeed, Q( 2r√
2)⊗Q is isomorphic to the direct product of 2r-copies of Q so
L⊗QQ is the union of a sequence of subrings each being a direct product of a
steadily increasing number of copies of Q.
7.8 Base change
The fiber product is in constant use in algebraic geometry, and it is an aston-
ishingly versatile and flexible instrument. In different situations it serves quite
different purposes and appears under different names. We shall comment on
some of the most frequently encountered applications, and we begin with notion
of base change.
In its simples and earliest appearances base change is just extending the
field over which one works; e.g., in Galois theory, or even in the theory of real
polynomials, when studying an equation with coefficients in a field k one often
finds it fruitful to study the equation over a bigger field K. Generalizing this to
extensions of algebras over which one works, and then to schemes, one arrives
naturally at the fiber product.
113
7.9. Scheme theoretic fibres
If X is a scheme over S and T → S is map. Considering T → S as change of
base schemes one frequently writes XT = X ×S T and says that XT is obtained
from X by base change (basisforandring) or frequently that XT is the pull back
(tilbaketrekningen) of X along T → S. This is a functorial construction, since
if φ : X → Y is a morphism over S, there is induced a morphism φT = φ× idTfrom XT to YT over T , and one easily checks that (φ φ′)T = φT φ′T . The
defining properties of φT are πY φT = φ πX and πT φ = πT , as depicted in
the diagram:
T
XT = X ×S TφT //
πT55
φπX ))
Y ×S TπY
πT
OO
Y.
Exercise 24. Verify in detail that (φ φ′)T = φT φ′T .
If P is a property of morphisms, one says that P is stable under base change
if for any T over S, the map fT has the property P whenever f has it. For
example, another way of phrasing lemma 7.3 on page 107 is to say that being an
open immersion is stable under base change.
7.9 Scheme theoretic fibres
In most parts of mathematics, when one studies a map of some sort, a knowledge
of what the fibres of the map are, is of great help. This is also true in the theory
of schemes.
Suppose that φ : X → Y is a map of schemes and that y ∈ Y is a point. On
the level of topological spaces, we are interested in the preimage φ−1(y), and
we aim at giving a scheme theoretic definition of the fiber φ−1(y). Having the
fiber product at our disposal, nothing is more natural than defining the fiber
to be the fiber product φ−1(Y ) = Spec k(y)×Y X, where Spec k(y)→ Y is the
map corresponding to the point y. Recall that k(y) = OY,y/my and the map is
the composition Spec k(y)→ Spec OY,y → Y . For diagrammoholics, the scheme
theoretic fiber of φ over y fits into the cartesian diagram
φ−1(y) = X ×Y Spec k(y) //
X
φ
Spec k(y) // Y
.
As the next lemma will show, the underlying topological space of φ−1(y) is
the topological fiber, but in addition there is a scheme structure on it. In in
114
Chapter 7. Fiber products
many cases it is not reduced, and this a mostly a good thing since it makes
certain continuity results true.
Proposition 7.12. The inclusion Xy → X of the scheme theoretic fiber is a
homeomorphism onto the topological fiber φ−1(y).
Proof. We start with the affine case. Obviously Y can always without loss of
generality be assumed to be affine, say Y = SpecB, but to begin with we adopt
the additional assumption that X is affine as well, let’s say X = SpecA.
The map φ of affine schemes is induced by a map of rings α : B → A. Let
p ⊆ B be a prime ideal. We have the following equality between sets
q ⊆ A | q prime ideal , α−1(q) ⊇ p = q ⊆ A | q prime ideal , q ⊇ pA .
In the particular case that p is a maximal ideal, the inclusion α−1(q) ⊇ q is
necessarily an equality, and the sets above describe the fiber set-theoretically:
φ−1(p) = q ⊆ A | q ⊇ pA ' SpecA/pA.
But this also describes the good old embedding of SpecA/pA into SpecA identi-
fying it with the closed subscheme V (pA), and therefore this yields a homeomor-
phism between SpecA/pA and the topological fiber φ−1(p). On the other hand
by standard equalities between tensor products one has
A/pA = A⊗B B/pB = A⊗B k(y),
and so the scheme theoretical fiber φ−1(y) = Xy = X×Y Spec k(y) = SpecA⊗B k(y)
is in a canonical way homeomorphic to the topological fiber.
If p is not a maximal ideal, the set SpecA/pA is strictly bigger than the fiber,
the superfluous prime ideals being those for which α−1q strictly bigger than p.
When localising in the multiplicative system S = B − p ⊆ B, these superfluous
prime ideals go non-proper, since they all contain elements of the form α(s) with
s ∈ S. Hence the points in the fiber correspond to the primes in the localized
ring (A/pA)p. Standard formulas for the tensor product gives on the other hand
the equality
(A/pA)p = A⊗B Bp/pBp = A⊗B k(y).
The topologies coincides as well, since Spec(A/pA)p naturally is a subscheme
of SpecA/pA; induced topology being the one a prime spectrum.
In the general case, i.e., when X is no longer affine, we cover X by open, affine
Ui’s. By the universal property of fiber products, we know that U ∩Xs = Us (see
the diagram below). This shows that the scheme theoretical and the topological
115
7.9. Scheme theoretic fibres
fiber coincide as topological spaces.
Us //
U
Xs
//
X
Spec k(y) // Y
Example 7.13. We take a look at a simple but classic example: The map
Spec k[x, y]/(x− y2)→ Spec k[x]
induced by the injection B = k[x] → k[x, y]/(x − y2) = A. Geometrically one
would say it is just the projection of the parabola onto the x-axis.
If a ∈ k computing the fiber yields, where k(a) denotes the field k(a) =
k[t]/(t− a) (which of course is just a copy of k).
k[x, y]/(x− y2)⊗k[x] k[x]/(x− a) ' k[y]/(y2 − a).
(Here we are using the isomorphism R/I ⊗RM ' M/IM for an ideal I in
an R-module M .)
Several cases can occur, apart from the characteristic two case being special.
• If a does not have a square root in k, the fiber is Spec k(√a) where k(
√a)
is a quadratic field extension of k.
• In case a has a square root in K, way b2 = a, the polynomial y2−a factors as
(y − b)(y + b), and the fiber becomes Spec k[y]/(y − b)× Spec k[y]/(y + b),
the disjoint union of two copies of Spec k
• Finally, the case appears when a = 0. The the fiber is not reduced, but
equals Spec k[y]/y2.
We also notice that the generic fiber is the quadratic extension k(x)(√x) of
the function field k(x).
Over perfect fields k of characteristic two, the picture is completely different.
Then a is a square, say a = b2 and as (y2 − b2) = (y − b)2 non of the fibers are
reduced, they equal Spec k[y]/(y − b)2, except the generic one which is k(x)(√x).
One observes interestingly enough, that all the non-reduced fibers deform into a
field!
116
Chapter 7. Fiber products
Example 7.14. A similar example can be obtained from the map
f : SpecA→ SpecB
where A = Spec k[x, y, z]/(xy−z) and B = k[z] (The map k[z]→ k[x, y, z]/(xy−z) is the obvious one). As before, we pick an element a ∈ SpecB, and consider
the fiber
Xa = Spec (A⊗B k(a)) = Spec k[x, y]/(xy − a)
Again, two cases occur. If a 6= 0, in which case xy−a is an irreducible polynomial,
and so Xa is an integral scheme. This is intuitive, since it corresponds to the
hyperbola xy = a. If a = 0, we are left with X0 = Spec k[x, y]/(xy), which is
not irreducible; it has two components corresponding to V (x) and V (y). (X0 is
reduced however).
For good measure, we also consider the fiber over the generic point η of
SpecB. This corresponds to
k[x, y, z]/(xy − z)⊗k[z] k(z) = k(z)[x, y]/(xy − z)
which is an integral domain. Hence Xη is integral.
7.10 Separated schemes
We have previously seen that the topology on schemes behaves very differently
from the usual euclidean topoology. In particular, schemes are not Hausdorff,
except in trivial cases – the open sets in the Zariski topology are just too large.
Still we would like to find an analogous property that can serve as a substitute
for this property. The route we take is to impose that the diagonal should be
closed.
7.10.1 The diagonal
Let X/S be a scheme over S. There is a canonical map ∆X/S : X → X ×S X of
schemes over S called the diagonal map or the diagonal morphism (diagonalav-
bildningen). The two components maps of ∆ are both equal to the identity idX ;
that is, the defining properties of ∆X/S are πi ∆X/S = idX for i = 1, 2 where
the πi’s denote the two projections.
In the case X and S are affine schemes, the diagonal has a simple and natural
interpretation in terms of algebras; it corresponds to most natural map, the
multiplication map:
µ : A⊗B A→ A.
It sends a⊗ a′ to the product aa′ and then extends to A⊗B A by linearity. The
117
7.10. Separated schemes
projections correspond to the two maps ιi : A → A⊗B A sending a to a⊗ 1
respectively to 1⊗ a. Clearly it holds that µ ιi = idA, and on the level of
schemes this translates into the defining relations for diagonal map. Moreover µ
is clearly surjective, so we have established the following:
Proposition 7.15. If X an affine scheme over the affine scheme S, then the
diagonal ∆X/S : X → X ×S X is a closed imbedding.
This is not generally true for schemes and shortly we shall give examples,
however from the proposition we just proved, it follows readily that the image
∆X/S(X) is locally closed—i.e., the diagonal is locally a closed embedding:
Proposition 7.16. The diagonal ∆X/S is locally a closed embedding.
Proof. Begin with covering S by open affine subset and subsequently cover each
of their inverse images in X by open affines as well. In this way one obtains a
covering of X by affine open subsets Ui whose images in S are contained in affine
open subsets Si. The products Ui×Si Ui = Ui×SUi are open and affine, and their
union is an open subset containing the image of the diagonal. By proposition
7.15 above the diagonal restricts to a closed embedding of Ui in Ui ×Si Ui.
Exercise 25. In the setting of the previous proof, show that ∆X/S|Ui = ∆Ui/S.
7.10.2 Separated schemes
Definition 7.17. The scheme X/S is separated (separert) over S, or that the
structure map X → S is separated if the diagonal map is a closed imbedding. If
X is separated over SpecZ one says for short that it is separated.
Example 7.18. The simplest example of a scheme that is not separated is
obtained by glueing the prime spectrum of a discrete valuation ring to itself
along the generic point.
To give more details let R be the DVR with fraction field K. Then SpecR =
x, η where x is the closed point corresponding to the maximal ideal, and η
is the generic point corresponding to the zero ideal. The generic point η is an
open point (the complement of η is the closed point x) and the support of
the open subscheme η = SpecK. By by the glueing lemma, we may glue one
copy of SpecR to another copy of SpecR by identifying the generic points—that
is, the open subschemes SpecK—in the two copies.
In this manner we construct a scheme ZR together with two open embeddings
ιi : SpecR→ ZR. They send the generic point η to the same point, which is an
open point in ZR, but they differ on the closed point x.
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Chapter 7. Fiber products
It follows that the diagonal is not closed. Indeed, the subscheme of SpecR×SpecR where the two maps ιi agree is the preimage of the diagonal. But this
subscheme has exactly one point which is open.
Exercise 26. Show that ZR × ZR is obtained by glueing four copies of SpecR
together along their generic points. Show that the diagonal is open and not
closed.
Example 7.19. The affine line with two origins from Chapter 3 is not separated.
One of the nice properties affine schemes enjoy, is the following:
Proposition 7.20. Assume that X is separated and that U and V are to
open affine subscheme. Then the intersection U ∩ V is affine and the map
Γ(U,OU)⊗ Γ(V,OV )→ Γ(U ∩ V,OU×V ) is surjective
Proof. The product U × V is an open and affine subset of X × V , and U ∩ V =
∆X(X) ∩ (U × V ). So if the diagonal is closed, U ∩ V is a closed subset of the
affine set U ×V hence affine. It is a general fact about products of affine schemes
that one has
Γ(U × V,OU×V ) = Γ(U,OU)⊗ Γ(V,OV ),
and as U ∩ V is a closed subscheme of U × V , the restriction map
Γ(U × V,OU×V )→ Γ(U ∩ V,OU∩V )
is surjective.
7.11 The valuative criterion
In some sense, these tiny schemes ZR together with some of their bigger cousins
are always at the root of a non separated scheme. Using the properties of these
subschemes, one can prove the following theorem, which gives a convenient
description of separated schemes.
Proposition 7.21. Assume that X is a quasi-compact scheme. Then X/S is
non-separated if and only if it contains a subscheme ZR/S for some valuation
ring R.
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7.11. The valuative criterion
This is usually formulated as follows:
Theorem 7.22 (Valuative criterion for separateness). A quasi-compact scheme
X is separated if and only if the following condition is satisfied: For any DVR
R with fraction field K, a map SpecK → X over S has at most one extension
to SpecR→ X.
In other words, a morphism X → S is separated if for every DVR R with
fraction field K and diagram
Spec(K) //
X
Spec(R) //
;;
S
there can be at most one extension SpecR → X (the dotted arrow in the
diagram) such that everything commutes.
This looks like a technical statement, but it is tremendously powerful for
proving theorems about separateness. Here is a sample:
Corollary 7.23. (i) Open and closed immersions are separated
(ii) A composition of two separated morphisms is again separated
(iii) Separatedness is stable under base change: If f : X → S is separated, and
S ′ → S is any morphism, then f ′ : X ×S S ′ → S ′ is separated.
The proof is straightforward using the Valuative Criterion.
For instance, consider item (iii). Taking a base change S ′ → S gives us a
diagram
Spec(K) //
X ×S S ′
g
// X
f
Spec(R) //
8888
S ′ // S
giving two arrows SpecR ⇒ X (by composition), which have to be equal, by
separatedness of f . Now, the two morphisms SpecR ⇒ X ×S S ′ have to be
equal by the universal property of the fiber product.
It is not true that maps SpecK → X like in the criterion always can be
extended—schemes with that property are called proper schemes—so the criterion
says there can never be two different extensions in case X is separated.
One doesn’t frequently encounter non-separated scheme in practice, but some
very nice properties are only true for separated schemes, and this legitimates the
notion. Of course, one needs good criteria to be sure we have a large class of
120
Chapter 7. Fiber products
separated schemes. We have already seen that all affine schemes are separated,
and when we come to that point projective schemes will turn out to be separated
as well.
7.11.1 Varieties vs schemes again
Finally we can state the definition of a variety:
Definition 7.24. A variety X is an integral, separated scheme of finite type
over an algebraically closed field.
This definition requires some explanation. Recall that we defined for a field
k a functor
t : Var/k → Sch/k
which associates a variety V to a scheme t(V ) over k, such that V is homeomorphic
to the subset of k-points of t(V ). We stated that this functor is fully faithful,
making the category of varieties Var/k equivalent to a full subcategory of Sch/k.
It is a theorem (Hartshorne II.4.10) that this subcategory is exactly the category
of schemes satisfying Definition 7.24. For the proof of this statement, see See
Hartshorne II.4.10.
Theorem 7.25. The functor t is fully faithful, and gives and equivalence between
the category of varieties Var/k and the subcategory of Sch/k of schemes satisfying
Definition 7.24.
121
Chapter 8
Quasi-coherent sheaves
When you study commutative algebra may be you are primarily interested in
the rings and ideals, but probably you start turning your interest towards the
modules pretty quickly; they are an important part of the world of rings, and to
get the results one wants, one can hardly do without them. The category ModAof A-modules is a fundamental invariant of a ring A; and in fact, is the principal
object of study in commutative algebra. This is true also with schemes; for
which the so called quasi-coherent Ox-modules form an important attribute of
the scheme, if not a decisive part of the structure. They form a category QCohXwith many properties parallelling those of the category ModA. In fact, in case
the schemeX is affine, i.e., X = SpecA, the two categories ModA and ModXare equivalent, as we will prove shortly. Imposing finiteness conditions on the
OX-modules one arrives at the category CohX of so called coherent OX-modules
that in the noetherian case parallel the finitely generated A-modules.
We start out by describing the much broader concept of an OX-module. The
theory here is presented for schemes, but the concept is meaningful for any ringed
space.
In the literature one finds different approaches to the quasi-coherent sheaves.
We follow EGA and Hartshorne and introduce the quasi-coherent module first for
affine schemes. If X = SpecA one defines an OX-module M associated with an
A-module M , and in the general case these modules serve as the local models for
the quasi-coherent ones. There is a notion of quasi-coherence for OX-modules on
a general locally ringed space. In some other branches of mathematics they are
important, e.g., analytic geometry, but we concentrate our efforts on schemes.
At the end of the chapter we will discuss locally free sheaves and vector
bundles, which are in many respects the most interesting coherent sheaves on a
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8.1. Sheaves of modules
scheme.
8.1 Sheaves of modules
A module over a ring is just an additive abelian group equipped with a multi-
plicative action of A. Loosely speaking we can multiply members of the module
by elements from the ring, and of course, the usual set of axioms must be
satisfied. In a similar way, if X is a scheme, an OX-module is an abelian sheaf
F whose sections can be multiplied by sections of OX ; the multiplicator and the
multiplicand of course being sections over the same open subset.
Formally, an OX-module structure on the abelian sheaf F is defined as a family
of multiplication maps Γ(U,F)×Γ(U,OX)→ Γ(U,F)—one for each open subset
U of X—making the space of sections Γ(U,F) into a Γ(U,OX)-module in a way
compatible with all restrictions. That is, for every pair of open subsets V ⊆ U ,
the natural diagram below—where vertical arrows are made up of appropriate
restrictions, and the horizontal arrows are multiplications—commutes
Γ(U,F)× Γ(U,OX)
// Γ(U,F)
Γ(V,F)× Γ(V,OX) // Γ(V,F).
Maps, or homomorphisms, of OX-modules are just maps α : F → G between
OX-modules considered as abelian sheaves, respecting the multiplication by
sections of OX . That is, for any open U the map αU : Γ(U,F) → Γ(U,G) is a
Γ(U,OX)-homomorphism.
We have now explained what OX-modules are and told what their homomor-
phisms should be, and this organizes the OX-modules into a category that we
denote by ModX .
The category ModX is an additive category : The sum of two OX-homomorphisms
as maps of abelian sheaves is again an OX-homomorphism. So for all F and
G the set HomOX (F ,G) is an abelian group, and the compositions maps are
bilinear. Moreover, the direct sum of two OX-modules as abelian sheaves has
an obvious OX-structure, with multiplication being defined componentwise. In
fact, this argument works for arbitrary direct sums (or coproducts as they also
are called). For any family Fi of OX-modules,⊕
i∈I Fi is an OX-module (see
Exercise 31 below).
The notions of kernels, cokernels and images of OX-module homomorphisms
now appear naturally. All the three corresponding abelian constructions are
invariant under multiplication by sections of OX , and therefore they have OX-
module structures. The respective defining universal properties (in the category
124
Chapter 8. Quasi-coherent sheaves
of OX-modules) come for free, and one easily checks that this makes ModX an
abelian category; i.e., the kernels of the cokernels equal the cokernels of the
kernels.
There is tensor product of OX-modules denoted by F ⊗OX G. As in many
other cases, the tensor product F ⊗OX G is defined by first describing a presheaf
that subsequently is sheafified. The sections of the presheaf, temporarily denoted
by F ⊗′OX G, are defined in the natural way by
Γ(U,F ⊗′OX G) = Γ(U,F)⊗Γ(U,OX) Γ(U,G).
There is also a sheaf of OX-homomorphisms between F and G. The definition
goes along lines slightly different from the definition of the tensor product. Recall
the sheaf Hom (F ,G) of homomorphisms between the abelian sheaves F and Gwhose sections over an open set U is the group Hom(F|U ,G|U ) of homomorphisms
between the restrictions F|U and G|U . Inside this group one has the subgroup
of the maps being OX-homomorphisms, and these subgroups, for different open
sets U , are respected by the restriction map. So they form the sections of a
presheaf, that turns out to be a sheaf, and that is the sheaf HomOX (F ,G) of
OX-homomorphisms from F to G.
Example 8.1 (Modules on spectra of DVR’s). Modules on the prime spectrum
of a discrete valuation ring R are particularly easy to describe.
Recall that the scheme X = SpecR has only two non-empty open sets, the
whole space X it self and the singleton η where η denotes the generic point.
The singleton η is underlying set of the open subscheme SpecK, where K
denotes the fraction field of R.
We claim that to give an OX-module is equivalent to giving an R-module M ,
a K-vector space N and a homomorphism ρ : M ⊗R K → N .
Indeed, given an OX module F , we get an R module M = Γ(X,F), and
a vector space N = Γ( η ,F) over K. The homomorphism ρ is just the
restriction map F(X)→ F( η ). Conversely, given the data M,N, ρ, we can
define F(X) = M and F(η) = N . The map ρ : F(X)→ F( η ) makes F into
an OX-module.
Note that the restriction map can be just any R-module homomorphism
M → N . In particular, it can be the zero homomorphism, and in that case M
and N can be completely arbitrary modules. Again this illustrates the versatility
of general OX-modules.
Example 8.2 (A bunch of wild examples). The OX-modules (or at least some
of them) play a leading leading role in the theory of schemes, and shortly we
shall see a long series of examples. These will all be so called quasi-coherent
sheaves. The examples we now describe are a bunch of wild examples, intended
125
8.1. Sheaves of modules
to show that OX-modules without any restrictive hypothesis are very general
and often unmanageable objects.
Recall the Godement construction from Chapter 1. Given any collection of
abelian groups Ax x∈X indexed by the points x of X. We defined a sheaf Awhose sections over an open subset U was
∏x∈U Ax, and whose restriction maps
to smaller open subsets were just the projections onto the corresponding smaller
products. Requiring that each Ax be a module over the stalk OX,x makes A into
an OX-module; indeed, the space of sections Γ(U,A) =∏
x∈U Ax is automatically
an Γ(U,OX)-module, the multiplication being defined componentwise with the
help of the stalk maps Γ(U,OX) → OX,x. Clearly this module structures is
compatible with the projections, and thus makes A into an OX-module.
Exercise 27. Assume that F and G are OX-modules and that α : F → G is a
map between them. Show that the kernel, cokernel and image of α as a map
of abelian sheaves indeed are OX-modules, and that they respectively are the
kernel, cokernel and image of α in the category of OX-modules as well. Show
that a complex of OX-modules is exact if and only it is exact as a complex of
abelian sheaves.
Exercise 28. Let F and G be two OX-modules on the scheme X. Show that
the stalk (F ⊗OX G)x at the point x ∈ X is naturally isomorphic to the tensor
product Fx⊗OX,x Gx of the stalks Fx and Gx. Show that the tensor product is
right exact in the category of OX-modules.
Exercise 29. Show that the sheaf-hom HomOX (F ,G) of two OX-modules as
defined above is a sheaf. Show that HomOX (F ,G) is right exact in the second
variable and left exact in the first.
Exercise 30 (Adjunction between Hom and ⊗). Show that for OX-modules
F ,G,HHomOX (F ,HomOX (G,H) ' HomOX (F ⊗OX G,H)
(this is easier than it looks – reduce to the usual tensor product for modules over
rings).
Exercise 31. Show that the category ModX has arbitrary products and coprod-
ucts, by showing that the products and coproducts in the category of abelian
sheaves ShX are OX-modules and are the products, respectively the coproducts,
in the category ModX .
Exercise 32. Assume that p1, . . . , pr is a set of primes, and let Z(pi) as usual
denote the localization at prime ideal (pi). Let X be the scheme obtained by
gluing the schemes Xi = SpecZ(pi) together along their common open subschemes
SpecQ. Describe the OX-modules on X.
Exercise 33. Let A =∏
1≤i≤nKi be the product of finitely many fields. Describe
the category ModX .
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Chapter 8. Quasi-coherent sheaves
8.1.1 The support of a sheaf
Definition 8.3. Let (X,OX) be a ringed space. Let F be a sheaf of OX-modules.
The support of F , Supp(F), is the set of points x ∈ X such that Fx 6= 0. For
s ∈ Γ(X,F) we define the support of s as the set of points x ∈ X such that the
image sx ∈ Fx of s is not zero. We denote this by Supp(s).
In general the support of a sheaf of modules is not closed. Indeed, as before,
we can get strange sheaves by taking any non-closed Z set of your favorite ringed
space, and define a Godement sheaf A by setting Ax 6= 0 if and only if x ∈ Z.
However, for sheaves of rings, the support always closed: this is because a
ring is 0 if and only if 1 = 0, and so the support of a sheaf of rings is the support
of the section ‘1’ - which is closed.
8.2 Pushforward and Pullback of OX-modules
8.2.1 Pushforward
Let f : (X,OX)→ (Y,OY ) be a morphism of schemes. If F is a sheaf on X the
pushforward f∗F is naturally a f∗OX-module via the addition and multiplication
maps
f∗F × f∗F → f∗F , f∗OX × f∗F → f∗F .
Via f# : OY → f∗OX we obtain the structure of a OY -module on f∗F .
Definition 8.4. The above OY -module is called the direct image of F under f .
This condition is clearly functorial in F and so we obtain a functor f∗ :
ModX → ModY . By the properties of f∗ in Chapter 1, we see that this is left
exact.
8.2.2 Pullback
The pullback of a sheaf of OY -modules is a little bit more subtle to define. Recall
that we in Chapter 1 defined the inverse image f−1G of a sheaf G by sheafifying
the presheaf given by assigning an open U to the direct limit of all G(V ) where V
contains f(U). This sheaf is naturally a f−1OY -module. We can make it into an
OX-module using the map f−1OY → OX (which makes OX an f−1OY -algebra),
and taking the tensor product:
f ∗G = OX ⊗f−1OY f−1G
The association G 7→ f ∗G is functorial, so we get a functor f ∗ : ModOY → ModOX .
The above OX-module is called the pullback of G under f . Note in particular
that f ∗OY = OX .
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8.2. Pushforward and Pullback of OX-modules
Proposition 8.5. For a point x ∈ X we have the following expression for the
stalk
(f ∗G)x = Gf(x) ⊗OY,f(x) OX,x.
Proof. This follows from the facts that taking stalks commutes with tensor
products, and (f−1G)x = Gf(x).
8.2.3 Adjoint properties of f∗, f∗
At first sight, the definition of the pullback might seem a bit out of the blue.
It is defined from f−1G, tensoring with OX over f−1OY to rig it into being a
OX-module. However, as like in the case of the inverse image functor f−1, the
important point is what the sheaf does, rather than how it is explicitly defined.
Proposition 8.6. The functors f∗, f∗ between ModOX ,ModOY are adjoint. In
particular, if F ∈ ModOX ,G ∈ ModOY , there is a functorial isomorphism
HomOX (f ∗G,F) ' HomOY (G, f∗F).
Proof. Suppose φ : G → f∗F is an OY -module homomorphism. Then by the
adjoint property of f∗ and f−1 (in the categories ShX and ShY ), we get a map
f−1G → F , which is f−1OY -linear. Now F is an OX-module, so we get a
OX-linear map f−1G ⊗f−1OY OX → G by the universal property of the tensor
product.
Conversely, let φ : f ∗G → F be OX-linear. There is a map f−1G → f ∗Gwhich is f−1OY -linear. Consequently there is a f−1OY -linear map f−1G → F .
This induces a OY -linear map G → f∗F by the earlier adjointness property of f∗and f−1.
In particular, we the maps idf∗G and idf∗F provide us with the canonical
maps
η : G → f∗f∗G, ν : f ∗f∗F → F
We already saw previously that f∗ is a left-exact functor. Now f ∗ is right-
exact, by general properties of adjoint functors.
Corollary 8.7. f ∗ is right exact and f∗ is left-exact.
8.2.4 Pullback of sections
We can also pull back sections of G. If G is an OY -module, and s ∈ Γ(V,G),
then we get a section f ∗(s) = η(s) ∈ Γ(f−1(V ), f ∗G) by the map η : G → f∗f∗G.
If we think of sections of G as local maps from X into the various stalks, the
section f ∗s simply associates to each x ∈ X the germ sy ⊗ 1 ∈ Gy ⊗Oy OX,x for
y = f(x).
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Chapter 8. Quasi-coherent sheaves
8.2.5 Tensor products
From its definition, it is straightforward to check that applying f ∗ commutes
with taking tensor products of sheaves. On the level of presheaves, we have
f ∗(G ⊗H) = f−1(G ⊗OY H)⊗f−1OY OX
= (f−1G ⊗f−1OY f−1H)⊗f−1OY OX
= (f−1G ⊗f−1OY OX)⊗OX ⊗(f−1H)⊗f−1OY OX)
= f ∗G ⊗OX f∗H.
and sheafifying, we get an isomorphism of the corresponding sheaves.
However, the pushforward f∗ rarely commutes with taking tensor products of
sheaves (we will see several examples of this later). There is however, at least, a
map f∗(F)⊗ f∗(G)→ f∗(F ⊗ G) for for OX-modules F ,G: If U ⊆ Y is an open,
and s ∈ f∗(F), t ∈ f∗(G), then s ⊗ t is an element of F ⊗ G over f−1(U), and
hence s⊗ t defines a section of f∗(F ⊗ G) over U .
8.3 Quasi-coherent sheaves
In The Oxford English Dictionary there several nuances of the word coherent
are given, but the one at top is:
That sticks or clings firmly together; esp. united by the force of cohesion. Said
of a substance, material, or mass, as well as of separate parts, atoms, etc.
So the coherence of a sheaf should mean that there are some strong relations
between the sections over different open sets, at least over sets from some
sufficiently large collection of open sets. In our context the open affine subsets
stand out as obvious candidates to form such a collection, and indeed, a quasi-
coherent sheaf on the scheme X will turn out1 to have the following coherence
property: If V ⊆ U are two affine open sets in X there is a canonical map
Γ(U,F)⊗Γ(U,OX) Γ(V,OV )→ Γ(V,F) (8.3.1)
sending s⊗ f to ρUV (s) · f , where ρUV as usual indicates the restriction maps
in OX . The salient point is that for a quasi-coherent sheaf F this map is an
isomorphism. In fact the converse holds true as well, so the map in (8.3.1) being
an isomorphism for all pairs V ⊆ U of open affine sets is equivalent to F being
quasi-coherent.
To pin down the quasi-coherent sheaves, one first establishes a collection
of model-sheaves on affine schemes. To each A-module M one associates an
1One might have used this coherence property as the definition, but because of obscurereasons we choose another definition.
129
8.4. Quasi-coherent sheaves on affine schemes
OX-module M on X = SpecA, and on a general scheme X, a quasi-coherent
module F is going to be one that locally is of the form M .
The coherent modules are the quasi-coherent modules that satisfy a certain
finiteness condition that in the noetherian case boils down to the OX-module
being finitely generated. These coherent OX-modules are the OX-modules most
frequently encountered in algebraic geometry and hence they have gotten the
shortest name, even though it is being quasi-coherent that means satisfying a
coherence property.
8.4 Quasi-coherent sheaves on affine schemes
For each A-module M we shall exhibit an OX-module M ; the construction of
which completely parallels what we did when constructing the structure sheaf
OX on X = SpecA. Letting B again be the base of the topology of distinguished
open subsets on X, we can define a B-presheaf M as follows:
(M)(D(f)) = Mf
The restriction maps are defined as before: D(g) ⊆ D(f) we have a canonical
localization map Mf →Mg. The same proof as for OX shows that this is actually
a B-sheaf, and hence gives rise to a unique sheaf M on X. By the properties of
sheafification, this is functorial in M : A homomorphism φ : M → N gives rise
to a canonical morphism of OX-modules M → N .
For any A-module homomorphism α : M → N there is an obvious way
of obtaining an OX-module homomorphism α : M → N ; indeed, the maps
α⊗ idΓ(U ;OX) are Γ(U,OX)-homomorphisms from Γ(U, M) to Γ(U, N) compatible
with the restrictions, and thus induce a map between M and N . The map α is
the associated map between the sheafifications. Clearly one has φ ψ = φ ψ,
and the “tilde-operation” is therefore a functor ModA → ModOX .
The three main properties of M are listed below. They are completely
analogous to the statements in proposition 2.36 on page 56 in Chapter 2 about
the structure sheaf OX ; and as well, the proofs are mutatis mutandis the same.
• Stalks: Let x ∈ SpecA be a point whose corresponding prime ideal is p.
The stalk Mx of M at x ∈ X is Mx = Mp = M ⊗AAp.
• Sections over distinguished open sets: If f ∈ A one has Γ(D(f), M) =
Mf = M ⊗AAf . In particular it holds true that Γ(X, M) = M .
• Sections over arbitrary open sets. For any open subset U of SpecA covered
130
Chapter 8. Quasi-coherent sheaves
by the distinguished sets D(fi) i∈I , there is an exact sequence
0 // Γ(U, M) α //∏
iMfi
ρ //∏
i,jMfifj . (8.4.1)
These statements follow formally from the construction. The first follows
since both the stalks Mx and the localizations Mp are direct limits of the same
modules over the same inductive system (indexed by the distinguished open
subsets D(f) containing x); the second follows from the way we defined M ; and
the third is just the general exact sequence expressing the space of sections of a
sheaf over an open set in terms of the space of sections over members of an open
covering.
Lemma 8.8. For any two A-modules M and N , the association φ → φ is an
isomorphism HomA(M,N) ' HomOX (M, N).
In the lingo of category theory the lemma is expressed by saying that the
tilde-operation is a fully faithful functor (trofast, full funktor), fully meaning
the map in the lemma is surjective and faithful that it is injective. Loosely
speaking, the “tilde-operation” gives an isomorphism of ModA with a subcategory
of ModOX . It is a strict subcategory; most of the OX-modules are not of the
form M , but those that are, form the category of quasi-coherent OX-modules,
that we shortly return to.
Assume that an OX-module F is given on X = SpecA, and let M denote
the global sections of F ; that is, M is the A-module M = Γ(X,F). There is
a natural map M → F of OX-modules. As usual, it suffices to tell what the
map does to sections over members of a basis for the topology, e.g., over open
distinguished sets. The sheaf F being an OX-module, multiplication by f−1
in the space of sections Γ(D(f),F) has a meaning since Γ(D(f),OX) = Af .
Hence we may send the section mf−n ∈Mf of M to the section of F over D(f)
obtained by multiplying the restriction of m to D(f) by f−n; i.e., we send m to
f−n ·m|D(f) . For later reference we state this observation as a lemma, leaving
the task of checking the details to the zealous student:
Lemma 8.9. Given a quasi-coherent sheaf F on the affine scheme X = SpecA.
Then there is a unique OX-module homomorphism
θF : Γ(X,F ) → F
inducing the identity on the spaces of global sections. Moreover, it is natural in the
sense that if α : F → G is a map of OX-module inducing the map a : Γ(X,F)→Γ(X,G) on global sections, one has θG a = α θF .
Exercise 34. Check that this is a well defined map (there is a choice involved
in the definition).
131
8.4. Quasi-coherent sheaves on affine schemes
Lemma 8.10. In the canonical identification of the distinguished open subset
D(f) with SpecAf , the OX-module M restricts to Mf .
Proof. As Γ(D(f), M) = Mf , there is a map Mf → M |D(f) that on distinguished
open subsets D(g) ⊆ D(f) induces an isomorphism between the two spaces of
sections, both being equal to the localization Mg.
The “tilde-functor” is really a no-nonsense functor having almost all properties
one can desire. It is fully faithful, as we have seen, but it also is an additive as
well as an exact functor. Moreover, it takes the tensor product M ⊗AN of two
A-modules to the tensor product M ⊗OX N of the corresponding OX-modules,
and if M is of finite presentation, the A-module of homomorphisms HomA(M,N)
to the sheaf of homomorphisms HomOX (M, N). However, this is not true if M
is not of finite presentation, the only lacking desirable property of the functor
(−).
To verify the first of these claims, assume given an exact sequence of A-
modules:
0 //M ′ //M //M ′′ // 0.
That the induced sequence of OX-modules
0 // M ′ // M // M ′′ // 0
is exact is a direct consequence of the three following facts. The stalk of a tilde-
module M at the point x with corresponding prime ideal p is Mp, localization is
an exact functor, and finally, a sequence of abelian sheaves is exact if and only if
the sequence of stalks at every point is exact.
For the tensor product, let T denote the presheaf U 7→ M(U)⊗OX(U) N(U).
We have the map on presheaves T → M ⊗A N (on U = D(f) it is the isomor-
phism Mf ⊗Af Nf ' (M ⊗AN)f ; it sends m/fa⊗ n/f b to (m⊗ n)/fa+b). After
sheafifying, we get a the desired map of sheaves
M ⊗OX N → M ⊗A N.
This is an isomorphism, since it is an isomorphism over every stalk.
When it comes to hom’s however, the situation is somehow more subtle. If
M is not of finite presentation, it is not true that HomA(M,N) localizes. There
is always a canonical map
HomA(M,N)⊗AAf → HomAf (Mf , Nf ), (8.4.2)
but without some finiteness condition (like being of finite presentation) on M ,
it is not necessarily an isomorphism. Even in the simplest case of a infinitely
132
Chapter 8. Quasi-coherent sheaves
generated free module M =⊕
i∈I Aei that map is not surjective. An element
in HomA(M,N)⊗AAf is of the form α⊗ f−n. An element in HomAf (Mf , Nf),
however, is given by its values mi⊗ f−ni on the free generators ei, and the salient
point is that the ni’s may tend to infinity, and no n working for all i’s can be
found.
Exercise 35. Show that the map (8.4.2) above is an isomorphism when M is of
finite presentation. Hint: First observe that this holds when M = A. Then use
a presentation of M to reduce to that case.
IfM is of finite presentation, the global sections of the sheaf-hom HomOX (M, N)
equals HomA(M,N), and there is a map
HomA(M,N)˜→HomOX (M, N).
In case M is of finite presentation, the maps in (8.4.2) are isomorphisms, and
the map above induces isomorphisms between the spaces of sections of the two
sides over any distinguished open subset. One concludes that the map is an
isomorphism, and one has HomA(M,N)˜'HomOX (M, N).
In short, we have established the important proposition:
Proposition 8.11. Assume that A is a ring and let X = SpecA. The functor
from the category ModA of A-modules to the category ModOX of OX-modules
given by M → M enjoys the following three properties
• It is a fully faithful additive and exact functor.
• One has a canonical isomorphism (M ⊗AN ) ' M ⊗OX N .
• If M is of finite presentation, one has a canonical isomorphism HomA(M,N)˜'HomOX (M, N).
Exercise 36. Show that the tilde-functor is additive; i.e., takes direct sums of
modules to the direct sum and sums of maps to the corresponding sums.
Example 8.12. Assume that A is an integral domain and that K is the field of
fractions of A. Show that the OX-module K is a constant sheaf in the strong
sense, that is Γ(U, K) = K for any non-empty open U ⊆ X and the restriction
maps all equal the identity idK .
8.4.1 Functoriality
Suppose we are given a morphism between the two affine schemes X and Y ; say
φ : X → Y . We let X = SpecA and Y = SpecB and we let φ# : B → A be the
map corresponding to φ.
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8.5. Quasi-coherent sheaves on general schemes
If M is an A-module, can one describe the sheaf φ∗M on Y ? The answer
is not only yes but the description is very simple. The A-module M can be
considered a B-module via the map B → A and as such is denoted by MB.
Clearly this is a functorial construction in M . In this setting one has
Proposition 8.13. One has φ∗M = MB.
Proof. There is an obvious map MB → φ∗(M) as Γ(Y, φ∗(M)) = Γ(X, M) = M .
The argument that follows shows it is an isomorphism, and it is as usual sufficient
to verify that sections over distinguished open sets of the two sides coincide. The
crucial observation is that φ−1D(f) = D(φ#f)—indeed, a prime ideal p ⊆ A
satisfies f ∈ φ#−1(p) if and only if φ#(f) ∈ p. As f acts on MB as multiplication
by φ#(f), one clearly has (MB)f = Mφ#(f) = Γ(D(φ#(f)), M).
8.5 Quasi-coherent sheaves on general schemes
Having established the sheaves on affine space that serve as local models for the
quasi-coherent sheaves, we are now ready for the general definition.
Definition 8.14. If X is a scheme and F an OX-module, one says that F is
quasi-coherent (kvasikoherent) OX-module, or quasi-coherent sheaf for short, if
there is an open affine covering Ui i∈I of X, say Ui = SpecAi, and modules
Mi over Ai such that F|Ui ' Mi.
Phrased in slightly different manner, OX-module F is quasi-coherent if the
restriction F|Ui of F to each Ui is of type tilde of an Ai-module. In particular,
the modules M on affine schemes SpecA are all quasi-coherent.
The restriction of a quasi-coherent sheaf F to any open set U ⊆ X is quasi-
coherent. Indeed, it will suffices to verify this for X an affine scheme, and by
lemma 8.10 the restriction of a sheaf of tilde-type to a distinguished open set is
of tilde-type. As any open U in an affine scheme is the union of distinguished
open subsets, it follows that F|U is quasi-coherent.
For F to be quasi-coherent, we require that F be locally of tilde-type for just
one open affine cover. However, it turns out that this will hold for any open
affine cover, or equivalently, that F|U is of tilde-type for any open affine subset
U ⊆ X. This is a much stronger than the requirement in the definition, and it
is somehow subtle to prove. As a first corollary we arrive at the a priori not
obvious conclusion that the modules of the form M are the only quasi-coherent
OX-modules on an affine scheme. We shall also see that quasi-coherent modules
enjoy the coherence property (8.3.1) on page 129 that was the point of departure
for our discussion.
134
Chapter 8. Quasi-coherent sheaves
The story starts with a lemma that establishes the coherence property (8.3.1)
in a very particular case; i.e., for sections over distinguished open sets of a quasi-
coherent OX-module on an affine scheme X = SpecA. For any distinguished
open set D(f) ⊆ X it holds that Γ(D(f),OX) = Af , and consequently there
is for any OX-module a canonical map Γ(X,F)⊗AAf → Γ(D(f),F) sending
s⊗ af−n to af−n · s|D(f). It turns out to be an isomorphism whenever F is
quasi-coherent:
Lemma 8.15. Suppose that X = SpecA is an affine scheme and that F is a
quasi-coherent OX-module. Let D(f) ⊆ X be a distinguished open set. Then one
has
• Γ(D(f),F) ' Γ(X,F)⊗AAf .
• Let s ∈ Γ(X,F) be a global section of F and assume that s|D(f) = 0, then
sufficiently big powers of f kill s, that is, for sufficiently big integers n one
has fns = 0.
• Let s ∈ Γ(D(f),F) be a section. Then for a sufficiently large n, the section
fns extends to a global section of F . That is, there exists an n and a global
section t ∈ Γ(X,F) such that t|D(f) = fns
Proof. The first statement is by the definition of localization equivalent to the
two others.
The sheaf F being quasi-coherent by hypothesis, and the affine scheme
X = SpecA being quasi-compact, there is a finite open affine covering of X by
distinguished sets D(gi) such that F|D(gi) ' Mi for some Agi-modules Mi. The
section s of F restricts to sections si of F|D(gi) over D(gi), that is, to elements
si of Mi.
Further restricting F to the intersections D(f) ∩D(gi) = D(fgi) yields the
equality F|D(fgi) = (Mi)f , and by hypothesis, the section s restricts to zero in
Γ(D(fgi),F) = (Mi)f . This means that the localization map sends si to zero
in (Mi)f . Hence si is killed by some power of f , and since there is only finitely
many gi’s, there is an n with fnsi = 0 for all i; that is, (fns)|D(gi) = 0 for all i.
By the locality axiom for sheaves, it follows that fns = 0.
Assume now a section s ∈ Γ(D(f),F) is given. We are to see that fns extends
to a global section of F for large n. Each restriction s|D(fgi) ∈ Γ(D(fgi),F) =
(Mi)f is of the form f−nsi with si ∈Mi = Γ(D(gi),F), and by the usual finiteness
argument, n can be chosen uniformly for all i. This means that si = fns and
sj = fns mach on the intersection D(f)∩D(gi)∩D(gj), and by the first part of
the lemma applied to SpecAgigj , one has fN (si − sj) = 0 on D(gi) ∩D(gj) for a
sufficiently large integers N . Hence the different fNsi’s patch together to give
the desired global section t of F .
135
8.5. Quasi-coherent sheaves on general schemes
Theorem 8.16. Let X be a scheme and F an OX-module. Then F is quasi-
coherent if and only if for all open affine subsets U ⊆ X = SpecA, the restriction
F|U is isomorphic to a an OX-module of the form M for an A-module M .
Proof. As quasi-coherence is conserved when restricting OX-modules to open sets,
we may surely assume that X itself is affine; say X = SpecA. Let M = Γ(X,F).
We saw in lemma 8.9 on page 131 that there is a natural map M → F that
on distinguished open sets sends mf−n to f−nm|U , but by the fundamental
lemma 8.15 above, this is an isomorphism between the spaces of sections over
the distinguished open sets. Hence the two sheaves are isomorphic.
Applying this to an affine scheme, yields the important fact that any quasi-
coherent sheaf F on an affine scheme X = SpecA is of the form M for an
A-module M .
Proposition 8.17. Assume that X = SpecA. The tilde-functor M 7→ M is an
equivalence of categories ModA and QCohX with the global section functor as an
inverse.
When speaking about mutually inverse functors one should be very careful;
in most cases such a statement is an abuse of language. Two functors F and
G are mutually inverses when there are natural transformations, both being an
isomorphisms, between the compositions F G and G F and the appropriate
identity functors. In the present case one really has an equality Γ(X, M) = M ,
so that Γ (−) = idModA . On the other hand, the natural transformation
Γ(X,F ) → F from lemma 8.9 on page 131 furnishes the required isomorphism
of functors.
Theorem 8.18. Let X be a scheme and let F be an OX-module on X. Then
F is quasi-coherent of and only if for any pair V ⊆ U open affine subsets, the
natural map
Γ(U,F)⊗Γ(U,OX) Γ(V,OX)→ Γ(V,F) (8.5.1)
is an isomorphism.
Proof. We may clearly assume that X is affine, say X = SpecA. Assume that
the maps (8.5.1) are isomorphisms. We may take V = D(f) and U = X and
M = Γ(X,F). Then from (8.5.1) it follows that Γ(D(f),F) = Mf which shows
that the canonical map M → F is an isomorphism over all distinguished open
subsets, and therefore an isomorphism. Hence F is quasi-coherent.
To argue for the implication the other way, assume that U = SpecB and that
F is quasi-coherent; that is, F = M for some A-module M after theorem 8.18.
The restriction of a quasi-coherent sheaf is quasi-coherent, so M |U = N for some
B-module N , and the map in (8.5.1) is just a map have map M ⊗AB → N . It
136
Chapter 8. Quasi-coherent sheaves
induces an isomorphism over all local rings Ap = Bp (where p ∈ U = SpecB)
since M |U ' N and therefore is an isomorphism of B-modules.
Example 8.19. The example of an discrete valuation ring is always useful to
consider, and we continue exploring example 8.1 above. The OX module given
by the data M,N, ρ. We claim that F is quasi-coherent if and only if ρ is an
isomorphism.
If F is quasi-coherent, then every point has a neighbourhood on which Fis the˜of some module. The only neighbourhood of the unique closed point is
X itself, and so F = M . Therefore, N = F(U) = M(0) = M ⊗R K and ρ is an
isomorphism. Conversely, of ρ : M ⊗R K → K is an isomorphism, then F is
given by F(X) = M and F( η ) = M ⊗R N , and so F ' M is quasi-coherent.
8.6 Coherent sheaves
Let A be a ring and let M be an A-module. The module M is of finite presentation
if for some integers n and m there is an exact sequence
An // Am //M // 0 .
One says that M is coherent if the following two requirements are fulfilled
• M is finitely generated.
• The kernel of every surjection An →M is finitely presented.
The second statement is equivalent to every finitely generated submodule N ⊆M
being of finite presentation. In the case A is a noetherian ring—which frequently
is the case in algebraic geometry—a module M being coherent is equivalent
to M being finitely generated. The condition comes from the theory analytic
functions where coherent non-noetherian rings are frequent.
When A is noetherian, then the three conditions of coherence, finitely gen-
eration and being of finite presentation on an A-module M coincide. The key
point is that for a finitely generated module M over a noetherian ring A, every
submodule N ⊆M is also finitely generated. Indeed, the set V of finitely gener-
ated submodules N ′ ⊆ N has a maximal element N, which has to equal N : If
not, there is an n ∈ N −N, and a finitely generated submodule N ′′ = N ′ +An
which is strictly bigger than N.
So to show that M is finitely presented, we can take a presentation Am →M → 0 and let L be the kernel, regarded as a submodule of An. Then L is
finitely generated, so there is a surjection An → L, and we get a presentation
An → Am → M → 0. Applying this argument to any finitely generated
submodule N ⊆M shows that M is coherent.
137
8.7. Functoriality
Exercise 37. Fill in the details of the proof that if A is a noetherian ring,
then the three conditions of coherence, finitely generation and being of finite
presentation on an A-module M coincide.
Definition 8.20. On a scheme X a quasi-coherent OX-module is coherent if
there is a covering of X by open affine sets Ui = SpecAi such that F|Ui = Mi
with the Mi’s being coherent Ai-modules. F is finitely presented if each Mi are
finitely presented as Ai-modules.
So if X is noetherian (or locally noetherian), the condition that M is coherent
is equivalent to the weaker condition that M is finitely generated.
Remark 8.21. The definition above is the one that appears in EGA and the
Stacks project. However, it differs slightly from the notation used in Hartshorne’s
book. In that book, one says that F is coherent if there is a covering by open
affines Ui = SpecAi such that F|Ui = Mi with the Mi’s being finitely generated
Ai-modules (that is, the condition about kernels is left out). In the case X is
(locally) noetherian, these two notions coincide, but they are not equivalent in
general.
The notion of coherent sheaf was actually introduced by Henri Cartan in
the theory of holomorphic functions of several varables around 1944. In 1946
Oka proved that OCn is coherent and this is a very difficult theorem (although it
seems like a trivial statement if the other definition of ‘coherence’ is used).
One benefit of using coherent modules rather than finitely generated ones
is that the category of coherent modules is an abelian category, even in the
non-noetherian setting. However, a problem is that coherence is very difficult to
check in general and actually for some schemes, even affine ones, the structure
sheaf OX is not coherent!
8.7 Functoriality
Proposition 8.22. Suppose that α : F → G is a map of quasi-coherent sheaves
on the scheme X. The kernel, cokernel and the image of α are all quasi-coherent.
The category QCohX is closed under extensions; that is, if
0 //M ′ //M //M ′′ // 0 (8.7.1)
is a short exact sequence of OX-modules with the two extreme sheaves M ′ and
M ′′ being quasi-coherent, the middle sheaf M is quasi-coherent as well.
Proof. If α : F → G is a map of quasi-coherent OX-modules, on any open affine
subsets U = SpecA of X it may be described as α|U = a where a : M → N
is a A-module homomorphism and M and N are A-modules with F|U = M
138
Chapter 8. Quasi-coherent sheaves
and G|U = N . Since the tilde-functor is exact, one has kerα|U = (ker a) .
Moreover, by the same reasoning, it holds true that cokerα|U = (coker a) and
imα|U = (im a) .
Suppose now that an extension like (8.7.1) is given. The leftmost sheaf M ′
being quasi-coherent lemma 13.4 entails that the induced sequence of global
sections is exact; that is, the upper horizontal sequence in the diagram below.
The three vertical maps in the diagram are the natural maps from lemma 8.9 on
page 131. Since M ′ and M ′′ both are quasi-coherent sheaves, the two flanking
vertical maps are isomorphisms, and the snake lemma implies that the middle
vertical map is an isomorphism as well. Hence M is quasi-coherent.
0 // Γ(X,M ′) //
Γ(X,M ) //
Γ(X,M ′′) //
0
0 //M ′ //M //M ′′ // 0
The category QCohX is an abelian category with tensor products and internal
hom’s.
8.7.1 Quasi-coherence of pullbacks
Recall the notion of pullback of a sheaf via a morphism f : X → Y . This
is a relatively complicated operation, since it involves taking a direct limit, a
tensor product, and finally a sheafification. The next result tells us that for G a
quasi-coherent sheaf on Y , we have a much simpler description of the pullback
f ∗G, which will allow us to do local computations more easily.
Theorem 8.23. Let f : X → Y be a morphism of schemes.
(i) If X = SpecB, Y = SpecA, and M is an A-module, then
f ∗(M) = (M ⊗A B)
(ii) If G is a quasi-coherent sheaf on Y , then f ∗G is quasi-coherent on X.
(iii) If X and Y are noetherian, then f ∗G is coherent if G is.
Proof. (ii) and (iii) follows by (i), since quasi-coherence is a local property, and
since M ⊗A B is a finitely generated B if M is a finitely generated A-module.
So let us proceed to prove the first point.
First, note that the theorem holds in the special case when M = AI is a free
module (here the index set I is allowed to be infinite) – this is simply because
139
8.7. Functoriality
f ∗OY = OX and f ∗ commutes with taking direct sums. To prove it in general,
we pick a presentation of M of the form
AJγ−→ AI →M → 0
Applying ∼ and then f ∗ we get a sequence
f ∗AJν−→ f ∗AI → f ∗M → 0
which is exact on the right, since f ∗ is right-exact. From this we get that
f ∗M = coker ν = coker ˜(γ ⊗A B)
= ((coker γ)⊗A B))∼ = (M ⊗A B)∼.
It is instructive to see the first point via the adjoint property of f∗ and f ∗. If
F = M for a B-module M and G = N for an A-module N , we have the following
isomorphisms induced by the adjoint property of Hom and ⊗:
HomX(f ∗G,F) = HomY (f ∗N , M)
= Hom((M ⊗A B)∼, N)
= HomA(M ⊗A B,N)
= HomB(M,HomA(B,N))
= HomB(M,NB)
= HomY (M, NB)
= HomY (G, f∗F)
8.7.2 Quasi-coherence of the direct image
Recall that we showed that for a map φ : SpecA → SpecN , the pushforward
φ∗F is quasi coherent, of F is quasi-coherent (since φ∗M = MB). The following
result applies to more general morphisms:
Theorem 8.24. Let φ : X → Y be a morphism of schemes and that F is a
quasi-coherent sheaf on X. If X is noetherian, then the direct image φ∗F is
quasi-coherent on Y .
Proof. We may assume that Y = SpecA. Then since X is quasi-compact, we
may cover it by open affines Ui. The intersection Ui ∩Uj is again quasi-compact,
so we can cover it with open affines Uijk.
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Chapter 8. Quasi-coherent sheaves
For any open V ⊆ Y , one has the exact sequence
0 // Γ(φ−1V,F) //∏
i Γ(Ui ∩ φ−1V,F) //∏
i,j,k Γ(Uijk ∩ φ−1V,F).
(8.7.2)
The sequence is compatible with the restriction maps induced from an inclusion
V ′ ⊆ V , hence gives rise to the following exact sequence of sheaves on X:
0 // φ∗F //∏
i φi∗F|Ui //∏
i,j,k φijk∗F|Uijk (8.7.3)
where φi = φ|Ui and φijk = φ|Uijk . Now, each of the sheaves φi∗F|Ui and φij∗F|Uijare quasi-coherent by the affine case of the theorem (proposition 8.13 on page
134). They are finite in number as the covering Ui is finite. Hence∏
i φi∗F|Ui and∏i,j φij∗F|Uij are finite products of quasi-coherent OX-modules and therefore
they are quasi-coherent. Now the φ∗F is the kernel of a homomorphism between
two quasi-coherent sheaves, and so the theorem the follows from proposition 8.22
on page 138.
The following example shows that some of the hypotheses are neccessary for
this statement to hold:
Example 8.25. Let X =∐
i∈I SpecZ be the disjoint union of countably in-
finitely many copies of SpecZ and let φ : X → SpecZ be the morphism that
equals the identity on each of the copies of SpecZ that constitute X. Then
φ∗OX is not quasi-coherent. Indeed, the global sections of φ∗OX satisfy
Γ(SpecZ, φ∗OX) = Γ(X,OX) =∏i∈I
Z.
On the other hand if p is any prime, one has
Γ(D(p), φ∗OX) = Γ(φ−1D(p),OX) =∏i∈I
Z[p−1].
It is not true that Γ(D(p), φ∗OX) = Γ(SpecZ, φ∗OX)⊗Z Z[p−1] hence φ∗OX is
not quasi-coherent. Indeed, elements in∏
i∈I Z[p−1] are sequences of the form
(zip−ni)i∈I where zi ∈ Z and ni ∈ N. Such an element lies in (
∏i∈I)⊗Z Z[p−1])
only if the ni’s form a bounded sequence, which is not the case for general
elements of shape (zip−ni)i∈I when I is infinite.
Exercise 38. Show that X =⋃i∈I SpecZ is not affine. Show that in the
category Sch of schemes X is the coproduct of the infinitely many copies of
SpecZ involved . Show that Spec∏
i Z is the coproduct of the copies of SpecZin the category Aff of affine schemes. Show that these gadgets are substantially
different.
141
8.8. The categories of coherent and quasi-coherent sheaves
8.7.3 Coherence of the direct image
For morphisms of schemes f : X → Y , it is not expected that the pushforward
of a coherent sheaf is again coherent, even for ‘nice’ morphisms f . A simple
example is the following:
Example 8.26. Let X = Spec k[t] and consider the structure morphism f :
X → Spec k (induced by k ⊆ k[t]). The sheaf OX is of course coherent, but
f∗OX is not. Indeed, this is k[t], and k[t] is clearly not finitely generated as a
k-module.
However, for finite morphisms, we have a positive result:
Lemma 8.27. Let f : X → Y be a finite morphism of schemes. If F be a
quasi-coherent sheaf on X, then f∗F is quasi-coherent on Y . If X and Y are
noetherian, f∗F is even coherent if F is.
Proof. Since f is finite, we can cover Y by open affines SpecA such that each
f−1 SpecA = SpecB is also affine, where B is a finite A-module. We then have
f∗F(SpecA) = F(SpecB). Now, since F is quasicoherent, we have F|SpecB = M
for some B-module, which we can view as an A-module via f . Hence f∗F is
quasi-coherent. If X and Y are noetherian, and F is coherent, the module M is
finitely generated as an B-module, and hence as an A-module, since B is a finite
A-module.
8.8 The categories of coherent and quasi-coherent
sheaves
Our work in the previous sections imply the following statement
Theorem 8.28. The category QCoh(X) is an abelian category and is closed
under direct limits.
By definition, being an abelian category entails that the hom-sets Hom(F ,G)
are abelian groups; finite direct sums exist; kernels and cokernels of morphisms
exist; every monomorphism F → G is the kernel of its cokernel; every epimor-
phism is the cokernel of its kernel; and every morphism can be factored into an
epimorphism followed by a monomorphism. The hard part is thus in the last
part of the statement, that any direct limit of quasi-coherent sheaves is again
quasi-coherent.
One reason why we prefer the notion of ‘coherence’ used here (rather than
the one in Harthorne) is that the category of coherent sheaves Coh(X) is also an
abelian category, even in the non-noetherian case. Note that it does not contain
142
Chapter 8. Quasi-coherent sheaves
all its direct limits, simply because an arbitrary product of coherent A-modules
is typically not coherent (not even finitely generated!)
Coherent sheaves can still be regarded as the building blocks of the category
QCoh(X). In fact, a common technique is to prove statements about quasi-
coherent sheaves by approximating them with coherent sheaves. This is justified
by the following
Theorem 8.29. Any quasi-coherent sheaf on a noetherian scheme is the direct
limit of its coherent subsheaves.
We will not go into details about this statement here, but remark that the
proof is not too difficult (see EGA I, Section 6.9).
8.9 Closed immersions and subschemes
If A is a ring and I is an ideal of A, we have seen that there is a morphism of
schemes f : Spec(A/I) → SpecA, inducing an homeomorphism of Spec(A/I)
onto the closed subset V (I). In other words, f is what’s known as a closed
immersion of topological spaces. Conversely, such a morphism f : SpecB →SpecA which is an homeomorphism onto a closed subset, the induced map
on rings φ : A → B is surjective and SpecB is isomorphic to the scheme
Spec(A/ kerφ).
The aim of this section is to globalize these observations for general schemes.
This will involve replacing ideals with (quasi-coherent) ideal sheaves.
Lemma 8.30. Let X be a scheme and let I ⊆ OX be a quasi-coherent sheaf
of ideals. Then the ringed space Z = (Supp(OX/I),OX/I) is a scheme with a
canonical morphism i : Z → X.
Indeed, to prove this, we may assume that X = SpecA is affine. In this case
I is the ∼ of some ideal I ⊆ A, and the support of this is exactly the primes p
such that (A/I)p 6= 0, or equivalently p ∈ V (I). Hence Z is the closed subset
V (I), which is homeomorphic to Spec(A/I). The sheaf of rings on Spec(A/I) is
the same as OX/I on Z and hence Z is the scheme SpecA/I. The map i is just
induced by the inclusion Z ⊆ X and the natural map OX → i∗(OX/I) is just ∼of the quotient map A→ A/I.
Definition 8.31. A closed subscheme of a schemeX is a scheme (Supp(OX/I),OX/I)
for some quasi-coherent sheaf of ideals I on X
Definition 8.32. A morphism of schemes f : Y → X is called an closed
immersion if there is an isomorphism Y ' Z onto a closed subscheme Z =
143
8.9. Closed immersions and subschemes
(Supp(OX/I),OX/I) for some quasi-coherent sheaf of ideals I on X, with a
commutative diagram
Yf //
X
Zi
>>
If φ : A→ B is a surjective homomorphism of rings, then SpecB → SpecA
is a closed immersion. Conversely, we have the following:
Lemma 8.33. Let f : Z → X be a closed immersion. If X is affine, then so is
Z and in fact Z ' SpecA/I for some ideal I of A.
Indeed, we can view Z as a closed subscheme of X = SpecA, with the locally
ringed structure being given by OX/I for some quasi-coherent sheaf of ideals
I on X. We have seen that this is the sheaf associated to an ideal I of A.
Since Z = Supp(OX/I), we have that Z = V (I) and in fact Z is isomorphic to
Spec(A/I), because the sheaves of rings are isomorphic.
Proposition 8.34. Let f : Y → X be a morphism of schemes. Then f is a
closed immersion if and only if f is a homeomorphism of the topological space of
Y onto a closed subset of X and f# : OX → f∗OY is surjective.
Proof. A closed immersion by a quasi-coherent sheaf of ideals clearly satisfies
the two conditions, so we need only prove the ‘if’ direction.
We want to associate to f a quasi-coherent ideal sheaf I on X. Since
OX → f∗OY is surjective, it is natural to take the kernel of this map. However,
we have to work a bit to prove that this is actually quasi-coherent.
Let us first show that f is affine, i.e., that for any x ∈ f(Y ), there is an
open affine neighborhood U of y with f−1(U) affine. Let y ∈ Y be the (uniquely
defined) preimage of x and choose affine open subsets V ⊆ Y,W ⊆ X with
f(V ) ⊆ W containing y and x respectively. f is a homeomorphism, so f(V )
can be written as W ′ ∩ f(X) for some W ′ open set W ′ ⊆ W containing y. Now
choose a section s of OX(W ) such that D(s) ⊆ W ′ and contains x. Then the
open set f−1(D(s)) is contained in the affine set V (it is the intersection of V
with D(f ∗s)). So f−1(D(s)) is also affine, and hence we see that f is affine.
Now we claim that f∗OY is quasi-coherent on X. By the above, X is covered
by open affines U = SpecA such that f−1(U) = SpecB is also affine. In this case,
f∗OY |U is the ∼ of B considered as an A-module, and hence is quasi-coherent.
Now, letting I be the kernel of f# (which is quasi-coherent!), we have an
exact sequence
0→ I → OX → f∗(OY )→ 0.
In particular, we see that f∗(OY ) is isomorphic to the sheaf OX/I. Moreover,
the OX/I has support on f(Y ). So the map f : Y → X is isomorphic to
144
Chapter 8. Quasi-coherent sheaves
the inclusion (f(Y ),OX/I)→ (X,OX), which proves that it is indeed a closed
immersion.
8.9.1 Morphisms to a closed subscheme
If f : Y → X is a map of schemes, it is natural to ask when it factors through a
closed immersion of X. Here we need only work up to isomorphism, so we can
assume that the closed immersion is given by (Supp OX/I,OX/I)→ (X,OX).
Proposition 8.35. Let Z be a closed subscheme of X given by sheaf of ideals I.
Suppose f : Y → X is a morphism of schemes. Then f factors through a map
g : Y → Z if and only if
(i) f(Y ) ⊆ Z.
(ii) I ⊆ ker(OX → f∗(OY )).
Proof. The condition (i) is clearly necessary. If there is a sequence Y → Z → X,
then there is a sequence of sheaves OX → OX/I → f∗(OY ), which means that
the map OX → f∗(OY ) factors through OX/I, and so also (ii) holds.
Conversely, we define the map g on topological spaces by the inclusion (i).
To define it on sheaves, we use the map OX → f∗(OY ). This annihilates I, so
we thus get a map OX/I → f∗(OY ) = g∗(OY ). This gives us the map g : Y → Z
factoring f .
Remark 8.36 (Scheme-theoretic image). Let f : Y → X be a morphism of
schemes. Then we can define the scheme-theoretic image of f as a subscheme
Z ⊆ X satisfying the universal property that if f factors through a subscheme
Z ′ ⊆ Z, then Z ⊆ Z ′. To define Z it is is tempting to use the ideal sheaf
I = ker(OX → f∗(OY )) – but this may fail to be quasi-coherent for a general
morphism f . However, one can show that there is a largest quasi-coherent sheaf
of ideals J contained in I, and we then define Z to be associated to J .
8.9.2 Induced reduced scheme structure
For an open subscheme U ⊆ X, we saw that there was a natural scheme structure
on U induced from that of X. For W ⊆ X a closed subset, there can be several
quasi-coherent ideal sheaves I corresponding to W . For instance, Spec k[x]/x
is naturally a subscheme of Spec k[x]/x2, but of course they have the same
underlying topological space. So, in contrast with the ‘open subschemes’ the
underlying topological space does not determine the scheme structure. However
there is one, which is in some sense the ‘smallest’ one:
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8.9. Closed immersions and subschemes
Proposition 8.37 (Induced reduced scheme structure). Let X be a scheme. Let
W ⊆ X be a closed subset. There exists a unique closed subscheme Z ⊆ X such
that
• Z is reduced;
• The underlying topological space of Z is W .
Proof. Let I ⊆ OX be the sheaf of ideals defined by
I(U) = f ∈ OX(U) | fx ∈ mx ⊆ OX,x for all x ∈ W ∩ U
We claim that this is quasi-coherent. On U = SpecA, we have W ∩U = V (I) for
a unique radical ideal I ⊆ R. Here we have I(U) = I: If f ∈ A is an element such
that f/1 ∈ mp ⊆ Ap then f ∈ p, and so if f ∈ I(U) then f ∈ ∩p∈V (I)p =√I = I.
Moreover, for D(g) ⊆ U , we have I(D(g)) = Ig by the same argument, and so I
and I are equal as sheaves on U , and hence I is a quasi-coherent sheaf of ideals.
Now define Z to be the closed subscheme associated with the ideal sheaf
I. Z is reduced, and has same underlying topological space as W ; this holds
because it is true on the open affines U = SpecA as above. Finally, if Z,Z ′ are
two subschemes satisfying the two points, then their ideal sheaves I, I ′ define the
same radical ideal I(U) = I ′(U) = I on U = SpecA, and so they are equal.
146
Chapter 9
Vector bundles and locally free
sheaves
The most important examples of quasi-coherent sheaves are the locally free
sheaves.
Definition 9.1. A locally free sheaf F is an OX-module such that there exists
an open cover Ui and an index set I such that F|Ui '⊕
I OUi for each i. We
call Ui a trivialization of F . If I is finite, we say that F has finite rank, and
that r = |I| is called the rank of F . A locally free sheaf of rank 1 is called an
invertible sheaf.
If F is a locally free sheaf, the stalk Fx at a point x ∈ X is a free OX,x-module,
i.e., Fx ' OrX,x where r is the rank of F . In fact the converse is also true:
Lemma 9.2. Suppose that X is locally noetherian. A coherent sheaf F having
the property that Fx ' OrX,x for some fixed r is locally free.
Proof. We can assume that X = Spec(A), where A is noetherian, and F = M ,
where M is a finitely generated A-module. Let x1, . . . , xn be generators for M
as an A module. We have Fx = Mp, for the prime ideal p ⊆ A corresponding to
x ∈ X. By assumption, Mp ' Arp is free, so let m1, . . . ,mr be a basis of Mp as
an Ap-module. We can write in Mp
xi =∑
cijmj
Clearing denominators, we see that some multiple of dixi (with di ∈ A − p)
is generated by the elements mi with coefficients in A. Let s = d1 · · · dr, and
147
consider the open set D(s) ⊆ X. s is invertible in Ms, so there is a surjective
map Ars → Ms. This is also injective, since any relation between the mi in
Ms must survive in Mp (since s 6∈ p). Hence Ms ' Ars. It then follows that
F |D(s) ' Ms ' OrX |D(s), is free on an open neighbourhood of x.
The coherence condition in this lemma is essential. Indeed, if X is a scheme,
and ix : x→ X denotes the inclusion of a point, the sheaf
F =⊕x∈X
ix∗Ox
is naturally an OX-module with Fx = Ox, but yet it is certainly not locally free.
Example 9.3. Here is an example of a locally free sheaf of rank 2. Consider
the morphism f : P1k → P1
k given by k[u] 7→ k[t] u 7→ t2 on U = Spec k[t], and
k[u−1]→ k[t−1] u−1 7→ t−2 on V = Spec k[t−1]. Then f∗OP1 is a locally free sheaf.
Indeed, on U , f∗OP1|U is the ∼ of k[t] as a k[u]-module, which equals k[u]⊕k[u]t.
We get a similar expression on V = Spec k[t−1]. It follows that f∗OP1 is locally
free of rank 2.
However, the pushforward of a locally free sheaf is not locally free in general.
For instance, if i : Y → is a closed immersion, then F = i∗OY has Fy = OY,y for
y ∈ Y , but zero stalks outside of Y .
For pullbacks however, we have the following:
Lemma 9.4. Let f : X → Y be a morphism of schemes. If G is a locally free
OY -module, then f ∗G is a locally free OX-module.
Proof. Let Ui be trivialization of F on Y , such that F|Ui '⊕
I OUi . Then, since
f ∗OY = OX , we see that f−1(Ui) is a trivialization of f ∗F .
Example 9.5. (The tangent bundle of the n-sphere) Let X = SpecA where
A = R[x0, . . . , xn]/(x20 + · · ·+x2
n−1), and consider the A-module homomorphism
f : An+1 → A given by f(ei) = xi. Then M = ker f gives rise to a quasi-coherent
sheaf F = M . Any element in the kernel corresponds to a vector of elements
a = (a0, . . . , an) ∈ An+1 so that
a0x0 + · · ·+ anxn = 0
On U = D(x0) we may divide by x0, and solve for a0, so a is uniquely determined
by the elements (a1, . . . , an). Conversely, given any such an n-tuple of elements
in A, we may define an element a ∈Mx0 using the above relation. In particular,
Mx0 ' An. A similar argument works for the other xi, showing that F is locally
free.
It is a non-trivial theorem that F 6= OnX if n 6∈ 0, 1, 3, 7.
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Chapter 9. Vector bundles and locally free sheaves
9.1 Locally free sheaves and projective mod-
ules (work in progress)
On an affine scheme X = SpecA, any quasi-coherent OX-module F is isomorphic
to M for an A-module M . In this section we investigate which A-modules give
rise to locally free sheaves. The main result is that F is locally free if and only
if M is finitely generated and projective.
Defn. projective module
Defn. Locally free module
Lemma 9.6. Let (A,m) be a local ring. If M is a finitely generated projective
A-module, then M is free.
Let M be a projective A-module, so there is a Q so that M⊕Q = F is finitely
generated, and free. Taking localizations, we then see that Mp is a projective
Ap-module for every prime p. Hence Mp is free for every p.
Lemma 9.7. Let M be a finitely generated A-module. then M is projective if
and only if M is finitely presented and locally free.
Theorem 9.8. Let M be a finitely generated A-module, and assume A is a
Noetherian ring. Then F = M is locally if and only if M is projective.
Corollary 9.9. Let X be a noetherian scheme and let F be a coherent sheaf.
Then the following are equivalent:
(i) F is locally free
(ii) Fx is a free OX,x-module for every x ∈ X.
(iii) For every open affine U = SpecA ⊆ X, we have F|U ' M , for some
finitely generated projective A-module M .
Proof. Suppose Fx is free for every x ∈ X. Fix x ∈ X and let U = SpecA be
an open subset containing x. We have F|U ' M for some finitely generated
A-module M . Then Mq is free for every q ∈ SpecA, and hence M is projective.
Hence M is locally free, and hence so is F , since x was arbitrary.
9.2 Invertible sheaves and the Picard group
An invertible sheaf on a scheme X is a locally free sheaf of rank 1. We usually
write L for such sheaves (for reasons that will become clear later). This means
that there exists a covering U = Ui and isomorphisms φi : OUi → L|Ui . We say
149
9.2. Invertible sheaves and the Picard group
that gi = (φi)Ui(1) ∈ L(Ui) is a local generator for L and that Ui is a trivialization
of L. By Lemma 9.2, a coherent OX-module L is invertible if and only if Lx is
isomorphic to OX,x for every x ∈ X.
Proposition 9.10. For L,M invertible sheaves, we have
(i) L⊗M is also an invertible sheaf. If g, h are local generators for L and M
respectively, then g ⊗ h a local generator for L⊗OX M .
(ii) Hom(L,OX) is invertible and Hom(L,OX) ⊗ L ' L. If g is a local
generator for L, then ψg defined by ψg(ag) = a is a local generator for
Hom(L,OX).
(iii) Hom(L,M) 'Hom(L,OX)⊗M
Proof. (i) We can find a common trivialization of L and M , such that X is covered
by open sets U where we have isomorphisms φ : OU → L|U and ψ : OU →M |U .
Over such a U , we have an isomorphism OU ' OU ⊗ OY ' L|U ⊗M |U given by
1 7→ 1⊗ 1 7→ φ(1)⊗ ψ(1) (all tensor products in this section are over OX . This
shows (i).
For (ii), as above the fact that Hom(L,OX) is invertible can be seen by re-
stricting to an open where L|U ' OU . The identity for the tensor product follows
from (iii) below. Finally, the fact that ψg is a local generator for Hom(L,OX)
follows from the isomorphisms
OX |U 'Hom(OX ,OX)|U 'Hom(L|U ,OX)
given by 1 7→ idOX and α 7→ α ψ−1. This means that 1 maps to ψ−1 = ψg.
(iii) By the universal property of the tensor product, we have a canonical
isomorphism
HomOX(U)(L(U),M(U))) ' HomOX(U)(HomOX(U)(L(U),OX(U))⊗OX(U) M(U))
This shows (iii) on the level of presheaves. Sheafify to get the result.
This proposition explains the term ‘invertible’. Indeed, the tensor product
acts as a sort of binary operation on the set of invertible sheaves; L ⊗M is
invertible if L and M are. Tensoring an invertible sheaf by OX does nothing,
so OX serves as the identity. Moreover, for an invertible sheaf L we will define
L−1 = Hom(L,OX); by the proposition, L−1 is again invertible, and serves as a
multiplicative inverse of L under ⊗. We can make the following definition:
Definition 9.11. Let X be a scheme. The Picard group Pic(X) is the group of
invertible sheaves L modulo isomorphism.
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Chapter 9. Vector bundles and locally free sheaves
Notice it is the set of isomorphism classes of line bundles that form a group –
not the line bundles themselves (L⊗ L−1 is isomorphic but strictly speaking not
equal to OX).
Note also that Pic(X) is also an abelian group, because L⊗M is canonically
isomorphic to M ⊗ L.
Example 9.12 (Invertible sheaves on the affine line). If F is a coherent sheaf on
A1k, then F = M for some finitely generated k[t]-module. The structure theorem
of finitely generated modules over a PID, tells us that M ' k[t]r ⊕ T where
r ≥ 0 and T is a torsion module (which is in turn a direct sum of modules of the
form k[t]/(t− a)n). From this we find
Proposition 9.13. Any invertible sheaf over A1k is trivial; Pic(A1
k) = 0.
More generally, in the correspondence between quasi-coherent sheaves on
X = SpecA and A-modules M , one can show that an OX-module is locally free
if and only if it is associated to a projective module M , that is, there exists N
such that M ⊕N is free.
9.2.1 Operations on locally free sheaves
Most constructions for vector spaces and modules globalize and carry over to
locally free sheaves. For instance, for a locally free sheaf F we can define
the tensor algebra T (F) as the graded algebra⊕
n≥0 Tn(F) where T n(F) =
F⊗n. We let Sym(F) = T (F)/I where I is the ideal generated by expressions
m⊗m′ −m′ ⊗m where m,m′ ∈ F . This inherits a Z-grading from T (F), and
we let SnF denote the n-th symmetric power of F .
Similarly, we define∧F as the OX-module T (F)/J where J is the ideal
generated by all products m ⊗m where m ∈ F . This is sometimes called the
exterior algebra. This is again graded, and a graded piece∧nF is called the
n-th wedge product. When F has rank r,∧r+1F = 0 and
∧r F is called the
determinant of F - it is a locally free sheaf of rank 1, i.e. an invertible sheaf.
Proposition 9.14 (Facts about locally free sheaves). (i) Locally free sheaves are
closed under direct sums, tensor products, symmetric products, exterior products,
duals, and pullbacks.
If F is locally free of rank r then
• T n(F) is locally free or rank rn (spanned by all elements m1 ⊗ · · · ⊗mn
where mi ∈ F)
• Sn(F) is locally free or rank(n+r−1r−1
)(spanned by all elements mn1
1 · · · ⊗mr
where mi ∈ F and ∼ ni = n)
151
9.3. Vector bundles
•∧nF is locally free of rank
(rn
)(spanned by elements mi1 ∧ · · ·min where
i1 < i2 < · · · < in.
(ii) Let 0→ F ′ → F → F ′′ → 0 be an exact sequence of locally free sheaves
of ranks n′, n, n′′ respectively. Then
∧nF ' ∧n′F ′ ⊗OX ∧n′′F ′′
9.3 Vector bundles
Locally free sheaves are in many respects the most basic sheaves on a scheme X;
they are obtained by locally gluing together copies of the structure sheaf OX in
various ways. What makes these sheaves particularly interesting is the link to
the theory of vector bundles.
A vector bundle is essentially a family of vector spaces parameterized over
a base scheme X. This means that we have a morphism π : E → X, such
that the scheme theoretic fibers π−1(x) are isomorphic to an affine space Ar.
The prototype example of a vector bundle is obtained by simply taking the
product E = X × Ar and letting π be the first projection. This is the so called
trivial bundle on X. A vector bundle is more generally a scheme together with a
morphism π : E → X which is locally isomorphic to the trivial bundle:
Definition 9.15. Let X be a scheme. A vector bundle E of rank r on X is a
scheme with a morphism π : E → X and an open cover U = Ui of X such that
for each i, there is an isomorphism φi : π−1(Ui)→ Ui × Ar making the following
diagram commutative:
π−1(Ui)φi //
π|π−1(Ui) ((
Ui × Ar
pr1
Ui
such that for each affine V = SpecA ⊆ Uij the automorphism φj φ−1i of V ×Ar
is linear; i.e., induced by an automorphism θ : A[x1, . . . , xn] → A[x1, . . . , xn]
such that θ(a) = a,∀a ∈ A, and θ(xi) =∑aijxj for aij ∈ A.
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Chapter 9. Vector bundles and locally free sheaves
So a vector bundle is obtained by gluing together trivial bundles Ui × Ar via
linear gluing maps. A convenient way of rephrasing this is in terms of transition
functions: Given a vector bundle π : E → X defined by the data (Ui, φi) we have
for each i, j an element gij ∈ GLr(Γ(Uij,OX)) such that the diagram
Uij × Ar
π−1(Uij)φj |Uij
&&
φi|Uij88
Uij × Ar
id×gij
OO
commutes.
Definition 9.16. The elements gij are the transition functions of E .
The gluing axioms for E shows that the transition functions satisfy the
following compatibility conditions (or ‘cocycle conditions’)
gik = gij gjk on Uijk (9.3.1)
gij = g−1ji on Uij
Proposition 9.17. Let X be a scheme with an open cover Ui and assume that
gij are a collection of elements of GLr(Γ(Uij,OX)) satisfying the compatibility
conditions (9.3.1). Then there is a vector bundle π : E → X, unique up to
isomorphism, whose transition functions are the gij.
Proof. The compatibility conditions ensure that the maps (id×gij) glue the affine
schemes Ui × Ar along (Ui ∩ Uj)× Ar to a scheme E . Moreover, the projection
maps Ui × Ar → Ui glue to give a morphism π : E → X. By construction, the
open set of E corresponding to Ui×Ar is identified with π−1(Ui), which gives an
isomorphism φi : π−1(Ui)→ Ui×Ar. Hence E is a vector bundle with transition
functions gij.
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9.3. Vector bundles
Example 9.18 (Facts about vector bundles). Let π : E → X be a vector bundle
or rank r associated with the transition functions gij ∈ Γ(Uij,O∗X) and let Odenote the trivial rank 1 bundle.
(i) There is a vector bundle E ∗, the dual bundle of E , with corresponding
transition functions g−1ij = gji. E ∗ = Homvb(E ,O) is also a vector bundle
(ii) If E ,E ′ are vector bundles with transition functions gij hij respectively,
one can define the tensor product bundle E⊗E ′ as the vector bundle corresponding
to the transition functions gij⊗hij . If E and E ′ have ranks r, r′, then E ⊗E ′ has
rank rr′. As a special case of this, we have the tensor power E ⊗n which is a vector
bundle of rank rn, with corresponding transition functions gij ⊗ gij ⊗ · · · ⊗ gij.
9.3.1 The sheaf of sections of a vector bundle
A section of a vector bundle π : E → X over U ⊆ X is a morphism s : U → Esuch that π s = idU . So s picks out a single vector in each fiber π−1(x) for
x ∈ X a closed point.
Proposition 9.19. Let π : E → X be a vector bundle given by the data
(Ui, φi, gij). A section s : X → E is determined uniquely by a collection of
r-tuples si ∈ OrX such that for each i, j
si|Uij = gijsj|Uij .
Proof. Given s : X → E , φi s|Ui is a section of Ui×Ar → Ui. Hence φi s|Ui =
(idU ×si) for si ∈ O(Ui)r. By construction, these satisfy the given compatibility
conditions si|Uij = gijsj|Uij . Conversely, any such section s : X → E defines a
set of such sections si = s|Ui satisfying this condition.
Given E , we can define a sheaf OX(E ) by defining
OX(E )(U) = sections s : U → E
The above proposition shows that this set is naturally a group: If s : U → E ,
t : U → E are two sections given by the data si and ti respectively, we can define
s+ t, by si + ti ∈ OrX . In fact, by multiplying si with elements of OX(U), we see
that OX(E )(U) has the structure of an OX(U)-module. This shows that OX(E )
is a sheaf of OX-modules, locally free of rank r.
9.3.2 Equivalence between vector bundles and locally free
sheaves
A vector bundle π : E → X gives rise to a locally free sheaf F of rank r. In fact,
there is a way to reverse this process; that is, to a locally free sheaf F one can
154
Chapter 9. Vector bundles and locally free sheaves
associate a scheme, denoted V(F) with a morphism π : V(F)→ X making V(F)
into a vector bundle under which F is the sheaf of sections. In particular, there
is an equivalence between vector bundles and locally free sheaves of finite rank.
The construction goes as follows. If F is locally free of rank r. Choose an
open cover Ui of X such that there is an isomorphism fi : F|Ui → OUi for
each i. If we take two fi : F|Ui → OUi and fj : F|Uj → OUi and restrict to
Uij = Ui ∩ Uj, we get two different isomorphisms gi, gj : F|Uij → OrUij
. Let
gij = gj g−1i , which gives an automorphism of Or
Uij. We can identity this with
an r × r matrix of regular functions on Uij.
Now we glue Ui × Ar and Uj × Ar along Uij × Ar by the map sending
(x, v) ∈ Uij × Ar to (x, gij(v)). The resulting scheme which we denote by
V(F) comes with a morphism π : V(F) → X (obtained by gluing all the first
projections Ui×Ar → Ui), whose fibers are linear spaces. Over each Ui ⊆ X the
maps gi also give isomorphisms from π−1(Ui) to Ui × Ar. The maps gj g−1i are
all linear - this follows from the fact that gj g−1i is a morphism of OX-modules.
Finally, it is a matter of checking that F coincides with the sheaf of sections of
π.
In particular, there is a correspondence between vector bundles and locally
free sheaves of finite rank. Under this correspondence the trivial bundle X × A1
corresponds to the structure sheaf OX .
Remark 9.20. There is also an alternate construction of V(F) using the relative
spectrum of the symmetric algebra of a sheaf. We direct the interested reader to
Exc. II.5.16 and II.5.18 in Hartshorne’s book for more details.
9.4 Extended example: The tautological bun-
dle on PnkLet k be a field, and let Pnk denote the projective n-space over k. The closed
points of Pnk parameterizes lines through the origin in An+1k . We define the
tautological sheaf, denoted by O(−1) as follows. Let L ⊆ Pn ×k An+1 denote the
variety of pairs ([l], v), where l ⊆ An+1 is a line, and v ∈ l. In terms of equations
L is defined by the 2× 2 minors of the matrix(x0 x1 . . . xny0 y1 . . . yn
)where x0, . . . , xn are homogeneous coordinates on P1 and y0, . . . , yn are affine
coordinates on An+1.
Let π : L → Pnk be the projection to the first factor. Then for a closed
point [a] = (a0 : . . . , an) ∈ Pnk , the preimage π−1([l]) is exactly the line l ⊆ An+1
155
9.4. Extended example: The tautological bundle on Pnk
corresponding to [a]. We claim that L is a rank 1 vector bundle on Pn.
On Ui = D(xi) the equations for L become
xjxiyi = yj
for each j 6= i and hence (x0 : . . . , xn, y0, . . . , yn) 7→ (x0 : . . . , xn, yi) defines an
isomorphism
φi : π−1(Ui)→ Ui × A1
In other words, yi is a local coordinate on the A1 on the right hand side. On the
intersection Ui ∩ Uj, yi is related to yj via
xixjyj = yi
This means that the transition function
Uij × A1 → Uij × A1
is given by idUij × gij where gij = xixj
. Hence π is a vector bundle. We denote
the corresponding sheaf of sections by O(−1).
What are the sections of O(−1) ? Note that a section s : Pn → L is given by
a set of n+ 1 sections si, such that for each i, j we have
si|Uij = gijsj|Uij =xixjsj|Uij (9.4.1)
The functions si : D(xi)→ L can be represented by polynomials in x0xi, . . . , xn
xi.
Note that on the right hand side of (9.4.2), the Laurent polynomial xixjsj has only
xj in the denominator, wheras si, on the left, has only xi’s. It follows that the
only way this equation in polynomials can hold is that both sides are identically
0. In particular,
Proposition 9.21. Γ(Pnk ,O(−1)) = 0.
In particular, the sheaf O(−1) is invertible, but not isomorphic to OPnk .
Consider now the vector bundle σ : L∗ → X with transition functions
gij =xjxi
Note that these transition functions are almost as before, only that we have
inverted the gij . The corresponding vector bundle is the dual bundle of π : L→ X;
the fiber σ−1(x) is identified with the dual vector space (π−1(x))∗.
In this case we find that the bundle does in fact have global sections: A
156
Chapter 9. Vector bundles and locally free sheaves
section s : Pn → L∗ is given by a set of sections si : Ui → W such that
si|Uij = gijsj|Uij =xjxisj|Uij (9.4.2)
As before, the left-hand side is a a polynomial in x0xi, . . . , xn
xi- in particular it is
a Laurent polynomial with xi in the denominators. This is ok with respect to
the right-hand side, as long as si has degree 1, that is it has a pole of order at
most one at xi = 0. Conversely, you can start with any section s0 which is linear
in x1x0, . . . , xn
x0, and use (9.4.2) to define s1, . . . , sn as Laurent polynomials. These
glue together to a global section s : Pn → L∗.
Proposition 9.22. The space of global sections of L∗ can be identified with the
vector space of linear forms in n+ 1 variables.
157
Chapter 10
Sheaves on projective schemes
Projective schemes are to affine schemes what projective varieties are to affine
varieties. The construction of the projective spectrum ProjR was similar to that
of Spec: the underlying topological space is defined via prime ideals and the
structure sheaf from localizations of R. However, there were some fundamental
differences between the two: We only consider graded rings R, and we only
consider homogeneous prime ideals that do not contain the irrelevant ideal R+.
As we saw, this reflects the construction of ProjR as a quotient space
π : SpecR− V (R+)→ ProjR
Given this, we can pull back a quasi-coherent sheaf to SpecR − V (R+) and
extend it to a sheaf on SpecR via the inclusion map. Thus, it is natural to
expect that quasi-coherent sheaves on ProjR should be in correspondence with
equivariant modules on SpecR, i.e., with graded R-modules. The irrelevant
subscheme V (R+) complicates the picture a little bit, so the classification is a
little bit more involved than the one for affines schemes. In particular, we will
see that different graded R-modules can correspond to the same quasi-coherent
sheaf on ProjR.
Another important feature of ProjR is that it comes equipped with a canonical
invertible sheaf that we will denote by O(1). This is the geometric manifestation
of the fact that R is graded. Unlike the case of affine schemes, ProjS can
typically not be recovered from the global sections of the structure sheaf. It is
the sheaf O(1), or rather, the various tensor powers O(d) = O(1)⊗d, that will
play the role of the affine coordinate ring in the affine case.
159
10.1. The graded ∼-functor
10.1 The graded ∼-functor
Let R be a graded ring and let GrModR denote the category of graded R-modules.
Just like in the case of SpecA we will define a tilde-construction to construct
sheaves on ProjR from graded S-modules, giving a functor GrModR to ModOX .
However, as we will see, this is not an equivalence of categories.
Recall that for f, g ∈ R+, such that D+(g) ⊆ D+(f), there is a canonical
localization homomorphism ρf,g : M(f) →M(g) where as before M(f) denotes the
degree 0 part of the localization 1, f, f 2, . . . −1M . It follows that we can define
a B-presheaf M by defining for each D+(f),
M(D+(f)) = M(f).
Note that M |D(f) ' (M(f)) via the isomorphism of D+(f) with SpecR(f). It
follows that this in fact gives a B-sheaf, and hence a sheaf on ProjR. Moreover,
as in the Spec case, the assignment M 7→ M is functorial. The following
proposition summarizes the properties of M :
Proposition 10.1. The contravariant functor M 7→ M has the following prop-
erties:
• ∼ is exact, commutes with direct sums and limits.
• The stalks satisfy Mp = M(p) for each p ∈ ProjR.
• If R is noetherian, and M is finitely generated, then M is coherent.
Proving these properties is straightforward, since most of them can be checked
locally on stalks. Using the isomorphisms between D+(f) and SpecR(f) we reduce
immediately to the affine case.
However, it is important to note that, unlike the affine case, the functor is not
faithful, as several modules can correspond to the same module. For instance,
take any graded R-module M such that Md = 0 for all large d. Then M(f) = 0
for all f ∈ R+, and so M = 0, even though M is not the 0-module. We will
however see shortly that it is only the modules of this form that cause the lack
of faithfulness.
The following is useful for working with the localization of M . It says
essentially that we are allowed to ‘substitute in 1’ when restricting a module to
an affine chart D+(f) ⊆ ProjR,
Lemma 10.2. Suppose that M is a graded R-module and f ∈ R homogeneous
of degree 1. Then
M(f) 'M/(f − 1)M 'M ⊗R R/(f − 1)
160
Chapter 10. Sheaves on projective schemes
Proof. Define a map M → M(f) by m 7→ m/fdegm. The submodule (f − 1)M
is annihilated by this map, so we get an induced map φ : M/(f − 1)M →M(f).
We can construct an inverse to this map as follows. Let m/fn ∈ M(f) for
m ∈ Md. Map this to the class of m in M/(f − 1)M . This is well-defined: If
m/fn = m′/fn′, then there is an N > 0 with
fN(mfn′ −m′fn) = 0.
so since f gets reduced to 1 modulo (f − 1), we find that they have the same
images in M/(f − 1)M .
Let us use this to compare M ⊗R N with M ⊗OX N . Let f ∈ R be a
homogeneous element. We have a map M(f) × N(f) → (M ⊗R N)(f) sending
m/fa × n/f b to (m⊗ n)/fa+b. As this is R(f)-bilinear, we get an induced map
M(f) ⊗R(f)N(f) → (M ⊗R N)(f). Since a map of B-sheaves induces a map of
sheaves, we get a natural map
M ⊗OX N → M ⊗R N. (10.1.1)
Proposition 10.3. Suppose R is generated in degree 1. Then the natural map
(10.1.1) is an isomorphism.
Proof. By assumption, X = ProjR is covered by open affines of the form D+(f)
where f has degree 1. For such an f , the functor M → M(f) is the same as
tensoring with R/(f − 1) ' R(f) by the previous lemma. Furthermore,
(M ⊗R R(f))⊗R(f)(N ⊗R R(f)) ' (M ⊗R N)⊗R R(f)
This isomorphism provides the inverse to the natural map M(f) ⊗R(f)N(f) →
(M ⊗R N)(f) defined above. Then, since the map (10.1.1) restricts to an isomor-
phism on all D+(f) for f ∈ R1, it is an isomorphism.
10.2 Serre’s twisting sheaf O(1)
Arguably the most interesting sheaf on X = ProjR is the so-called Serre’s
twisting sheaf, denoted by OX(1). This is a generalization of the tautological
sheaf on Pnk , and constitutes a geometric manifestation of the fact that R is a
graded ring.
For an integer n, we will define an R-module R(n) as follows: As an underlying
module R(n) is just R, but with the grading shifted by n:
R(n)d = Rd+n
161
10.2. Serre’s twisting sheaf O(1)
This is naturally a graded R-module and hence gives rise to a quasi-coherent
OX-module on X.
Definition 10.4. For an integer n, we define
OX(n) = R(n).
For a sheaf of OX modules F on X, we define the twist by n by F ⊗OX OX(n).
For an element r ∈ Rd, there is a corresponding section in Γ(X,OX(d)). This
is because we can think of an element of Γ(X,OX(d)) as a collection of elements
(rf , D+(f)) with rf ∈ (Rf )d matching on the overlaps D+(f)∩D+(g). Hence we
can define an R0-module homomorphism
Rd → Γ(X,OX(d))
by r 7→ (r/1, D+(f)). (On the overlaps D+(fg) it is clear that the two elements
(r/1, D+(f)) and (r/1, D+(g)) become equal, so this defines an actual global
section of O(d)). Abusing notation, we will also denote the section by r.
Note that if f ∈ R1, then R(n)(f) = fnR(f). Thus, on the affine D+(f), we
have OX(n)|D+(f) = fnOX |D+(f). In particular, OX(n)|D+(f) ' OD+(f). In other,
words OX(n) is a locally free sheaf of rank 1, that is, an invertible sheaf. By the
previous chapter, we know that this gives rise to a line bundle L→ ProjR.
Proposition 10.5. If R is generated in degree 1, then OX(n) is an invertible
sheaf for every n. Moreover, there is a canonical isomorphism
OX(m+ n) ' OX(m)⊗OX OX(n)
Indeed, if R is generated in degree 1, Proposition 10.3 shows that OX(m)⊗OX(n) is the sheaf associated to R(m)⊗R R(n) ' R(n+m), i.e., OX(n+m).
So this is a big difference between affine schemes and projective schemes:
ProjR comes equipped with lots of invertible sheaves.
Example 10.6. Consider again the example of projective space Pnk = ProjR
whereR = k[x0, . . . , xn]. We use the covering Ui = D(xi) ' Spec(k[x0, . . . , xn](xi)
)=
Spec(k[x0
xi, . . . , xn
xi])
.Then R (l)xi = (k[x0, . . . , xn]xi)l = xlik[x0xi, . . . , xn
xi], and so
Γ (Ui,O (l)) = xlik[x0
xi, . . . ,
xnxi
]
On the overlaps,(R (l)xi
)xjxi
= R (l)xixj =(R (l)xi
)xixj
we find that two regular
sections
xlisi
(x0
xi, . . . ,
xnxi
)162
Chapter 10. Sheaves on projective schemes
and
xljsj
(x0
xj, . . . ,
xnxj
)restrict to the same section of O (l) |Ui∩Uj if and only if
sj
(x0
xj, . . . ,
xnxj
)=
(xixj
)lsi
(x0
xi, . . . ,
xnxi
)You should compare this to the descriptions of the transition functions of the
tautological bundle in Chapter 7. In particular, we see that the invertible sheaf
O(1) coincides with the tautological sheaf defined there.
10.3 Extending sections of invertible sheaves
Let F be an OX-module and let x ∈ X be a point. We define the fiber of F at
x to be the k(x)-vector space
F(x) : Fx/mxFx ' Fx ⊗OX,x k(x)
If U is an open set containing x and s ∈ Γ(U,F), we denote by s(x) the image
of the germ sx ∈ Fx in F(x).
Definition 10.7. Let L be an invertible sheaf and f ∈ Γ(X,L). We define the
open set Xs as
Xs = x ∈ X|s(x) 6= 0.
Equivalently, Xf is the set of points where f 6∈ mxLx.
Xs is indeed an open set of X: L is locally free, so through every point there
is a neighbourhood U such that L|U ' OX |U . To show that it is open, we can
therefore assume that L = OX . If x ∈ Xs there exists a tx ∈ OX,x such that
sxtx = 1. Choose an open neighbourhood V of x such tx is represented by a
section t ∈ Γ(V,OX). By shrinking V we can assume that (s|V )t = 1 on V , and
so V ⊆ Xs is open.
Lemma 10.8. Suppose X is a noetherian scheme. Let F be a quasi-coherent
sheaf on X, and L an invertible sheaf. Suppose f ∈ Γ(X,L). Then:
1. If a section t ∈ Γ(X,F) restricts to zero on Xf , then there is an integer
N such that t⊗ fN ∈ Γ(X,F ⊗ LN) is zero (on all of X).
2. Suppose t ∈ Γ(Xf ,F). Then there is an integer N such that t⊗fN extends
to a global section of F ⊗ LN .
163
10.4. The associated graded module
Proof of the lemma. (i) Suppose t restricts to zero on Xf . We may cover X
by finitely many open affines Ui such that there is a collection of trivialization
isomorphisms φi : L|Ui → OX |Ui . Then t maps to zero in Γ(Ui,F)f because Ui is
affine (regarding f as an element of OX(Ui) via the isomorphism φi). So there is
a power of f that annihilates t on Ui. In other words, this says that t⊗ fNi = 0
on Ui. Taking N = maxNi, we find that t⊗ fN is zero on all of X, as desired.
(ii) For each i, we know that some t⊗ fNi extends to a section over all of Ui(because Ui is affine and taking sections over basic open subsets corresponds to
localization). Take M = maxNi. Then t⊗fM extends to sections ti ∈ Γ(Ui,F ⊗LM). A potential problem is that the ti might not necessarily glue on Ui ∩ Uj.However, ti = tj on Ui ∩ Uj ∩Xf since they are extending something defined on
Xf (namely t). Since Ui∩Uj∩Xf is also noetherian, the first part (i) now implies
that there are powers Mij such that (ti− tj)⊗fMij = 0 ∈ Γ(Ui∩Uj,F⊗LM+Mij ).
Taking N = M + maxMij now does the trick: the ti ⊗ fmaxMij will glue and
extend t⊗ fN .
Remark 10.9. The same proof works also for an invertible sheaf L on a quasi-
compact and separated scheme X.
10.4 The associated graded module
We have associated to a graded R-module M a sheaf M on X = ProjR. To
classify quasi-coherent sheaves on X we would, like in the case of affine schemes,
give some sort of inverse to this assignment. However, unlike the case for
X = SpecA, we can not simply take the global sections functor. Indeed, even for
F = OX on X = P1k, Γ(X,OX) = k, form which we cannot recover F . However,
the remedy is to look at the various Serre twists F(m) – in fact all of them at
once:
Definition 10.10. Let R be a graded ring and let F be an OX-module on
X = ProjR. We define the graded R-module associated to F , Γ∗(F) as
Γ∗(F) =⊕n∈Z
Γ(X,F(n)).
This has the structure of an R-module: If r ∈ Rd, we have a corresponding
section r ∈ Γ(X,OX(d)) (abusing notation, as before). So if σ ∈ Γ(X,F(n))
then we define r · σ ∈ Γ(F(n+ d)) by the tensor product r ⊗ σ, and using the
isomorphism F(n)⊗ O(d) ' F(n+ d). In particular, Γ∗(OX) is a graded ring.
164
Chapter 10. Sheaves on projective schemes
10.4.1 Sections of the structure sheaf
Proposition 10.11. Let R be a graded integral domain, finitely generated in
degree 1 by elements x1, . . . , xn, and let X = ProjR. Then
1. Γ∗(OX) =⋂ni=1R(xi) ⊆ K(R)
2. If each xi is a prime element, then R = Γ∗(OX).
Proof. Cover X by opens Ui = D+(xi). We have, since Γ(D+(xi),O(m)) '(Rxi)m, that the sheaf axiom sequence takes the following form
0→ O(m)→n⊕i=0
(Rxi)m →⊕i,j
(Rxixj)m
Taking directs sums over all m, we get
0→ Γ∗(OX)→n⊕i=0
(Rxi)→⊕i,j
(Rxixj)
So a section of Γ∗(OX) corresponds to an (n+ 1)-tuple (t0, . . . , tn) ∈⊕n
i=0(Rxi)
such that ti and tj coincide in Rxixj for each i 6= j. Now, the xi are not zero-
divisors in R, so the localization maps R → Rxi are injective. It follows that
we can view all the localizations Rxi as subrings of Rx0...xn , and then Γ∗(OX)
coincides with the intersection
n⋂i=0
Rxi ⊆ A[x±10 , . . . , x±1
n ].
In the case each xi is prime, this intersection is just R.
Corollary 10.12. Let X = PnA = ProjA[x0, . . . , xn] for a ring A. Then
Γ∗(OX) ' A[x0, . . . , xn]
In particular we can identify Γ(PnA,O(d)) with the A-module generated by homo-
geneous degree d polynomials.
When R is not a polynomial ring, it can easily happen that Γ∗(OX) is different
than R. Here is an explicit example:
Example 10.13 (A quartic rational curve). Let k be a field and let R =
k[s4, s3t, st3, t4] ⊆ k[s, t]. Note that the monomial s2t2 is missing from the
generators of R. Define the grading such that R1 = k · s4, s3t, st3, t4.
165
10.4. The associated graded module
We can also think of R as the graded ring
R = k[x0, x1, x2, x3]/〈x20x2 − x3
1, x1x23 − x3
2, x0x3 − x1x2〉.
We have a covering ProjR = U0 ∪ U1, where
U0 = Spec(R(x0)) and U1 = Spec(R(x3)).
Here R(x0) = k[ ts, t
3
s3, t
4
s4] = k[ t
s] and R(x3) = k[ s
t]. So ProjR is simply P1. We
have shown that X embeds as a smooth, rational (degree 4) curve in P3.
What is Γ(X,OX(1)? On the opens we find OX(1)(U0) = k[ts
]· s4 and
OX(1)(U1) = k[st
]· t4. So using the sheaf sequence, we get
0→ Γ(X,OX(1))→ k[st
]s4 ⊕ k
[t
s
]t4 → k
[s
t,t
s
]u4
Note that the monomial s2t2 belongs to both k[st
]s4 and k
[ts
]t4, and so defines
an element in Γ(X,OX(1)). In fact,
Γ(X,OX(1)) = ks4, s3t, s2t2, st3, t4
even though R1 = ks4, s3t, st3, t4.In this example, the graded ring Γ∗(OX) is the integral closure of R. We will
see later that this is not a coincidence.
10.4.2 The homomorphism α.
Let X = ProjR, where R be a graded ring and let M be a graded R-module.
We will define a homomorphism of graded R-modules called the saturation map
α : M → Γ∗(M)
As before, it is useful to think of elements in Γ(X, M(n)) as a collection
of elements (mf , D+(f)) for m ∈ (M(f))n and f ∈ R matching on the various
overlaps.
Proposition 10.14. When R is generated in degree 1, there is a graded R-module
homomorphism
α : M → Γ∗(M)
Indeed, we can define α by sending an element m ∈Md to the collection given
by (m/1, D+(f)), where f ranges over R1. On the overlaps D+(f) ∩D+(g) =
D+(fg) it is clear that the two elements (m/1, D+(f)) and (m/1, D+(g)) become
166
Chapter 10. Sheaves on projective schemes
equal so this defines an actual global section of M(n). We see that this is a
graded homomorphism. Moreover, it is functorial in M .
Lemma 10.15. If R is a noetherian integral domain generated in degree 1. Then
R′ = Γ∗(OX) is integral over R.
Proof. Let x1, . . . , xr be degree 1 generators of R. Let α : R→ Γ∗(OX), be the
map above, and let s ∈ R′ be a homogeneous element of non-negative degree.
We can find an n > 0, so that α(xni )s ∈ α(R) for every i. Rm is generated by
monomials in xi of degree m, so α(Rm)s ⊆ α(R) for m large (e.g., m ≥ rn. Let
R≥rn be the ideal of R generated by elements of degree ≥ rn. We have that
α(R≥rn)s ⊆ α(R≥rn). Moreover, since R is noetherian, R≥rn is finitely generated,
so applying Cayley–Hamilton, we get that s satisfies an integral equation over R.
Hence R′ is integral over R.
10.4.3 The map β
Let F be an OX-module. We will define a map of OX-modules
β : Γ∗(F)→ F
as follows. Let f ∈ R1. We will define β over D+(f). A section of Γ∗(F) is
represented on D+(f) by a fraction m/fd where m ∈ Γ(X,F(d)). If we think
of f−d as a section in O(−d)(D+(f)), then we can consider the tensor product
m⊗ f−d which is a section of F via the isomorphism F(d)⊗ O(−d) ' F . This
is compatible with the module structures, so we obtain a homomorphism of
OX-modules
β : Γ∗(F)→ F
by associating m/fd to m⊗ f−d.
Proposition 10.16. Suppose R is a graded ring, finitely generated in degree 1
over R0. Suppose F is a quasi-coherent sheaf on ProjR. Then the map
β : Γ∗(F)→ F (10.4.1)
is an isomorphism.
Proof. Since R is generated by R1 over R0, the open sets D+(f) with f ∈ R1
cover X. To show that (10.4.1) is an isomorphism, it is sufficient to prove it on
such an open.
Let f ∈ R1, and consider it as a section of Γ(X,O(1)). Then taking L = O(1)
in Lemma 10.8, (i) there says that if an element s of Γ(D+(f),F) is given, we
167
10.4. The associated graded module
can find some element t of Γ∗(F)N (for N sufficiently large) that t ⊗ f−N ∈Γ(D+(f),F) equals s. This implies that the map (10.4.1) is surjective.
For injectivity, suppose s ∈ Γ(X,F(n)) is such that s⊗ f−n = 0 on D+(f),
i.e. s/fn ∈ Γ∗(F)(f) is in the kernel of (10.4.1) on the D+(f)-sections. Then
the lemma implies that there is a power fN with s⊗ fN ∈ Γ(X,F(n+N)) = 0.
This states that s/fn = 0 in Γ∗(F)(f) by the definition of localization and so the
map is injective.
We have defined two functors
∼: GrModR → QCohX
and
Γ∗ : QCohX → GrModR
Since β : Γ∗(F)→ F is an isomorphism, it follows that ∼ is essentially surjective.
However, unlike the affine case, the functors do not give mutual inverses. This is
because, as we have seen, that ∼ is not faithful; the ∼ of any module M which
is finite over R0 is the zero sheaf.
It turns out that if we restrict to a subcategory of GrModR, the category of
saturated R-modules, we do get an equivalence. Here we say that a graded R
module is saturated if α : M → Γ∗(M) is an isomorphism. This makes sense, as
the module M = Γ∗(F) is a saturated R-module. We denote this category by
GrModsatR . We then have a diagram
GrModRM 7→M
&&α
QCohX
F7→Γ∗(F)xxGrModsatR
KK
We have seen that the map β is an isomorphism, so any quasi-coherent sheaf
F is isomorphic to the ∼ of some module (Γ∗(F)).
Putting everything together, we find
Theorem 10.17. Let R be a graded ring, finitely generated in degree 1 over R0
and let X = ProjR. Then the functors
∼: GrModsatR → QCohX
and
Γ∗ : QCohX → GrModsatR
168
Chapter 10. Sheaves on projective schemes
are mutually quasi-inverse, and establish an equivalence of categories.
So we have a full classification of quasi-coherent sheaves on X in terms of
modules on R.
10.5 Closed subschemes of ProjR
The notion of a saturated module comes from commutative algebra and has a
more conceptual definition there. To give some flavour of what it entails, we will
explain what this means for a graded ideal I ⊆ R. Here the saturation of I with
respect to an ideal B is defined as the ideal
I : B∞ :=⋃i≥0
I : Bi = r ∈ R|Bnr ∈ I for some n > 0.
We say that I is B-saturated if I = I : B∞ and in our case, saturated if it is
R+-saturated. We will here denote I : (R+)∞ by I. Note that the ideal I is
homogeneous if I is.
Example 10.18. In R = k[x0, x1], the (x0, x1)-saturation of (x20, x0x1) is the
ideal (x0). Note that both (x0) and (x20, x0x1) define the same subscheme of
P1k, but in some sense the latter ideal is inferior, since it has a component in
the irrelevant ideal (x0, x1). This example is typical; the saturation is a process
which throws away components of I supported in the irrelevant ideal.
Proposition 10.19. Let A be a ring and let R = A[x0, . . . , xn].
(i) To each subscheme Y of PnA, there is a corresponding homogeneous saturated
ideal I ⊆ R. Such that Y corresponds to the subscheme Proj(R/I) →ProjR.
(ii) Two ideals I, J defined the same subscheme if and only if they have the
same saturation.
(iii) If Y ⊆ PnA is a closed subscheme with ideal sheaf I, then Γ∗(I) is a
saturated ideal of R. In fact, the ideal Γ∗(I) is the largest ideal that defines
the subscheme Y .
In particular, there is a 1-1 correspondence between closed subschemes i : Y → PnAand saturated homogeneous ideals I ⊆ R.
Proof. (i) Let i : Y → PnA be a subscheme of PnA = ProjR and let I⊆OPnAdenote the ideal sheaf of Y . Using the fact that global sections is left-exact,
we have Γ∗(I) ⊆ Γ∗(OPnA). This is naturally an R-module, so I = Γ∗(I) is a
(homogeneous) ideal of R.
169
10.6. The Segre embedding
Any such ideal I gives rise to a closed subscheme i′ : Proj(R/I)→ PnA and
hence an ideal sheaf J satisfying I = J . By Proposition 10.16, we also have
I = IY , so the two quasi-coherent sheaves coincide and i is indeed the same as
i′. By construction I = Γ∗(I), so I is saturated.
(ii) If I, J define the same subscheme, they have the same ideal sheaf I = Jon PnA. Let s ∈ Id, then on Ui = D+(xi), the fraction rx−di defines an element
of Γ(Ui, I) = Γ(Ui, J). Since also I corresponds to J , we have rx−di = tix−di for
some ti ∈ Jd of degree d. Hence there is a power ni such that xnii (r − ti) = 0 in
R. This shows that r is in the saturation of J . By symmetry, we have I = J .
(iii) Let r ∈ R be such that xnii r ∈ Γ∗(I) (that is r ∈ Γ∗(I)). Let m = maxni.
We want to show that r ∈ Γ∗(I)d = Γ(X, I(d)). On Ui = D+(xi), we see that
x−mi ⊗ xmi r defines a section of I(d + m) ⊗ O(−n). The latter is isomorphic
to I(d) and x−mi ⊗ xmi r = r via this isomorphism. So r ∈ Γ(Ui, I(d)). Hence
r ∈ Γ(X, I(d)) ⊆ Γ∗(I), and Γ∗(I) is saturated.
10.6 The Segre embedding
Theorem 10.20. Let R,R′ be graded rings with R0 = R′0 = A. Let S =⊕n≥0(Rn ⊗R′n). Then
ProjS ' ProjR×A ProjR′
Proof. Let X = ProjR, Y = ProjR′ and Z = ProjS. Let f ∈ R be a
homogeneous element. Define
Zf =⋃g∈R′
SpecSfdeg g⊗gdeg f
We claim that there is a natural isomorphism
Sf ′⊗g′ → R(f) ⊗A R′(g)r ⊗ r′
(f ′ ⊗ g′)s7→ r
f ′s⊗ r′
g′s
where f ′ = fdeg g and g′ = gdeg f . Indeed, the inverse is given by the map
R(f) ⊗A R′(g) → Sf ′⊗g′ defined by
r
f r⊗ r′
gt7→ rtdeg g ⊗ r′r deg f
(f ′ ⊗ g′)rt
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Chapter 10. Sheaves on projective schemes
Hence we see that
Zf =⋃g∈R′
Spec(R(f) ⊗A R′(g))
On the overlaps, we have
Spec(R(f) ⊗A R′(g)) ∩ Spec(R(f) ⊗A R′(h)) = Spec(Sfdeg g+deg h⊗(gh)deg f
)= Spec
(R(f) ⊗A R′gh
)From this is is clear that
Zf = D+(f)×R Y
Moreover, for any other f ′ ∈ R we have Zf ′ = Zff ′ = X(ff ′) ×R Y . Hence
Z =⋃f∈R
Zf = X ×R Y.
Corollary 10.21. Let A be a ring and let m,n ≥ 1 be integers. Then there is a
closed immersion
σ : PmA ×A PnA → Pmn+m+n
Proof. Consider theA-algebra S =⊕
n≥0(Rn⊗R′n) above, whereR = A[x0, . . . , xm]
and R′ = A[y0, . . . , yn] are the polynomial rings. Consider the following morphism
of graded A-algebras.
A[zij]0≤i≤m,0≤j≤n → A[x0, . . . , xm]⊗ A[y0, . . . , yn]
zij 7→ xi ⊗ yj.
It is clear that S is generated as an R0 ⊗R′0-algebra by the products xi ⊗ yj , so
the map is surjective and thus we get the desired closed immersion.
Example 10.22. Let R = k[x0, x1], R′ = k[y0, y1]. Then uij = xiyi defines an
isomorphism
S =⊕n≥0
(Rn ⊗R′n)→ k[u00, u01, u10, u11]/(u00u11 − u01u10)
This recovers the usual embedding of P1k ×k P1
k as a quadric surface in P3k.
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10.7. Two important exact sequences
A smooth quadric surface
10.7 Two important exact sequences
10.7.1 Hypersurfaces
Let R = k[x0, . . . , xn] and Pnk = ProjR. Let F ∈ R denote a homogeneous
polynomial of degree d > 0. F determines a projective hypersurface X = V (F ),
which has dimension n− 1. Note that I(X) = (F ) by the Nullstellensatz. We
then have an isomorphism
R(−d)→ I(X)
given by multiplication with F . Note the shift here: The constant ‘1’ gets sent
to F should have degree d on both sides! This gives the sequence of R-modules
0→ R(−d)→ R→ R/(F )→ 0
We have R(−d) = OPnk (−d) and (R/F ) = i∗OX , where i : X → Pnk is the
inclusion, so we get the exact sequence of sheaves
0→ OPnk (−d)→ OPnk → i∗OX → 0
10.7.2 Complete intersections
Let F,G be two homogeneous polynomials without common factors of degrees
d, e respectively. Let I = (F,G) and X = V (I) ⊆ Pnk . X is called a ‘complete
intersection’ – it is the intersection of the two hypersurfaces V (F ) and V (G). To
study X we have exact sequences
0→ R(−d− e) α−→ R(−d)⊕R(−e) β−→ I → 0
and applying ∼:
0→ OPnk (−d− e)→ OPnk (−d)⊕ OPnk (−e)→ IX → 0
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Chapter 10. Sheaves on projective schemes
The maps here are defined by α(h) = (−hG, hF ) and β(h1, h2) = h1F + h2G.
These maps preserve the grading. To prove exactness, we start by noting that α
is injective (since R is an integral domain) and β is surjective (by the defintion
of I). Then if (h1, h2) ∈ ker β, we have h1F = −h2G, which by the coprimality
of F,G means that there is an element h so that h1 = −hG, h2 = hF .
These sequences are fundamental in computing the geometric invariants from
X. We will see several examples of this later.
173
Chapter 11
Differentials
So far we have defined schemes and surveyed a few of their basic properties
(e.g., how to study sheaves on them). In this chapter, we introduce Kahler
differentials, which allow us in some sense to do calculus on schemes. This in
turn will allow us to define the most important sheaves in algebraic geometry,
namely, the cotangent sheaf, the tangent sheaf, and the sheaves of n-forms.
Differentials appear prominently throughout many areas of mathematics,
e.g., multivariable analysis, manifolds and differential geometry. In algebraic
geometry they are introduced algebraically using their formal properties and are
usually refered to as Kahler differentials after the German mathematician Erich
Kahler (1906–2000).
11.1 Tangent vectors and derivations
Consider a real manifold M of dimension n. To each point p ∈M , we can attach
its tangent space of TpM , which is a real vector space of dimension n. When M
is embedded as a submanifold in some RN , we can think of tangent vectors v
as vectors in the ambient space RN that ‘stick out’ from p, and naively define
TpM as the vector space of vectors that tangentially pass through p. However,
we do not like to think of manifolds as embedded in some RN , and then giving a
precise definition of TpM becomes a little bit subtle.
When defining tangent spaces in the abstract setting, there are a few basic
properties we want. First of all, each R-vector space TpM should have dimension
n, and the elements should have some sort of geometric interpretation. Moreover,
the assignment p→ TpM should vary continuously in p, so that we can be able
175
11.1. Tangent vectors and derivations
to talk about continuous families of tangent vectors (or vector fields). The most
precise way to phrase this is to say that the collection TM = TpMp∈M should
be a vector bundle, i.e., TM has the structure of a 2n-dimensional manifold,
and there is a projection map π : TM →M sending (p, v) tp p, which is locally
diffeomorphic to U ×Rn in a neighbourhood of p. This is what is usually known
as the tangent bundle. In this setting a vector field is simply a map ν : M → TM
of π so that π ν = idM .
There are a number of ways to rigorously define the tangent bundle TM .
The simplest, and perhaps most flexible method is using derivations. Here we
think about tangent vectors as ways to differentiate functions defined at p. So we
consider the local ring Op of C∞-functions defined at p and think about elements
of Tp as linear functionals d : Op → R, satisfying the Leibniz rule
d(fg) = fd(g) + gd(f).
Note that the set of such derivatives naturally form a real vector space; we can
add them and multiply them with real numbers and the result will be a new
derivation.
In the prototype example, when M ⊆ RN is an embedded submanifold, we
can consider the directional derivative
∇vf = limh→0
f(p+ hv)− f(p)
h.
which we may think of as a derivation C∞(M,R) → R. It is not hard to see
that every derivation arises this way, and that conversely a tangent vector v is
determined by the functional ∇v.
An alternative viewpoint to tangent spaces uses the maximal ideal m ⊆ Op
of functions vanishing at the point p. We would like to consider functions f
vanishing at p, but identify two functions f1, f2 if they have the same linear
Taylor expansion around p, i.e., if f1 − f2 ∈ m2. In this way we can define the
cotangent space m/m2, and then set Tp = (m/m2)∨ (vector space dual). That is,
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Chapter 11. Differentials
we can think about elements of Tp as linear functionals
v : m/m2 → R.
It is not hard to see that these two perspectives give the same vector space.
Given a derivation d : Op → R, we must have d(m2) = 0, and so d induces via
restriction a map v : m/m2 → R. Conversely, any element of (m/m2)∨ arises this
way.
So for instance, ifM has local coordinates x1, . . . , xn near p, we let dx1, . . . , dxndenote their images in m/m2. Each xi determines dually a derivation ∂
∂xi: Op →
R, so that ∂∂x1, . . . , ∂
∂xnforms a basis Tp. Note that by definition, ∂
∂xi(dxi) = 1
and ∂∂xi
(dxj) = 0 for i 6= j.
11.2 Regular schemes and the Zariski tangent
space
Let X be a scheme and let x ∈ X be a point. We have the local ring R = OX,x
with maximal ideal m. Motivated by the case of manifolds, we will then consider
mx/m2x, which is in a natural way a vector space over the residue field k(x) =
R/mx. This is what we would like to define as the cotangent space of X at x.
Intuitively, the cotangent space is what you get by differentiating functions which
vanish at that point, but differentiating functions that vanish at x at higher
order should give zero.
Definition 11.1. The cotangent space of X at x is the quotient mx/m2x.
The Zariski tangent space TX,x to X at x is the dual vector space of mx/m2x.
That is,
TX,x = Homk(x)(mx/m2x, k(x))
and an element of TX,x is called a tangent vector; it is a linear functional
mx/m2x → k(x).
X is called regular at x if the local ring OX,x is a regular local ring, i.e.,
dim OX,x = dimk(x) m/m2.
X is regular if this condition holds for every x ∈ X.
Example 11.2. A1k and P1
k are both regular schemes. Indeed, in both cases, the
local ring at a point is isomorphic to a localization of a polynomial ring, which
is regular.
A non-normal example is X = SpecA where A = k[x, y]/(x2 − y3). In
the point m = (x, y) corresponding to the origin, the local ring is not normal:
177
11.2. Regular schemes and the Zariski tangent space
x/y satisfies a monic equation, (x/y)3 − x = 0, but it is not contained in the
localization Ap. In this example, the vector space m/m2 is 2-dimensional at
the origin – it is spanned by the residue classes of x and y (which are linearly
independent since all relations in A happen in degree at least two).
11.2.1 Zariski tangent space and the ring of dual num-
bers
Let k be a field. The ring R = k[ε]/(ε2) is called the ring of dual numbers over k.
The spectrum of R is a very simple scheme: Its underlying topological space is
a single point. However, the non-reduced structure on SpecR shows that it is
more interesting than Spec k. We picture it as follows:
Proposition 11.3. Let X be a scheme over k. To give a k-morphism Spec(k[ε]/ε2)→X is equivalent to giving a point x ∈ X which is rational over k (meaning that
k(x) = k), and an element of TX,x.
Proof. Let Y = Spec(k[ε]/(ε2). The map Y → X is a map of schemes over k,
and so f# induces a map of k-algebras:
OX,x// k[ε]/ε2//
k
OO ::
This is a local homomorphism, so we can mod out by the ideals mx and (ε) to
get
k(x) // k
k
OO ==
where the diagonal arrow is an isomorphism; hence k(x) = k and x is a k-
rational point. Moreover, restricting f#x to mx ⊆ OX,x we get a k-algebra map
178
Chapter 11. Differentials
α : mx → (ε) ' k. This has to vanish on m2x, and hence induces an element in
TX,x.
Conversely, suppose that x ∈ X is a k-rational point and let α : m/m2 → k
be a k-linear map (m is the maximal ideal in OX,x). We want to construct
(f, f#) : (Y,OY ) → (X,OX). Define f((ε)) = x; this takes care of the map on
topological spaces. Now we define f#. Set for U ⊆ X not containing x, we
define f# : OX(U) → OY (∅) = 0 to be the zero map. For U containing x, we
argue as follows. The point is that each OX(U) has a structure of a k-algebra
(ie there is a map k → OX(U)), and so we can write any element a ∈ OX,x as
a = a(x) + (a− a(x)) where a(x) is the image of a via
OX,x → OX,x/mx → k → OX,x.
Note that the germ of a − a(x) in OX,x lies in mx. Define f# : OX(U) →OY (f−1(U)) = k[ε]/ε2 by sending a to the class of
a(x) + α((a− a(x))|x + m2)ε
One can check that this is a ring homomorphism, using k-linearity and the fact
that the square of (a− a(x))|x lies in m2. Everything here is compatible with
the inclusions V ⊆ U , and so sets up a map of sheaves f#.
11.3 Derivations and Kahler differentials
So far we have considered a scheme X and a point x ∈ X, and the cotangent
space m/m2 at x. We would like to generalize this construction, and instead of
fixing x, consider all points at once. That is, we would like to form a sheaf on X
with stalks m/m2 at each point. This is what will be the cotangent sheaf on X.
We will work over a base ring A. B will be an A-algebra and M an B-
module. The picture to have in mind is that A = k, where k is a field, and
X = SpecB → Spec k, where B is a k-algebra.
Definition 11.4. An A-derivation (from B with values in M) is an A-linear
map d : B →M satisfying the product rule:
d(b1b2) = b1 d(b2) + b2 d(b1).
Given that the product rule holds, it is easy to see that d is A-linear if and
only if d vanishes on all elements of the form a · 1 where a ∈ A. We can therefore
think of the elements in B of the form a · 1 as the ‘constants’ – however a
derivation can also be zero on other elements on B.
Example 11.5. A simple example is the classical derivation of polynomial ring
179
11.3. Derivations and Kahler differentials
B = k[x] over k to itself given by P (x) 7→ P ′(x). As usual, P (x) must be defined
formally by P (x) =∑
i iaixi−1 if P (x) =
∑i aix
i.
More generally, the differential operators ∂∂x1. . . . , ∂
∂xn, as well as their linear
combinations, define derivations on the polynomial ring k[x1, . . . , xn].
The set of derivations d : B →M is usually denoted with DerA(B,M). This
inherits a B-module structure from M and it is as such naturally a submodule of
HomA(B,M). This gives rise to a covariant functor DerA(B,−) from mod B to
itself. More precisely, if φ : M →M ′ is a B-module homomorphism, we can map
a derivation d ∈ DerA(B,M) to φ d : B →M ′, which is in turn an A-derivation
of B with values in M ′.
The derivation DerA(B,M) is also functorial in the base ringA and the algebra
B; in both of these cases, it is covariant. If A → A′ is a ring homomorphism,
any A′-derivation B → M is in turn an A-derivation. We therefore obtain an
inclusion DerA′(B,M) ⊆ DerA(B,M).
11.3.1 The module of Kahler differentials
The covariant functor Derk(A,−) on the category of A-modules is representable.
In down-to-earth terms, this just means that there exist a B-module ΩB/A and
an isomorphism of functors
DerA(B,−) ' HomB(ΩB/A,−).
By formal properties of functors and representability, this condition is equivalent
by having a universal derivation dB : B → ΩB/A, having the following property:
For any A-derivation d′ : B →M there exists a unique B-module homomorphism
α : ΩB/A →M such that d′ = α dB. In terms of diagrams, we have
BdB //
d′
ΩB/A
α
M
To see directly why such a module exists, we as construct it explicitly as follows.
ΩB/A is the A-module defined as the quotient of the free module⊕
b∈B Bdb by
the submodule generated by the expressions d(b+b′)−db−db′, d(bb′)−bdb′−b′db,da, for b, b′ ∈ B, a ∈ A. Then the universal derivation dB : B → ΩB/A is given
by b 7→ db.
It is not hard to see that this module indeed satisfies the property above:
Given an A-derivation D : B →M , we define the homomorphism α : ΩB/A →M
180
Chapter 11. Differentials
by α(db) = D(b) (which is well-defined precisely because D is a derivation!).
Conversely, given α : ΩB/A → M , we get a derivation D : B → M defined by
D(b) = φ(db).
Definition 11.6. The elements of the module ΩB/A are called the Kahler
differentials, or simply differentials of B over A.
Example 11.7. Let B = k[t] be the polynomial ring over k in the variable t.
Then ΩB/k is a free module over k generated by dBt, i.e., ΩB/k = B · dBt. This
follows since an element m in an arbitrary A-module M uniquely determines
a derivation A→ M with dt = m, and so there is an isomorphism of functors
Derk(B,−) = Hom(B,−). By the Yoneda lemma, it follows that ΩB/k = BdBt.
Alternatively, we can see this from the explicit construction of Ω1A/k, because
for any f ∈ k[t], we have df = f ′(t)dt.
More generally we have:
Proposition 11.8. Let A be any ring and let B = A[x1, . . . , xn]. Then ΩB/A is
the free B-module generated by dx1, . . . , dxn.
The universal derivation d : B → ΩB/A is the map f 7→∑ ∂f
∂xidxi. Since B is
generated as an A-algebra by the xi, ΩB/A is generated as a B-module by the
dxi and there is a surjection Br → ΩB/A taking the ith basis vector to dxi.
On the other hand, the partial derivative ∂/∂xi is an A-linear derivation from
B to B, and thus induces a B-module map ∂i : ΩB/A → B carrying dxi to 1 and
all the other xj to 0. Putting these maps together we get the inverse map.
Example 11.9. If B = S−1A is a localization of A, then ΩB|A = 0. Indeed, take
b ∈ B, and choose s ∈ S so that sb ∈ A, and hence sdb = d(sb) = 0 This implies
that db = 0, since s is invertible in B.
Example 11.10. If B = A/I, then ΩB|A = 0. More generally, if φ : A→ B is
surjective, then ΩB|A. This follows because if b = φ(a), then db = a · d(1) = 0 in
ΩB|A.
Example 11.11. Suppose k is a field of characteristic p > 0. Let B = k[x] and
let A = k[xp]. Then ΩB/A is the free B-module of rank 1 generated by dx.
Example 11.12. Let B = Q[√
2] and A = Q. Then 0 = d(2) = d(√
2√
2) =
2√
2d(√
2) so d(√
2) = 0. Thus ΩQ[√
2]/Q = 0.
11.4 Properties of Kahler differentials
There are a few useful ways for computing modules of differentials when changing
rings. The proofs of the following propositions are not difficult, and involve only
basic commutative algebra.
181
11.4. Properties of Kahler differentials
11.4.1 Localization
The aim of this section is to show that Kahler differentials behave very well
under localization.
Proposition 11.13. Let S ⊆ A be a multiplicative subset mapping into the
group of units in B. Then
ΩB|S−1A = ΩB|A
Proof. We have a natural mapA→ S−1A, so we get DerS−1(B,M) ⊆ DerA(B,M).
Conversely, if d : B → B is an A-derivation, then for each s ∈ S, we have
0 = d(1) = sd(1/s) + 1/sd(s) = sd(1/s)
As s is invertible in B, it follows that d(1/s) = 0, and hence by the product rule,
d is also an S−1A-derivation of B.
Proposition 11.14. Suppose S is a multiplicative system in B and let i : B →S−1B be the localization map. Then
di : ΩB/A⊗A S−1B → ΩS−1B/A
is an isomorphism
Proof. Using the usual formula of differentiating a fraction, we can define an
A-derivation d : S−1B → ΩB/A⊗A S−1A. This will give rise to an S−1B-module
homomorphism ΩS−1B/A → ΩB/A⊗A S−1B which will turn out to give the inverse
to di.
Let us first define the derivation d. We define
d(b/s) = (sdB(b)− adB(s))s−2
We need to check that this does not depend on the representative a/s. Using
the product rule for dB, we have
d(b1b2/st) = b1s−1d(b2/t) + b2t
−1d(b1/s), (11.4.1)
Suppose now that tb1 = sb2. Using this, the product rule, the identity tdBb1 +
b1dBt = sdBb2 + b2dBs, and solving for tdBb1 and substituting into (11.4.1), we
find
t2(sdBb1 − b1dBs) = ts(sdBb2 + b2dBs− b1dBt)− t2b1dBs = s2(tdBb2 − b2dBt)
where the last equality follows by tb1 = sb2. This shows that d(a/s) = d(b/t).
Finally, if b1s−1 = b2t
−1, there is an element u ∈ S such that utb1 = usb2. We
182
Chapter 11. Differentials
then find
d(b2/t)u−1 + b2t
−1d(1/u) = d(b2/tu) = d(b1/su) = d(b1/s)u−1 + b1s
−1d(1/u)
and since b1s−1 = b2t
−1, it follows that d(b1/s) = d(b2/t), and d is well defined.
Moreover, it is an A-derivation by (11.4.1).
11.4.2 Base change
Proposition 11.15. Let A,B be as above and let A′ be an A-algebra. Define
B′ = B ⊗A A′. Then there is a canonical isomorphism
ΩB′|A′ ' ΩB|A ⊗B B′
Proof. The universal derivation d : B → ΩB|A induces an A′-derivation
d′ = d⊗ idA′ : B′ → ΩB|A ⊗A A′ = ΩB|A ⊗B B′
One can check that this is the required universal derivation of ΩB′|A′ which gives
the claim.
11.4.3 Exact sequences
Let A be a ring and let ρ : B → C be a homomorphism of A-algebras. Then
there are universal homomorphisms of (C-modules)
α : ΩB|A ⊗B C → ΩC|A
defined by α(db⊗ c) = cdρ(b). And
β : ΩC|A → ΩC|B
Proposition 11.16. Then there is an exact sequence of C-modules
ΩB/A ⊗B Cα−→ ΩC/A
β−→ ΩC/B → 0.
Proof. It is sufficient to show that for any C-module N , the dual sequence
0→ HomC(ΩC|B, N)→ HomC(ΩC|A, N)→ HomC(ΩB/A ⊗B C,N)
is exact. Note that HomC(ΩB/A ⊗B C,N) = HomB(ΩB/A, N), so the sequence
can be written as
0→ DerB(C,N)→ DerA(C,N)→ DerA(B,N)
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11.4. Properties of Kahler differentials
The map on the right hand side is the one induced by composition A→ B → C.
The fact that it is exact on the left is clear: If d : C → N maps to the zero
derivation, it is the zero homomorphism and hence 0 in DerB(C,N). To check
exactness in the middle, note that if d : C → N is mapped to the zero derivation
in DerA(B,N), then d : C → N is in fact B-linear: This follows by the Leibniz
rule d(bc) = b ·dc+db · c = b ·dc+ 0 = bdc. Hence d is in fact a B-derivation.
Applying this to C = S−1B, we obtain a new proof of Proposition 11.14 that
S−1ΩB/A ' ΩS−1B|A
for the localization in a multiplicative subset S ⊆ B.
Proposition 11.17 (Conormal sequence). Suppose that B is an A-algebra, and
C = B/I for some ideal I ⊆ B. Then there is an exact sequence of C-modules
I/I2 d−→ ΩB/A ⊗B Cα−→ ΩC/A → 0
Here the first map sends [f ] 7→ df ⊗ 1 and the second sends c⊗ db 7→ cdb.
Proof. Note that I/I2 = I ⊗B C. As in the previous proposition it suffices to
check that for each C-module N , the dual sequence
0→ DerA(C,N)→ DerA(B,N)→ HomC(I/I2, N) = Hom(I,N)
is exact. The rightmost map associates to a derivation d′ : B → N its restriction
to I. Note that this is indeed a homorphism of C-modules since IN = 0.
That this sequence is exact is a matter of checking, the case of exactness at
the left being easy. To check exactness in the middle, we suppose that d : B → N
restricts to 0 on I. Then d factors via a homomorphism d′ : B/I → N , which is
C-linear (since d is B-linear). This is the required element in DerA(C,N).
Corollary 11.18. Let A be a ring and let B be a finitely generated A-algebra
(or a localization of such). Then ΩB|A is finitely generated over B.
Proof. Write B = A[x1, . . . , xn]/I for some variables x1, . . . , xn and apply the
above propositions.
11.4.4 The diagonal and ΩB/A
Suppose now that B is an A-algebra. There is an exact sequence of A-modules
0→ I → B ⊗A B∆−→ B → 0
184
Chapter 11. Differentials
where ∆ is the diagonal map b1⊗b2 7→ b1b2 and I is the kernel of ∆. We can make
I/I2 into a B-module by letting B act on the first factor by b(x⊗ y) = (bx)⊗ y)
and then define a map d : B → I/I2 by db = 1⊗ b− b⊗ 1.
Proposition 11.19. The module I/I2 along with the map d is the module of
differentials for B over A. E
Proof. Let us check that the map d is indeed a derivation: Suppose b1, b2 ∈ B,
then
d(b1b2) = 1⊗ b1b2 − b1b2 ⊗ 1.
One the other hand,
b1db2 + b2db1 = b1(1⊗ b2 − b2 ⊗ 1) + b2(1⊗ b1 − b1 ⊗ 1)
= b1 ⊗ b2 − b1b2 ⊗ 1 + b2 ⊗ b1 − b1b2 ⊗ 1
The difference between these two elements is
1⊗ b1b2 − b1b2 ⊗ 1− b1 ⊗ b2 + b1b2 ⊗ 1− b2 ⊗ b1 + b1b2 ⊗ 1
= (1⊗ b1 − b1 ⊗ 1)(1⊗ b2 − b2 ⊗ 1) ∈ I2
Now one has to check that (I/I2, d) satisfies the required universal property. We
leave this as a task for the reader (See [Matsumura p.192] for details).
11.5 Some explicit computations
We can use the previous exact sequences to do explicit computations with
ΩB/A. If B is a finitely generated A-algebra, say B = A[x1, . . . , xr]/I where
I = (f1, . . . , fr). Then we have
B⊗A ΩA[x1,...,xr]/A 'n⊕i=1
Bdxi
Note that I/I2 is generated by the classes of the f1, . . . , fr modulo I2. This gives
a surjection Br → I/I2. Now, by the conormal sequence (Proposition 11.17), we
have
ΩB/A = coker(d : I/I2 →n⊕i=1
Bdxi)
= coker(d : Bn →n⊕i=1
Bdxi)
185
11.5. Some explicit computations
Explicitly, the map Bn →⊕n
i=1Bdxi is given by the n × r Jacobian matrix
J =(∂fj∂xi
).
Theorem 11.20. Let A be a ring, and let B = A[x1, . . . , xn]/(f1, . . . , fr). Then
ΩB|A =
⊕iBdxi∑
j B(∑
i∂fj∂xidxi
)together with the A-derivation dB|Af =
∑i∂f∂xidxi.
Example 11.21. Let k be a field and let X = SpecR where R = k[x, y]/(f).
Let us compute the module of differentials. Let fx, fy denote the (images of the)
partial derivatives of f in R. Then
ΩR|A =Rdx⊕Rdy
(fxdx+ fydy)
If X is non-singular, e.g., V (f, fx, fy) = ∅, then ΩR|A is locally free of rank 1.
The bases are given by
dy
fxon D(fx); and − dx
fyon D(fy).
(Note that on the overlap D(fx) ∩D(fy) we have fxdx+ fydy = 0 in ΩR|A).
Example 11.22. The curve X given by y2 = x2(x+1) in A2k is the so-called nodal
cubic. It has a singular point at the origin (0, 0). Let B = k[x, y]/(y2− x2(x+ 1).
Then
ΩB/k =Bdx⊕Bdy
(2ydy − (3x2 + 2x)dx)
In this case ΩX|k has rank 1 for every point (x, y) 6= (0, 0). At the origin, the
relation 2ydy − (3x2 + 2x)dx is identically zero, so ΩX|k has rank two there.
We can also view B as an algebra over A = k[x]. In that case, we get
ΩB/A =Bdy
(2ydy)
Example 11.23. Let B = k[x, y]/(x2 + y2). If k has characteristic 6= 2, then
ΩB|k = (Bdx+Bdy) /(xdx+ ydy)
If k has characteristic 2, then ΩB|k is the free B module Bdx+Bdy.
186
Chapter 11. Differentials
11.6 The sheaf of differentials
For us, the primary motivation for studying ΩB/A is that it gives us an intrisic
module ΩB/A associated to a ring homomorphism A→ B. By taking ∼, we thus
get an intrinsic sheaf on X = SpecB associated to the map of affine schemes
SpecB → SpecA. We would like to globalize this construction to an arbitrary
morphism of schemes f : X → S. This will lead us to form the sheaf of relative
differentials ΩX|S which will be a quasicoherent OX-module.
This sheaf is locally built out of the various ΩB|A on local affine charts. These
not arbitrary modules that just happen to glue together to a sheaf; each of them
come with a universal property of classifying derivations d : B →M . For this
reason, we would like to say that the ΩX|Y should satisfy a similar universal
property. We make the following definition:
Definition 11.24. Let F be an OX module. A morphism d : OX → F of
sheaves is an S-derivation if for all open affine subsets V ⊆ S and U ⊆ X
with f(U) ⊆ V , the map dU |V is a OS(V )-derivation of OX(U). The set of all
S-derivations is denoted by DerS(OX , F ).
Definition 11.25. The sheaf of relative differentials is a pair (ΩX|S, dX|S) of
an OX-module ΩX|S and a S-derivation dX|S : OX → ΩX|S that satisfies the
following universal property: For any OX-module F , and each S-derivation
d : OX → F there exists a unique OX-linear map φ : ΩX|S → F such that
d = φ dX|S.
When S = SpecA, we sometimes write ΩX|A for ΩX|S.
In other words, ΩX|S is a sheaf that represents the sets of S-derivations, in
the sense that
HomOX (ΩX|S,F) = DerS(OX , F ).
As usual, the universal property gives that this sheaf is unique up isomorphism,
if it exists.
Theorem 11.26. Let f : X → S be a morphism of schemes. Then there is a
sheaf of relative differentials ΩX|S, which is a quasicoherent sheaf on X.
Moreover, ΩX|S has the property that for each open affine open V = SpecA,
and open affine U = SpecB ⊆ f−1(V ),
ΩX|S|U ' ΩB|A
Also for each x ∈ X,
(ΩX|S)x ' ΩOX,x|OS,f(x)
187
11.6. The sheaf of differentials
Proof. Fix an open subset V = SpecA of S, and let U = SpecB be an affine
open subset in X so that f(U) ⊆ V . For these two, we define
ΩU |V = ΩB|A
on U . We first show that the ΩU |V glue to a Of−1V -module Ωf−1(V )|V . This
comes down to showing that if U ′ = SpecB′ is a distinguished open affine subset
of U , then
ΩU |V |U ′ ' ΩU ′|V .
But this follows from Proposition 11.14, as B′ is a localization of B.
Then we show that the sheaves Ωf−1V |V for all affine opens V ⊆ S glue to
a OX-module ΩX|S. This amounts to showing that for each distinguished open
V ′ = SpecA′ ⊆ V , and all open U = SpecB of f−1(V ′), we have
ΩU |V = ΩU |V ′
But this follows from Proposition 11.13, as A′ is a localization of A in a single
element (which maps to an invertible element in B).
This means that we get an OX-module ΩX|S. Let us check that it satisfies
the above universal property. So we need to define the universal derivation
dX|S : OX → ΩX|S.
Let V = SpecA ⊆ S and U = SpecB ⊆ X be an affine open subset such
that f(U) ⊆ V . Define dX|S(U) = dB|A. By the gluing construction above, this
map does not depend on the chosen affine open V , and it can be checked that
the rule is compatible with restriction maps. Hence this gives a morphism of
sheaves dX|S : OX → ΩX|S, which by construction is a S-derivation.
To check that this is universal, we again work locally. Let d : OX → Fbe a S-derivation, where F is an OX-module. Let U = SpecA ⊆ S and
V = SpecB ⊆ X so that f(U) ⊆ V . By the universal property of ΩB|A,
we get an A-derivation d(V ) : B → F(V ), and hence a unique B-linear map
φ(V ) : ΩX|S(V ) = ΩB|A → F such that d(V ) = φ(V )dX|V (V ). One has to check
that these maps are compatible with restriction maps (use the universal property
of ΩB/A), but after that, we obtain a unique OX-linear map φ : ΩX|S → F so
that d = φ dX|S.
Note that the sheaf ΩX|S is always quasi-coherent (it is by definition locally
of the form M for some module). Moreover, when X is of finite type over a field,
ΩB/A is finitely generated, and so ΩX|k is even coherent.
Example 11.27. Let A be a ring and let X = AnS = SpecS[x1, . . . , xn] be affine
n-space over S = SpecA. Then ΩX/S ' OnX is the free OX-module generated by
dx1, . . . , dxn.
188
Chapter 11. Differentials
Remark 11.28. If X is a separated scheme over S then one could also define
ΩX/S as follows. Let ∆ : X → X ×S X be the diagonal morphism and let I∆
be the ideal sheaf of the image of ∆. Then ΩX/S = ∆∗(I∆/I2∆). This does in
fact give the sheaf above, since these two definitions coincide when X and S are
both affine (Section 11.4.4). This definition gives a quick way of obtaining the
sheaf ΩX/S, but it is not very well suited for computations.
The properties of ΩB/A translate into the following results for ΩX|Y :
Proposition 11.29 (Base change). Let f : X → S be a morphism of schemes
and let S ′ be a S-scheme. Let X ′ = X×S S ′ and let p : X ′ → X be the projection.
Then
ΩX′|S′ ' p∗ΩX|S
Proposition 11.30. Let X, Y , and Z be schemes along with maps Xf−→ Y
g−→ Z.
Then there is an exact sequence of OX-modules
f ∗(ΩY/Z)→ ΩX/Z → ΩX/Y → 0.
Proposition 11.31 (Conormal bundle sequence). Let Y be a closed subscheme
of a scheme X over S. Let IY be the ideal sheaf of Y on X. Then there is an
exact sequence of sheaves of OX-modules
IY /I2Y → ΩX/S ⊗OY → ΩY/S → 0.
11.7 The sheaf of differentials of PnAWe have seen that the cotangent sheaf of An is trivial, i.e., isomorphic to On
An . In
this sections we give a concrete description of the cotangent bundle of projective
space, suitable for explicit computations.
Theorem 11.32. Let X = PnA, then there is an exact sequence
0→ ΩX/A → OX(−1)n+1 → OX → 0.
Proof. Let R = A[x0, . . . , xn] so that X = ProjR. Let T be the R-module
R(−1)n+1 with basis elements e0, . . . , en. Define the map α : T → R by sending
ei 7→ xi and let M = kerα. We have an exact sequence
0→M → Tα−→ R
Taking ∼ this gives
0→ M → OX(−1)n+1 → OX → 0
189
11.8. Relation with the Zariski tangent space
which is surjective on the right, because the map α is surjective in degrees 0
(the cokernel C of α is a sheaf with Cd = 0 for large d, and so it has C = 0). We
claim that M = ΩX/A.
Let Ui = D(xi) = SpecR(xi) denote the standard open affines. We have
ΩX/A|Ui = ΩUi/A = ΩR(xi)/A
and ΩR(xi)/A is generated as an R(xi)-module by the expressions d
(xjxi
)for i 6= j.
Now define the maps
φi : ΩX/Y |Ui → M |Uid(xjxi
)7→ 1
x2i(xiej − xjei)
Note that xiej − xjei ∈ kerα. The factor 1x2i
is included to obtain an element of
degree zero. This is an isomorphism on Ui, so we are done if we can make sure
that the φi glue. Now, on the overlaps Ui ∩ Uj, we have xkxi
= xkxj
xjxi
, which gives
d
(xkxi
)=xkxjd
(xjxi
)+xjxi
(xkxj
)or, in other words,
xjxid
(xkxj
)= d
(xkxi
)− xkxjd
(xjxi
)Applying φi to both sides, gives the same answer ( 1
xixj(xjek−xkej)), so the maps
φi are compatible.
Since ΩPnA injects into OPnA(−1)n+1 (which has no global sections), we get:
Corollary 11.33. Γ(PnA,ΩPnA) = 0
11.8 Relation with the Zariski tangent space
Let X be a scheme over an algebraically closed field k. Recall that for a point
x ∈ X, the Zariski tangent space is defined as m/m2 where m is the maximal
ideal in the local ring B = OX,x. In this section we relate this to the module of
differentials ΩR|k. Note that since k is algebraically closed, the Nullstellensatz
gives that B/m = k.
Proposition 11.34. Suppose (B,m) is a local ring with residue field k = B/m
190
Chapter 11. Differentials
and there is an injection k → B. Then there is an exact sequence
m/m2 d−→ ΩB/k ⊗B k → 0
and d is actually an isomorphism.
Proof. The exact sequence is obtained from the conormal sequence by letting
A = C = k.
The above map sends x ∈ m to dx. We construct an inverse ψ : ΩB|k ⊗B k →m/m2 as follows. Constructing such a map is equivalent to constructing a map
of B-modules ΩB|k → m/m2, or equivalently, a derivation ∂ : B → m/m2. We
define ∂ by
∂(a+ x) = x
for a ∈ k and x ∈ m. (This is well-defined since k + m = B). To complete the
proof we need only prove that ∂ is a derivation:
∂((a+ x)(a′ + x′)) = ∂(aa+ (ax′ + a′x) + x′x)
= ∂(aa) + ∂(ax′ + a′x) + ∂(x′x)
= ax′ + a′x
We get the same answer when we expand
(a′ + x′)∂(a+ x) + (a+ x)∂(a′ + x′)
(since xx′ ∈ m2). Hence the map is a derivation, and we get a map ψ. This
is indeed an inverse to the map d, since via the identification DerA(B,M) =
HomB(ΩB|A, A), ψ sends dx to x
Corollary 11.35. Let (B,m) be as above. Then B is a regular local ring if and
only if
dimBm = dimk ΩB|k ⊗B k
Definition 11.36. Let X be a scheme over a field k and let x ∈ X be a point.
We say that X is smooth at x if ΩX|k is locally free at x. X is called smooth if it
is smooth at every point.
Theorem 11.37. Let X be a variety over an (algebraically closed) field k of
characteristic 0 and let x ∈ X. Then the following are equivalent:
(i) X is smooth at x
(ii) (ΩX|k)x is free of rank dimxX
(iii) X is non-singular at x.
191
11.8. Relation with the Zariski tangent space
Proof. (i)⇐⇒ (ii) is clear.
(ii) =⇒ (iii). If ΩB|k is free of rank n = dimR, then by the above proposition,
we have dimk(x) m/m2 = n and so Bm is a regular local ring.
(iii) =⇒ (i). This direction requires some commutative algebra of ΩB|A we
haven’t covered. See [Matsumura p. 191] or [Hartshorne p. 174] for details.
Definition 11.38. When X is smooth and ΩX|k is a locally free sheaf on X,
and we refer to it as the cotangent bundle or cotangent sheaf of X.
The sheaf of p-forms is defined as
ΩpX =
p∧ΩX|k
In particular, if X has dimension n, ωX = ΩnX is called the canonical bundle of
X. As ΩX|k has rank n, ωX is locally free of rank 1, i.e., an invertible sheaf.
The tangent sheaf, or tangent bundle is the sheaf
TX = HomOX (ΩX|k,OX)
Thus for X = P1, we get ΩP1 = O(−2) and TP1 = OP1(2).
Proposition 11.39. Let X = Pn. Then ΩnX = O(−n− 1).
Proof. Consider the Euler sequence for the cotangent bundle of Pn
0→ ΩPn → O(−1)n+1 → O → 0
In general, if 0→ E ′ → E → E ′′ → 0, we have∧e E =
∧e′ E ′ ⊗∧e′′ E ′′. Hence
we getn∧
ΩPn =n+1∧
O(−1) = O(−n− 1).
Note by the way that the tangent bundle TPn fits into the following sequence:
0→ OPn → OPn(1)n+1 → TX → 0
where the left-most map sends 1 to the vector (x0, . . . , xn).
Example 11.40. Let A be a ring and let X = P1A. Then we have ΩX|A '
OP1(−2). For this, we can use the standard covering of P1 = ProjA[x0, x1], given
by Ui = D+(xi). On U0, we have
ΩU0|A ' A[x1
x0
]d
(x1
x0
)192
Chapter 11. Differentials
, and similarly on U1. On the intersection D+(x0) ∩D+(x1), we have
x20d
(x1
x0
)= −x2
1d
(x0
x1
)This gives a non-vanishing section of ΩX(2)⊗ ΩX|A and furthermore an isomor-
phism ΩX|A ' OX(−2).
11.8.1 Smooth morphisms
We can also use the sheaves of differentials to define a notion of smoothness for
morphisms. In short, we say that a morphism f : X → S at a point x ∈ X if
smooth if ΩX|S is locally free of rank dimxX − dimf(x) S there. If this is not
the case, we say that f is ramified at x, and that x is a ramification point. f is
smooth if it is smooth at every point.
Example 11.41. Let A = k[x] and let B = k[x, y]/(x− y2) ' k[y] where k is a
field of characteristic not equal to 2. Let X = SpecB and let Y = SpecA. Let
f : X → Y be the morphism induced by the inclusion A → B (thus x 7→ y2).
Since B ' k[y] it follows that ΩX = Bdy, the free B-module generated by
dy. Similarly ΩY = Adx. The conormal sequence gives us
→ ΩY ⊗A B → ΩX → ΩX/Y → 0
|| || o|Bdx → Bdy → Bdy/B(2ydy)
The point is that ΩX/Y = (k[y]/(2y))dy is a torsion sheaf supported on the
ramification locus of the map f : X → Y . (The only ramification point is above
0.) Note that ΩX/Y is the quotient of ΩX by the submodule generated by the
image of dx in ΩX = Bdy. The image of dx is 2ydy.
193
11.9. The conormal sheaf
11.9 The conormal sheaf
Let Y ⊆ X be a closed subscheme defined by an ideal sheaf I. Then I/I2 is
naturally a OY -module via I/I2 = I ⊗ OX/I = I ⊗ OY . We call I/I2 the
conormal sheaf of Y . Its dual, NY = HomOY (I/I2,OY ) is the normal sheaf of
Y in X.
Proposition 11.42. When X and Y are non-singular schemes over a field k,
the sheaves I/I2 and NY are locally free of rank r = codim(Y,X).
In this case, they fit into the exact sequences
0→ I/I2 → ΩX|k|Y → ΩY |k → 0
0→ TY → TX |Y → NY → 0
(This the conormal bundle sequence and its dual respectively. The first sequence
is exact on the left, because I/I2 is locally free).
Taking∧
of these sequences we get the following result, which is very useful
for computing canonical bundles of subvarieties:
Proposition 11.43 (Adjunction formula). If X and Y are non-singular, we
have
ωY = ωX ⊗r∧NY = ωX ⊗ detNY
In particular, if Y ⊆ X is a smooth divisor, we get
ωY = ωX ⊗ OX(Y )|Y
Example 11.44. Let X ⊆ P2 be a non-singular plane curve of degree d. Then
IX = OP2(−d), and so I/I2 = I ⊗ OP2/I = OY (−d). Here∧1 I/I2 = OY (−d)
so the previous proposition shows that
ωY ' OY (d− 3)
For d = 1, this gives our previous computation that ΩP1|k = OP1(−2).
Also for d = 2, Y ' P1, and ΩY = OP2(−1)|Y . This is consistent with the
previous example, because OP2(1)|Y ' OP1(2).
For d ≥ 3, the invertible sheaf ωY has many global sections. In particular, we
recover the fact that a non-singular plane curve of degree d ≥ 3 is not rational
(i.e., not isomorphic to P1). In fact, it will follow from the results of the next
chapter that Γ(Y, ωY ) has dimension exactly 12(d− 1)(d− 2). So for d ≥ 3, no
two smooth curves of different degrees can be isomorphic.
194
Chapter 12
First steps in sheaf cohomology
We have seen several examples of the global sections functor Γ failing to be exact
when applied to a short exact sequence. On the other hand, we have a sequence
0 Γ(X,F ′) Γ(X,F) Γ(X,F ′′)
which is exact at each stage except on the right. In many situations in algebraic
geometry, knowing that Γ(X,F) → Γ(X,F ′′) is surjective of fundamental im-
portance. We will see several examples of this phenomenon later in this chapter.
Essentially, cohomology is a tool that allows us to continue this sequence, so that
the sequence can be continued as follows:
0 Γ(X,F ′) Γ(X,F) Γ(X,F ′′)
H1(X,F ′) H1(X,F) H1(X,F ′′)
H2(X,F ′) H2(X,F) H2(X,F ′′) −→ · · ·
In other words, the failure of surjectivity of the above is controlled by the group
H1(X,F ′), and the other groups in the sequence.
Cohomology groups can be defined in a completely general setting, for any
topological space X and a sheaf F on it. With that as input, we define the
cohomology groups Hk(X,F), which will capture the main geometric invariants of
F . These should also be functorial in F , which means that we want to construct
195
12.1. Some homological algebra
functors
Hq(X,−) : AbShX → Ab
F 7→ Hq(X,F)
satisfying the following properties:
(I) H0(X,F) = Γ(X,F) = F(X),
(II) A morphism of sheaves φ : F → G induces maps H i(X,F) → H i(X,G)
which are functorial and take the identity to the identity (e.g., H i(X,−−)
is a functor).
(III) Every short exact sequence 0 → F ′ → F → F ′′ → 0 gives a long exact
sequence as above.
There are several ways to define these groups. The modern approach, and
the one summarized in Hartshorne Chapter III, takes the approach of using
derived functors. This is in most respects this is the ‘right way’ define the groups
in general, but going through the whole machinery of derived functors and
homological algebra would take us too far astray. We take a more down-to-earth
approach using Cech cohomology and the Godement resolution. The Godement
resolution has the advantage that it is completely canonical, and we can prove the
main theorems we need by hand. On the other hand, the Cech resolution, which
depends on the choice of a covering of X, but is better suited for computations.
Of course, the two notions of cohomology of course turn out to be the same, see
section XX.
12.1 Some homological algebra
A complex of abelian groups A• is a sequence of groups Ai together with maps
between them
· · · di−2
−−→ Ai−1 di−1
−−→ Aidi−→ Ai+1 di+1
−−→ · · ·
such that di+1 di = 0 for each i. A morphism of two complexes A•fn−→ B• is a
collection of maps fn : An → Bn so that the following diagram commutes:
· · · Ai−1 Ai Ai+1 · · ·
· · · Bi−1 Bi Bi+1 · · ·
di−1
fi−1
di
fi
di+1
fi+1
di−1 di di+1
In this way, we can talk about kernels, images, cokernels, exact sequences of
complexes, etc.
196
Chapter 12. First steps in sheaf cohomology
Let ZnA• = ker di and BiA• = im di−1. From this condition on the di, we
have ZnA• ⊇ BnA•. We define the cohomology groups of A•, denoted HpA• as
HpA• = Zn(A•)/Bn(A•) = ker di/ im di−1
A resolution of a group G is a complex of the form
0→ G→ A0 → A1 → A2 → · · ·
where the maps are exact at each Ai, i.e., HpA• = 0 for each p ≥ 0.
Proposition 12.1. Suppose that 0→ F •f−→ G•
g−→ H• → 0 is an exact sequence
of complexes. Then there is a long exact sequence
· · · HnF • HnG• HnH•
Hn+1F • Hn+1G• Hn+1H• −→ · · ·Proof. Consider the commutative diagram
0 F n Gn Hn 0
0 F n+1 Gn+1 Hn+1 0
fn
dn
gn
dn dn
fn+1 gn+1
where the rows are exact. By the snake lemma, we obtain a sequence
0→ Zn(F •)fn−→ Zn(G•)
gn−→ Zn(H•)δ−→ F n+1/Bn(F •)
fn+1−−→ Gn+1/Bn(G•)gn+1−−→ Hn+1/Bn(H•)→ 0
Consider now the diagram
0 F n/Bn(F •) Gn/Bn(G•) Hn/Bn(H•) 0
0 Zn+1(F •) Zn+1(G•) Zn+1(H•)
fn
dn
gn
dn dn
fn+1 gn+1
where the rows are exact. Applying the snake lemma one more time, we get the
desired exact sequence.
A complex of sheaves F• is a sequence of sheaves with maps between them
· · · di−2
−−→ Fi−1di−1
−−→ Fidi−→ Fi+1
di+1
−−→ · · ·
such that di+1di = 0 for each i. Given such a complex, we define the cohomology
sheaves HpF• as ker di/ im di−1. A resolution of a sheaf F is a complex of the
197
12.2. Cech cohomology
form
0→ F → C 0 → C 1 → C 2 → · · ·
where the maps are exact at each C i, i.e., HpC • = 0 for each p ≥ 0.
12.2 Cech cohomology
Let X be a topological space, and let F be a sheaf on it. Let U = Ui be an
open cover of X, indexed by an ordered set I. As we saw previously, the sheaf
axioms gives that the following sequence is exact:
0→ F(X)→∏i
F(Ui)→∏i,j
F(Ui ∩ Uj).
The Cech complex is essentially the continuation of this sequence; it is a complex
obtained by adjoining all the groups F(Ui1 ∩ · · · ∩ Uir) over all intersections
Ui1 ∩ · · · ∩ Uir .
Definition 12.2. For a sheaf F on X, define the Cech complex of F (with
respect to U) as
C•(U ,F) : C0(U ,F)d0−→ C1(U ,F)
d1−→ C2(U ,F)d2−→ · · ·
where
C0(U ,F) =∏i
F(Ui)
C1(U ,F) =∏i0<i1
F(Ui0 ∩ Ui1)
...
Cp(U ,F) =∏|I|=p+1
F(Ui0 ∩ · · · ∩ Uip).
and the coboundary map d : Cp(U ,F)→ Cp+1(U ,F) by
(dpσ)i0,...,ip+1 =
p+1∑j=0
(−1)jσi0,...ij ,...,ip+1|Ui0∩···∩Uip
where i0, . . . ij, . . . , ip+1 means i0, . . . , ip+1 with the index ij omitted.
Example 12.3. Let us look at the first few maps in the complex:
198
Chapter 12. First steps in sheaf cohomology
In degree 0: d0 : C0(U ,F)→ C1(U ,F). If σ = (σi)i, then
(d0σ)ij = σj − σi
In degree 1: d1 : C1(U ,F)→ C2(U ,F). If σ = (σij)i, then
(d1σ)ijk = σjk − σik + σij
Notice that in the example that d1 d0 = 0 (all the σij cancel). This happens
also in higher degrees by a basic computation using the definition of di.
Lemma 12.4. dp+1 dp = 0.
In particular, the C•(U ,F) forms a complex of abelian groups. We say that
an element σ ∈ Cp is a cocycle if dpσ = 0, and a coboundary if σ = dp−1τ . By
the lemma, all coboundaries are cocycles. The cohomology groups of F are set
up to measure the difference between these two notions:
Definition 12.5. The p-th Cech cohomology of F with respect to U is defined
as
Hp(U ,F) = (ker dp)/(im dp−1)
One can show that a sheaf homomorphism F → G induces a mapping of
Cech cohomology groups, so we obtain functors F → Hp(U ,F).
Example 12.6. Again it is instructive to consider the group H0(U ,F). Here
the map d0 : C0(U ,F)→ C1(U ,F), which is simply the usual map∏F (Ui)→
∏F(Ui ∩ Uj)
which has kernel F (X) by the sheaf axioms. It follows that H0(U ,F) = F(X).
Example 12.7. X = S1, U = U, V the standard covering of S1 (intersecting
in two intervals) and let F = ZX (the constant sheaf).
Here we have
C0(U ,F) = ZU × ZV C1(U ,Z) = ZU∩V = Z× Z
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12.2. Cech cohomology
The map C0 → C1 is the map d0 : Z2 → Z2 given by d0(a, b) = (b − a, b − a).
Hence
H0(U ,ZX) = ker d = Z(1, 1) ' Z
and
H1(U ,ZX) = coker d = Z× Z/Z(1, 1) ' Z
Students familiar with algebraic topology, might recognize that this is gives the
same answer as singular cohomology.
Exercise 39. Let X = S1 and let U be the covering of X with three pairwise
intersecting open sets with empty triple intersection. Show that the Cech
-complex is of the form
Z3 d0−→ Z3 → 0
Compute the map d0 and use it to verify again that H i(U ,ZX) = Z for i = 0, 1
as above.
Exercise 40. Let X = S2 and let U be the covering of X with four pairwise
intersecting open sets with empty triple intersection. Show that the Cech
-complex takes the form
Z4 d0−→ Z6 d1−→ Z4 → 0
Compute the matrices d0, d1 and show that H i(U ,ZX) = Z for i = 0, 2 and
H i(U ,ZX) = 0 for i 6= 0, 2.
The following proposition shows that constant sheaves are not so interesting
for our purposes.
Proposition 12.8. Let X be an irreducible topological space (so that any open
set U ⊆ X is connected). Then for any covering U of X we have for a constant
sheaf AXH i(U , AX) = 0
for i > 0.
Proof.
At this point, there are good news and bad news. As seen in the examples
above, the groups Hp(U ,F) are easily computable, if one is given a nice cover of
X. Indeed, the maps in the Cech complex are completely explicit, and computing
their kernels and cokernels are usually just a computation in linear algebra.
On the other hand, the definition is a little bit unsatisfactory for various
reasons. First of all, the groups Hp(U ,F) depend on the cover U , whereas we
want something canonical that only depends on F . More importantly, it is not
clear that the definition above should capture enough of the desired information
200
Chapter 12. First steps in sheaf cohomology
of F . For instance, U could consist of the single open set X, and so H i(U ,F) = 0
for i ≥ 1! Finally, it is not at all clear if these groups satisfy the requirements
mentioned in the introduction.
There is a fix for these problems, which involves passing to finer and finer
‘refinements’ of the covering. We say that a covering V is a refinement of U if for
every V ∈ V there is a U ∈ U so that V ⊆ U . This defines an ordering on the
coverings which we denote by V ≤ U . One can then define a group Hp(X,F) to
be the direct limit of all Hp(U ,F) as U runs through all possible open covers Uordered by ≤. The resulting groups are indeed canonical, and turn out to give
the right answer:
Definition 12.9. The groups Hp(X,F) are called the cohomology groups of F .
In symbols,
Hp(X,F) = lim−→UHp(U ,F)
The main preoperties of Cech cohomology are summarized in the following
theorem:
Theorem 12.10. Let X be a topological space and let F be a sheaf on X.
• The Cech cohomology groups are functors H i(X,−) : ShX → Groups.
• H0(X,F) = Γ(X,F) = F(X),
• Short exact sequences of sheaves induce long exact sequences of cohomology.
• (Leray’s theorem) If F is a sheaf and U is a covering such that H i(Ui1 ∩· · ·Uip .F) = 0 for all i > 0 and multiindexes i1 < · · · < ip, then
H i(X,F) = H i(U ,F).
We have proved the first two of these properties. The next two require more
work, but the arguments are mostly formal. We will nevertheless postpone the
proof until Appendix 13.
Note that the last statement shows that it suffices to compute the Cech
cohomology groups on a covering which is ‘sufficiently fine’ in the sense that the
higher groups H i(Ui1 ∩ · · ·Uip .F) = 0 vanish for i > 0.
Remark 12.11. Let us at least say a few words about the first boundary map δ
in the long-exact sequence above. Let 0→ F ′ → F → F ′′ → 0 be a short exact
sequence of sheaves. We would like to construct the boundary map δ which is a
map
δ : H0(X,F ′′)→ H1(X,F ′)
201
12.3. Cohomology of sheaves on affine schemes
If each restriction map
Γ(Ui,F)→ Γ(Ui,F ′) (12.2.1)
is surjective, then we can explicitly describe δ as follows: Represent a class
σ ∈ H0(X,F ′′) by a collection of elements fi ∈ Γ(Ui,F ′′) so that fi = fj on
Ui∩Uj . If the above restriction map is surjective, we can choose lifts gi ∈ Γ(Ui,F)
and define f ′ij = gi − gj. The f ′ij then form a 2-cocycle, since f ′ij − f ′ik + f ′jk = 0.
So we get a well defined class δ(σ) ∈ H1(X,F ′) (of course one needs to check
that this is independent of the choice of lift, but this is straightforward). Hence
we get a sequence
0→ H0(F ′)→ H0(F)δ−→ H0(F)→ H1(F)→ H1(F ′′)
which, by another diagram chase is exact.
In general, it can certainly happen that the restriction map (12.2.1) is not
surjective – one can for instance take the open covering of X with just one open
set X. This explains why the Cech cohomology groups H i(U ,F) do not give long
exact sequences in general. However, by passing to smaller refinements V ≤ U ,
we can arrange that any section lifts and we can use the above to construct δ.
12.3 Cohomology of sheaves on affine schemes
Theorem 12.12. Let X = SpecA and let F be a quasi-coherent sheaf on X.
Then
Hp(X,F) = 0 for all p > 0.
Proof. Recall that we defined the groups H i(X,F) by taking the direct limit of
H i(U ,F) finer and finer coverings U of X. Since the distinguished opens subsets
form a basis for the topology on X. It suffices to prove that
Hp(U ,F) = 0 for all p > 0.
for a covering U = D(gi) where the gi are finitely many elements of A generating
the ring. (We can choose finitely many gi, by the quasi-compactness of X).
As F is quasi-coherent, F = M for some A-module M , and M = Γ(X,F).
With this setup, the fact that Cech cohomology vanishes of a quasi-coherent
module on an affine scheme corresponds to the observation that for some com-
mutative ring A, some finite sequence of elements (gi)i∈I of A generating A as
an ideal, and some A-module M , the following sequence is exact:
0→M →∏i∈I
Mgi →∏i,j∈I
Mgigj →∏i,j,k∈I
Mgigjgk → . . .
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Chapter 12. First steps in sheaf cohomology
Here, the boundary maps are given as alternating sums of localization maps. For
example,
d :∏i,j∈I
Mgigj →∏i,j,k∈I
Mgigjgk
maps (σij)i,j ∈Mgigj to (σjk − σik + σij)i,j,k.
Notice that the beginning of the exact sequence
0→M →∏i
Mgi →∏i,j
Mgigj
appeared in the construction of the quasi-coherent module M . The proof for the
exactness of this sequence is similar to the general case.
To prove that the cohomology groups vanish, we must to each cocycle σ (such
that dσ = 0) find an element τ such making σ = dτ a boundary. The proof is a
direct calculation; one constructs an element τ by hand.
To see how this can be done, let us for simplicity consider the case p = 1 first.
Let σ ∈∏
i,jMgigj be in the kernel of d. We may write
σij =mij
(gigj)rwhere mij ∈M
for some r (since the index set I is finite, we may choose this independent of
i, j). The relation dσ = 0 gives the relation
mjk
(gjgk)r− mik
(gigk)r+
mij
(gigj)r= 0
or in other words, we have, in Mgjgk ,
gr+li mjk
(gjgk)r=glig
rjmik
(gjgk)r− glig
rkmij
(gjgk)r(12.3.1)
for some l ≥ 0. As the sets D(gi) = D(gr+li ) cover X, we have a relation
1 =∑i∈I
higr+li
where hi ∈ A. Now, define τ ∈∏Mgj by
τj =∑i∈I
higlimij
grj
203
12.3. Cohomology of sheaves on affine schemes
In∏Mgjgk , we may write it as
τj =∑i∈I
higlig
rkmij
(gjgk)r
We want to show that dτ = σ. This is a basic computation, using the relation
(12.3.1) above:
(dτ)jk = τk − τj
=∑i∈I
higlig
rjmik
(gjgk)r−∑i∈I
higlig
rkmij
(gjgk)r
=∑i∈I
hi
(glig
rjmik
(gjgk)r− glig
rkmij
(gjgk)r
)
=∑i∈I
higr+li mjk
(gjgk)r=
mjk
(gjgk)r
∑i∈I
higr+li
=mjk
(gjgk)r= σjk
As desired. Hence H1(U ,F) = 0.
In the general case, let σ ∈∏
i0,...,ipMgi0 ···gip be in the kernel of d. We may
write
σi0,...,ip =mi0,...,ip
(gi0 · · · gip)rwhere mi0,...,ip ∈M
for some r (since the index set I is finite, we may choose this independent of
i, j). The relation dσ = 0 gives the relation in Mgi0 ···gip :
gr+li mi0,...,ip
(gi0 · · · gip)r=
p∑k=0
(−1)kglig
rikmii0,...,ik...ip
(gi0 · · · gip)r(12.3.2)
As the sets D(gi) = D(gr+li ) cover X, we have a relation
1 =∑i∈I
higr+li
where hi ∈ A. Now, define τ ∈∏Mgi0 ···gip−1
by
τi0,...,ip−1 =∑i∈I
higli
mii0,...ip
(gi0 · · · gip−1)r
204
Chapter 12. First steps in sheaf cohomology
Localizing to∏Mgi0 ,...,gip
, we may write this as
τi0,...,ip =∑i∈I
higligrik
mii0,...ip
(gi0 · · · gip)r
We want to show that dτ = σ. As before, we can check this using the relation
(12.3.2) above:
(dτ)i0···ip =
p∑k=0
(−1)kτi0...ik...ip
=∑i∈I
higr+li σi0...ip = σi0...ip
This completes the proof.
12.3.1 Cech cohomology and affine coverings
As a Corollary of the previous theorem, we see that affine coverings of schemes
satisfy the conditions of Leray’s theorem (see Theorem 12.10). This implies
Corollary 12.13. Let X be a noetherian scheme and let U = Ui be an affine
covering such that all intersections Ui0 ∩ · · · ∩ Uis are affine. Then
H i(X,F) = H i(U ,F)
In particular, the theorem applies to any affine covering on a noetherian separated
scheme.
12.4 Chomology and dimension
Lemma 12.14. Let X be a topological space and let Z ⊆ X be a closed subset.
Then for any abelian sheaf F on Z we have Hp(Z,F) = Hp(X, i∗F).
Theorem 12.15. Let X be a quasi-projective scheme of dimension n. Then X
admits an open cover U consisting of at most n+ 1 affine open subsets.
This implies that
H i(X,F) = 0 for i > n
for any quasi-coherent sheaf F on X.
Proof. Let X be a quasi-projective scheme, i.e., X = Y −W , where Y,W ⊆ PnAare (closed) projective schemes. Consider the irreducible decomposition Y = ∪iYiand observe that IW 6⊆ ∪IYi where IT ⊆ A[xo, . . . , xN ] denotes the ideal of the set
205
12.5. Cohomology of sheaves on projective space
T ⊆ PN . Pick a homogenous polynomial f of degree d such that f ∈ IW −(∪iIYi).Let H = Z(f). Then Pn −H is affine and hence so is Y −H.
By construction Y −H ⊆ Y −W = X and H 6⊇ Yi for any i by the choice of
f . Therefore dim(Yi ∩H) < dim Yi so we may use induction on dimX. Notice
that the affine subset is obtained as an affine subset of the ambient projective
space intersected with our scheme, so the affine schemes obtained subsequently
are restrictions of affine subschemes of the original X. This shows the first claim.
For the second, note that for ≤ n + 1 affines, there are at most n terms
Ci(X,F) in the Cech complex. From this it follows that H i(U ,F) = 0 for any
F and i > n.
In fact, this theorem holds in much greater generality:
Theorem 12.16. Let X be a topological space of dimension n, and let F be a
sheaf on X. Then
Hp(X,F) = 0
for all p > n.
12.5 Cohomology of sheaves on projective space
On projective space PnA, we can use the Cech complex associated with the
standard covering to compute the cohomology of any invertible sheaf on PnA.
Theorem 12.17. Let X = PnA = ProjR where R = A[x0, . . . , xn] where A is a
noetherian ring.
(i)
H0(X,OX(m)) =
Rm for m ≥ 0
0 otherwise
(ii)
Hn(X,OX(m)) =
A for m = −n− 1
0 for m > −n− 1
(iii) For m ≥ 0, there is a perfect pairing1 of A-modules
H0(X,O(m))×Hn(X,OX(−m− n− 1))→ Hn(X,OX(−n− 1)) ' A
1Recall that a bilinear map M × N → A is a perfect pairing if the induced map M 7→HomA(N,A) is an isomorphism
206
Chapter 12. First steps in sheaf cohomology
(iv) For 0 < i < n and all n ∈ Z, we have
H i(X,OX(m)) = 0
Proof. Consider the OX-module F =⊕
m∈Z O(m). We would like to show for
instance that H1(X,F) = 0 for i 6= 0, n; this is equivalent to H i(X,OX(m)) = 0
for all m, but F has the advantage that it is a graded OX-algebra. Consider
the Cech complex associated with the standard covering U = Ui where Ui =
D+(xi) = SpecR(xi). This is simply∏i
Rxid0−→∏i,j
Rxixjd1−→ . . .
dn−1
−−−→ Rx0···xn
We have a graded isomorphism of R-modules:
H0(X,F) = ker d0
= (ri)i∈I |ri ∈ Rxi , ri = rj ∈ Rxixj' R.
This isomorphism preserves the grading, so we get (i).
For (ii): Note that Rx0···xn is a free graded A-module spanned by monomials
of the form
xa00 · · ·xannfor multidegrees (a0, . . . , an) ∈ Zn+1. The image of dn−1 is spanned by such
monomials where at least one ai is non-negative. Hence
Hn(X,F) = coker dn−1
= A xa00 · · ·xann |ai < 0∀i ⊆ Rx0...xn
Hence
Hn(X,OX(m)) = Hn(X,F)m
= Axa00 · · ·xann |ai < 0∀i,
∑ai = m
⊆ Rx0...xn
In degree −n− 1 there is only one such monomial, namely x−10 · · ·x−1
n .
(iii): If we identify Hn(X,OX(−m− n− 1)) with
Axa00 · · ·xann : ai < 0∀i,
∑ai = m
and H0(X,O(m)) = Rm, we can define the pairing via multiplication of Laurent
207
12.5. Cohomology of sheaves on projective space
polynomials:
H0(X,O(m))×Hn(X,OX(−m− n− 1))→ Rx0···xn
(xm00 · · ·xmnn )× (xa00 · · ·xann ) 7→ xa0+m0
0 · · ·xan+mnn
Here the exponents satisfy mi ≥ 0, ai < 0∑ai = −m− n− 1,
∑mi = m. This
gives a map
H0(X,O(m))×Hn(X,OX(−m−n−1))→ Hn(X,OX(−n−1)) = Ax−10 · · ·x−1
n
sending (xm00 · · ·xmnn ) × (xa00 · · ·xann ) to zero if mi + ai ≥ 0 for some i. This
pairing is perfect: The dual of a monomial (xm00 · · ·xmnn ) is represented by
(x−m0−10 · · · x−mn−1
n ).
(iv) This point is more involved, and we proceed by induction on n. For
n = 1, there is nothing to prove. For n > 1, let H = V (xn) ' Pn−1 be the
hyperplane determined by xn. We have an exact sequence
0→ R(−1)·xn−−→ R→ R/(xn)→ 0 (12.5.1)
Applying ∼, we find
0→ OX(−1)→ OX → i∗OH → 0
where i : H → X is the inclusion. If we take the direct sum of all the twists of
this sequence, we get
0→ F(−1)→ F → i∗FH → 0
where FH =⊕
m∈Z OH(m). By induction on n, we have for 0 < i < n− 1 and
all m ∈ Z: H i(X, i∗OH(m)) = H i(H,OH(m)) = 0. So taking the long exact
seqence of cohomology, we get isomorphisms
H i(X,F(−1))·xn−−→ H i(X,F)
for 1 < i < n − 1. We claim that we have isomorphisms also for i = 1 and
i = n− 1. For i = 1, this follows because the sequence
0→ H0(X,F(−1))→ H0(X,F)→ H0(X, i∗FH)→ 0
is exact (this is the same sequence as (12.5.1)).
For i = n− 1 we need to show that
0→ Hn−1(X, i∗FH)δ−→ Hn(X,F(−1))
·xn−−→ Hn(X,F)
208
Chapter 12. First steps in sheaf cohomology
is exact. The kernel of ·xn, is generated by monomials xa00 · · ·xann with ai < 0 for
all i. So it suffices to show that the connecting map δ is just multiplication by
x−1n . Define R′ = R/xn. Writing the arrows in the Cech complex, vertically we
get the diagram
0 //
∏iR(−1)x0···xi···xn
·xn //
∏iRx0···xi···xn
//
R′x0···xn−1//
0
0 // Rx0···xn(−1)·xn // Rx0···xn
// 0
If xa00 · · ·xan−1
n−1 is a monomial in Hn−1(H,FH) with ai < 0 for all 1 ≤ i ≤ n− 1,
then it comes from an (n+1)-tuple in∏
iRx0···xi···xn which maps to ±xa00 · · ·xan−1
n−1
in Rx0···xn , which is in turn mapped onto by the monomial xa00 · · ·xan−1
n−1 x−1n in
Rx0···xn(−1). So δ(xa00 · · ·xan−1
n−1 ) is represented by the monomial xa00 · · ·xan−1
n−1 x−1n
in Hn(X,F(−1)).
Now we claim that we have an isomorphism H∗(X,F)xn = H∗(Un,F|Un).
Indeed, the Cech complex of F|Un with respect to the covering Ui∩Un is just the
localization of C•(X,F) at xn. Localization is exact, so it preserves cohomology,
which gives the claim.
We know that H i(X,F)xn = H i(Un,F|Ur) = 0 for all i > 0, since Un is affine.
Hence for l 0, xlnHi(X,F) = 0 as an A-module. However, we have shown
that ·xn gives an isomorphism of H i(X,F) for 0 < i < n. This implies that
H i(X,F) = 0.
In the above proof, we used the following lemma:
Lemma 12.18. Let X = PnA over a noetherian ring A and let F be a quasi-
coherent sheaf on X. Then H i(X,F) = 0 for all i > n.
Proof. PnA can be covered with n+1 open affine schemes, all of whose intersections
are affine. Hence the Cech complex associated with F has length n+ 1, and so
there is no cohomology in degrees higher than n.
12.6 Extended example: Plane curves
Let X = V (f) denote an integral subvariety of P2, defined by an irreducible
homogeneous polynomial f of degree d.
LetX ⊆ P2 be a plane curve, defined by a homogeneous polynomial f(x0,1 , x2)
of degree d. Let us compute H i(X,OX).. We have the ideal sheaf sequence
0→ IX → OP2 → i∗OX → 0
209
12.7. Extended example: The twisted cubic in P3
where the ideal sheaf IX is the kernel of the restriction OP2 → i∗OX . By Section
10.7.1, OP2(−X) ' OP2(−d). More explicitly, on Spec k[x1x0, x2x0
], we see IX is
a k[x1x0, x2x0
]-module generated by f(x1x0, x2x0, 1), and OP2(−d) is a k[x1
x0, x2x0
]-module
generated by x−d0 , and this isomorphism sends generator to generator. The above
exact sequence can thus be rewritten as
0 OP2(−d) OP2 OX 0.·f
From the short exact sequence, we get the long exact sequence as follows:
0 H0(P2,O(−d)) H0(P2,OP2) H0(X,OX)
H1(P2,O(−d)) H1(P2,OP2) H1(X,OX)
H2(P2,O(−d)) H2(P2,OP2) 0.
Using the results on cohomology of line bundles on P2, we deduce thatH0(X,OX) 'k and
H1(X,OX) ' k(d−12 ).
12.7 Extended example: The twisted cubic in
P3
Let k be a field and consider P3 = ProjR where R = k[x0, x1, x2, x3]. We will
consider the twisted cubic curve C = V (I) where I ⊆ R is the ideal generated
by the 2× 2-minors of the matrix
M =
(x0 x1 x2
x1 x2 x3
)i.e., I = (q0, q1, q2) = (x2
1 − x0x2, x0x3 − x1x2,−x22 + x1x3). Let us by hand
compute the group H1(X,OX). Of course we know what the answer should be,
since X ' P1, and H1(P1,OP1) = 0. Indeed, P1 = Proj k[s, t] is isomorphic to
the Veronese embedding Proj k[s, t](3), where k[s, t](3) = k[s3, s2t, st2, t3], and the
latter ring is isomorphic to S = R/I.
Now, to computeH1(X,OX) onX, it is convenient to relate it to a cohomology
group on P3. We have H1(X,OX) = H1(P3, i∗OX) where i : X → P3 is the
inclusion. This fits into the ideal sheaf sequence
0→ I → OP3 → i∗OX → 0.
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Chapter 12. First steps in sheaf cohomology
where I is the ideal sheaf of X in P3. Applying the long exact sequence in
cohomology we get
0 I OP3 i∗OX 0.
From the short exact sequence, we get the long exact sequence as follows:
· · · H1(P3, I) H1(P3,OP3) H1(P3, i∗OX)
H2(P3, I) H2(P3,OP3) · · ·
By our description of sheaf cohomology on P3, H1(P3,OP3) = H2(P3,OP3) = 0,
we find that H1(X,OX) = H2(P3, I). We can compute the latter cohomology
group as follows. Note that I = I. Consider the map R3 → I → 0 sending
ei 7→ qi. Let us consider the kernel of this map, that is, relations of the form
a0q0 + a1q1 + a2q2 = 0 for ai ∈ R. There are two obvious relations of this form,
i.e., the ones we get from expanding the determinants of the two matricesx0 x1 x2
x0 x1 x2
x1 x2 x3
x0 x1 x2
x1 x2 x3
x1 x2 x3
(So first matrix gives x0q2 − x1q1 + x2q2 = 0 for instance). These give a map
R2 ·M−→ R3, where M is the matrix above. This map is injective, and there is an
exact sequence
0→ R2 ·M−→ R3 → I → 0
If we want to be completely precise, and consider these as graded modules, we
must shift the degrees according to the degrees of the maps above
0→ R(−3)2 ·M−→ R(−2)3 → I → 0
Applying ∼, we get an exact sequence of sheaves
0→ OP3(−3)2 → OP3(−2)3 → I → 0
Now, this we can use to compute H2(X, I); taking the long exact sequence we
get
· · · H2(P3,O(−3)2) H2(P3,OP3(−2)3) H2(X, I)
H3(P3,O(−3)2) H3(P3,OP3(−2)3) H3(X, I)
211
12.7. Extended example: The twisted cubic in P3
Here H2(P3,O(−2)) = 0 and H3(P3,O(−3)) = 0 (!) by our previsous computa-
tions. Hence by exactness, we find H2(X, I) = 0, as expected.
Exercise 41. Using the sequences above, show that
• H0(P3, I(2)) = k3 (find a basis!)
• H1(P3, I(m)) = 0 for all m ∈ Z.
• H2(P3, I(−1)) = k.
212
Chapter 13
More on sheaf cohomology
13.1 Flasque sheaves
A sheaf F on a topological space X is said to be flasque if the restriction maps
F(X)→ F(U)
for every open subset U ⊆ X.
Example 13.1. If X is irreducible, then any constant sheaf is flasque.
Example 13.2. The skyskraper sheaves A(x) defined in Chapter 1 are flasque.
Example 13.3. OP1k
is not flasque, since Γ(P1,OP1k) = k, whereas Γ(U,OP1
k) is
typically much larger (e.g., for U = A1k ⊆ P1
k it is an infinite dimensional k-vector
space).
Recall that the global section functor Γ is left exact. The following lemma
says that it is also exact on the right, given that the left-most term in the
sequence is flasque:
Lemma 13.4. Given an exact sequence
0 // F // G //H // 0
of sheaves F , G where F is flasque. Then the corresponding sequence of global
sections
0 // F(U) // G(U) //H(U) // 0
is exact for every open set U ⊆ X.
213
13.1. Flasque sheaves
Proof. By restricting F ,G,H to U , it suffices to prove the statement for U = X.
We only need to check that the sequence is exact on the right. Let s ∈ H(X)
and let V be the family of open subsets U ⊆ X such that s|U is represented by
an element of G(X). We need to show that X ∈ V .
Since V contains a covering of X and X is quasi-compact, it suffices to show
that V is closed under finite unions. So let U1, U2 ∈ V and let ti ∈ Γ(Ui,F)
represent s on Ui, i = 1, 2. Note that by flasque property of F , t1 − t2 on
U1 ∩ U2 lifts to a global section u of F . Moreover, by adding u|U2 to t2, we may
assume that t1, t2 agree on U1∩U2, so by gluing we get a section t ∈ G(X) which
represents s, and hence U1 ∪ U2 ∈ V .
Lemma 13.5. Suppose we are given an exact sequence of sheaves
0 // F // G //H // 0.
If F and G are flasque, then so is H.
Proof. Let U ⊆ X be a subset of X. Then any section h ∈ H(U) is represented
by a section g ∈ G(U) by the previous lemma. Since G is flasque, this can be
extended to an element g of G(X). Then g maps to an element h ∈ H(X) which
has the property that h|U = h.
Lemma 13.6. Suppose we are given an exact sequence of flasque sheaves
0 // E → F0 → F1 → F2 → · · · , (13.1.1)
then for each open set U ⊆ X, the sequence
0 // E (U)→ F0(U)→ F1(U)→ F2(U)→ · · · , (13.1.2)
is exact.
Proof. We chop (13.1.2) into short exact sequences as follows:
0→ E → F0 → Q0 → 0
0→ Q0 → F1 → Q1 → 0
· · ·
By induction, it follows that the sheaves Qi are flasque, being quotients of flasque
sheaves. Applying Γ we then get exact sequences
0→ E (U)→ F0(U)→ Q0(U)→ 0
0→ Q0(U)→ F1(U)→ Q1(U)→ 0
· · ·
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Chapter 13. More on sheaf cohomology
If we connect these together, we obtain the exact sequence (13.1.2).
13.2 The Godement resolution
We will define the cohomology groups Hp(X,F) using a special resolution of Fknown as the Godement resolution. To explain how this works, we again recall
the canonical map κ : F → Π(F), which over an open set U ⊆ X sends a section
s ∈ F(U) to the element (sx)x∈U ∈ Π(F)(U) =∏
x∈U Fx. Defining C 0F = Π(F)
and Z1F as the cokernel of κ, we get a canonical exact sequence
0→ F → C 0F → Z1F → 0
Taking global sections, we get
0→ F(X)→ C 0F(X)→ Z1F(X).
As we have seen, we cannot expect to be exact on the right. In fact, this failure
of exactness is how we will define cohomology groups: We define H1(X,F) as
the cokernel of the right most map C 0F(X)→ Z1F(X), i.e.,
H1(X,F) = Z1(X)/B0(X).
where B = im(C 0F(X)→ Z1(X)).
We note that the assignments F 7→ C 0F , F 7→ Z1F and F 7→ H1(X,F) are
indeed functors. Indeed, given a map F → G of sheaves, we get canonical maps
between the two sequences
0 F(X) C 0F(X) Z1F(X) H1(X,F)→ 0
0 G(X) C 0G(X) Z1G(X) H1(X,G)→ 0
and from this we see that we have a canonical map H1(X,F)→ H1(X,G).
What about the higher cohomology groups? Well, we can continue, replacing
F with the new sheaf Z1. We again have a canonical map κ : Z1 → Π(Z1), and
we can define C 1F = C 0Z1 and Z2F = coker(Z1F → C 1F). We then have an
exact sequence
0→ Z1F → C 1F → Z2F → 0
and we can define H2(X,F) to be the cokernel of the map on global sections
C 1F(X)→ Z2F(X).
215
13.2. The Godement resolution
So to define cohomology groups in general, we define inductively the sheaves C i
C 0F = Π(F)
Z1F = coker(κ : F → C 0F)
C nF = C 0Zn for n ≥ 1
Zn+1F = Z1Zn = coker(κ : Zn → C n) for n ≥ 1
The way to visualise these is as follows:
0→ F C 0F C 1F C 2F · · ·
Z1F Z2F
κ d0 d1 d2
κ κ
(13.2.1)
Here the map dq : C q → C q+1 is defined by the composition
C qF → Zq+1F → C 0Zq+1 = C q+1F
The second map is injective, which means that the complex in (13.2.1) is exact.
It follows that the complex 0 → F → C 0 → C 1 → C 2 → · · · is actually a
resolution, and we call it the Godement resolution of F .
13.2.1 Functoriality
Given a morphism of sheaves φ : F → G, we have a canonical morphism
Π(F) → Π(G) which is compatible with the two maps κF : F → Π(F) and
κG : G → Π(G). Therefore, we get a morphism of quotient sheaves
C 0F/F → C 0G/G
or in other words, a morphism Z1F → Z1G. Taking again Godement sheaves,
we get
C 0Z1F → C 0Z1G
or, a morphism C 1F → C 1G. Thus, inductively, we construct canonical sheaf
maps
C kφ : C kF → C kG
These C k() define the Godement functors; they are functors from ShX to ShX .
The Godement resolution is also functorial, in the sense that we have a canonical
morphism of complexes C •F → C •G:
216
Chapter 13. More on sheaf cohomology
0 F C 0F C 1F C 2F · · ·
0 G C 0G C 1G C 2G · · ·
13.2.2 Sheaf cohomology
If we take global sections in the Godement resolution C •F , we get a sequence
0→ F(X) C 0F(X) C 1F(X) C 2(X) · · ·d0(X) d1(X) d2(X)
(13.2.2)
This may or may not be exact, but since di+1(X) di(X) = 0 we still have a
complex of abelian groups, and we can talk about its cohomology.
Definition 13.7. The p-th cohomology group Hp(X,F) is defined to be the
p-th cohomology group of the complex of abelian groups (13.2.2), i.e.,
Hp(X,F) = (ker dp(X))/(im dp−1(X))
Equivalently,
Hp(X,F) = Zp(X)/Bp(X)
where Bp(X) = (im(C p−1(X)→ Zp(X))).
So what is H0(X,F)? By definition this group is the kernel of the map d0,
which is defined as the composition
C 0F(X)→ Z1F(X) → C 1F(X)
Hence H0(X,F) = ker(C 0F(X)→ Z1F(X)). On the other hand, we have the
sequence 0→ F → C 0 → C 1; taking global sections, we get the exact sequence
0→ F(X)→ C 0(X)d0(X)−−−→ Z1F(X)
from which we see that H0(X,F) = ker d0(X) = Γ(X,F).
By the recursive definition of the sheaves C pF and Zp, we only need to know
how to define H1 of a sheaf in order to see the higher cohomology groups. Indeed,
from the complex (13.2.2), it is clear that the C q also give a resolution of Z1, so
that
H i(X,F) =
Γ(X,F ) if i = 0
coker(C 0(X)→ Z1(X)) if i = 1
H1(X,Z i−1) if i ≥ 2
The geometric meaning of the elements of these higher cohomology groups
217
13.2. The Godement resolution
Hp(X,F) is on the other hand far from obvious, and unfortunately this is a
general tendency when defining cohomology groups. What is important is what
sort of properties they have: at the moment these groups are completely canonical
in their definition, and again since κ and Π(F) are functorial, we find that the
assignments F 7→ H i(X,F) define functors from sheaves to groups. This takes
care of the two first properties of sheaf cohomology.
We now turn to prove the third property, that is, that we have a long exact
sequence. First we prove that the assignments F 7→ C iF and F 7→ Z iF are in
fact exact functors.
Lemma 13.8. Let 0→ F1 → F2 → F3 → 0 be a short exact sequence of sheaves.
Then we have exact sequences
0→ C iF1 → C iF2 → C iF3 → 0
and
0→ Z iF1 → Z iF2 → Z iF3 → 0
Proof. By definition we have C 0i (U) =
∏x∈U (Fi)x for an open set U ⊆ X. Since
the above sequence is exact on stalks, we get also that the sequence
0→ C 0F1 → C 0F2 → C 0F3 → 0
is also exact. (In fact this observation is one of main reasons why we use the
Godement sheaves in the definition of cohomology in the first place.) This row
fits into the following diagram:
0 0 0
0 F1 F2 F3 0
0 C 0F1 C 0F2 C 0F3 0
0 Z1F1 Z1F2 Z1F3 0
0 0 0
(13.2.3)
Since the two top rows are exact, the snake lemma implies that also the lower
row is exact. Now using the fact that the composition of exact functors is still
exact, and the fact that C iF = C 0Z i and Z i+1F = Z1Z iF , we conclude by
induction on i.
218
Chapter 13. More on sheaf cohomology
Lemma 13.9. The sheaves C iF and Z iF are flasque for every i ≥ 0.
Proof. Note first that Π(F) is flasque: For any open set U ⊆ X it is clear that∏x∈X
Fx →∏x∈U
Fx
is surjective. Hence C 0 is flasque. This implies that Z1 is flasque as well, being
a quotient of C 0 by a subsheaf. Now we can conclude by induction, using
C iF = C 0Z i and Z i+1F = Z1Z iF
Corollary 13.10. If F is flasque, then for every U ⊆ X we have
H i(U,F) = 0
In particular, F is acyclic.
Proof. We take the Godement resolution 0→ F → C 0F → C 1F →. Since Fand each C iF are flasque, we get
0→ F(U)→ C 0(U)→ C 1(U)→ C 2(U)→ · · ·
Since this complex is exact, we obtain that H i(X,F) = 0 by the definition of
cohomology.
We are now ready to prove property (iii), namely the long exact sequence of
cohomology.
Theorem 13.11. Let 0 → F1 → F2 → F3 → 0 be a short exact sequence of
sheaves. Then there is a long exact sequence
0→ H0(X,F1)→ H0(X,F2)→ H0(X,F3)δ−→ H1(X,F1)→ H1(X,F2)→ H1(X,F3)→ · · ·
Proof. First of all, by the previous lemma, there are exact sequences
C • : 0→ C iF1 → C iF2 → C iF3 → 0
Since each C iF is flasque, we get also that
0→ C iF1(X)→ C iF2(X)→ C iF3(X)→ 0
is exact. Hence we get an exact sequence of complexes of abelian groups
0→ C •F1(X)→ C •F2
(X)→ C •F3(X)→ 0
Then, since Hp(X,Fi) is defined as the cohomology of this complex, we conclude
by Proposition 12.1 that we have the above long exact sequence.
219
13.2. The Godement resolution
Exercise 42. Let X be a topological space and let i : Z → X be the inclusion
of a subset. Show that for a sheaf F on Z,
H i(X, i∗F) = H i(Z,F) (13.2.4)
for all i.
13.2.3 Acyclic sheaves
Definition 13.12. A sheaf F is called acyclic if H i(X,F) = 0 for all i > 0.
Given a resolution
0→ F → C 0 → C 1 → C 2 → · · ·
of F (which by definition means that the sequence is exact), the resulting
sequence
C •(X) : C 0(X)→ C 1(X)→ C 2(X)→ · · · ,
may fail to be exact, but is still a complex of abelian groups, so it makes sense
to ask about its cohomology.
Lemma 13.13. If the sheaves C i are acyclic, then there is a natural isomorphism
H i(X,F) ' H i(C •(X))
Proof. Define K−1 = F , and Ki = ker(C i+1 → C i+2) for i ≥ 0. By exactness of
C •, we have for each i ≥ 0 and exact sequence
0→ Ki−1 → C i → Ki → 0
Taking the long exact sequence, gives
0→ H0(Ki−1)→ H0(C i)→ H0(Ki)→ H1(Ki−1)→ H1(C i) = 0 (13.2.5)
where the right-most group it zero because C i is acyclic. Also, the same sequence
shows that Hp(Ki) = Hp+1(Ki−1) for every p ≥ 1. The maps in these sequences
fit into the diagram
220
Chapter 13. More on sheaf cohomology
From this, we see that
im(H0(C i)→ H0(K i)
)= im
(H0(C i)→ H0(C i+1)
)We furthermore have
H0(Ki−1) = ker(H0(C i)→ H0(C i+1)
).
Indeed, if σ ∈ H0(C i) maps to zero in H0(F i+1), then it maps to zero in H0(Ki),
and hence it lies in the image of H0(Ki−1). Hence
H0(F) = ker(H0(C i)→ H0(C i+1)
)In particular,
H0(F) = ker(H0(C 0)→ H0(C 1)
)= H0(C •(X))
and the theorem holds in degree p = 0. By the same token, we have
H0(Ki) = ker(H0(C i+1)→ H0(C i+2)
).
From (13.2.5), and the isomorphisms Hp(Ki) ' Hp+1(Ki−1) we therefore get
H i+1(C •(X)) = ker(H0(C i+1)→ H0(C i+2)
)/ im(H0(C i)→ H0(C i+1))
= H0(Ki)/ im(H0(C i)→ H9(Ki))
= H1(Ki−1)
= H2(Ki−2)
= · · ·= H i+1(K−1)
= H i+1(F)
221
13.2. The Godement resolution
13.2.4 The Cech resolution
We also have the following sheafified version of the Cech complex. Given a sheaf
F on X and an open cover U , we set
Cp(U ,F) =∏
i0<···<ip
i∗F|Ui0∩···∩Uip
where i : Ui0 ∩ · · · ∩ Uip → X denotes the inclusion.
We will now show that the sheaves Cp(U ,F) give a resolution of F . So we
have an exact sequence
C•(U ,F) : 0→ F → C0(U ,F)d0−→ C1(U ,F)
d1−→ C2(U ,F)→ · · · (13.2.6)
In particular, if we apply Γ, we get back the Cech complex from earlier.
We define the maps in the complex as follows. The first map F → C0(U ,F)
is takes a section and restricts it to Ui. The differentials di are constructed in the
same way as in the Cech complex. Thus if we apply the global sections functor
Γ to Cp(U ,F) we get the earlier defined Cech complex.
The main difference here is that the new complex is exact. We can check
exactness on stalks and the nicest way to do that is with a homotopy operator
k : Cp(U , F )→ Cp−1(U ,F) so that (dk + kd)(α) = α: If we are given such a k,
every cocycle is a coboundary and so HpC•(U ,F) = 0.
For each x ∈ X, and open set V 3 x, refine V to V ∩Uj for some Uj 3 x. So
now, V ⊆ Uj. Define k by
(kα)i0,...,ip−1 = αj,i0,...,ip−1
Then it is easy to check that (dk + fd)(α) = α.
Proposition 13.14. If F is flasque, then so are the sheaves Cp(U ,F) for p > 0.
Hence (13.2.6) is an acyclic resolution for F and
Hp(X,F) = Hp(C•(U ,F))
Proof. If F is flasque, then so is each restriction to each Ui0∩···∩Up , and products
of flasque sheaves are flasque, so∏
i0<···<ip i∗F|Ui0∩···∩Up is flasque.
222
Chapter 13. More on sheaf cohomology
13.3 Godement vs. Cech
It remains to see why these two definitions are equivalent. So let U = Ui be a
covering for F . We will assume that this is Leray in the sense that H i(UI ,F) = 0
for all multi-indices I and i > 0. We claim that there is a natural isomorphism
H i(X,F) ' H i(U ,F),
where we, in order to avoid confusion, let H i(U ,F) denote the Cech cohomology
group. The statement is clearly true for i = 0, since both coincide with Γ(X,F).
Lemma 13.15. Let 0→ F → G → H → 0 be an exact sequence. If U is Leray,
there is a long exact sequence
0 H0(U ,F) H0(U ,G) H0(U ,H)
H1(U ,F) H1(U ,G) H1(U ,H) −→ · · ·
Proof. Since U is Leray, we have H1(UI ,F) = 0 for all multi-indexes I (in fact,
this is the only property we need from the covering U). Hence the following
sequences are exact:
0→ F(UI)→ G(UI)→ H(UI)→ 0
Then applying the Cech complex, we get an exact sequence of complexes
0→ C•(U ,F)→ C•(U ,G)→ C•(U ,H)→ 0
Now the claim follows from Proposition 12.1.
Hence we get our desired theorem:
Theorem 13.16 (Leray). Suppose U is a cover of X and Hq(U1∩· · ·∩Up,F ) = 0
for all p, q > 0 and all U1, . . . , Up ∈ U . Then there is a natural isomorphism
between cohomology and Cech cohomology:
Hp(X,F) ' Hp(U ,F)
Proof. We use induction on p. For p = 0, the claim is clear. Note that we have
the exact sequence
0→ F → Π(F)→ Z1 → 0
and H1(X,F) = coker(Γ(X,Π(F))→ Γ(X,Z1F)), and
Hp(X,F) = Hp−1(X,Z1F)
223
13.3. Godement vs. Cech
for p ≥ 2. On the other hand, we also have the corresponding result for Cech
cohomology:
0 H0(U ,F) H0(U ,Π(F)) H0(U ,Z1)
H1(U ,F) H1(U ,Π(F)) = 0
where H1(U ,F) = 0 by Lemma 13.13, since Π(F) is acyclic. Hence also
H1(X,F) = coker(Γ(X,Π(F))→ Γ(X,Z1F)) = H1(X,F)
Hence the theorem also holds for p = 1.
We continue by induction on p. Since also H i(U ,Π(F )) = 0 for all i > 0,
same long exact sequence of Cech cohomology also shows that Hp(U ,F) =
Hp−1(U ,Z1). Moreover, the cover U is also Leray with respect to Z1: H i(UI ,Z1) =
H i+1(UI ,F) = 0). Hence replacing F with Z1, we get the desired conclusion.
224
Chapter 14
Divisors and linear systems
When studying a scheme it is natural to ask about its closed subschemes. We
have seen that such subschemes are in correspondence with quasi-coherent ideal
sheaves. For (integral) subschemes of codimension 1, these ideal sheaves tend
to be much simpler than for higher codimension. This is essentially because of
Krull’s Hauptidealsatz, which says roughly that (under certain hypotheses), this
is generated locally by one element. This is essentially the prototype of a Cartier
divisor; a subscheme which is locally cut out by one equation.
The prototype of a divisor is a hypersurface of projective space, that is, an
integral subscheme of Pn of codimension one. Each such subscheme is defined by
a homogeneous ideal I of k[x0, . . . , xn], and since the codimension is assumed to
be one, Krull’s theorem implies that I is principal, i.e., I = (f) for some degree
d polynomial f ∈ k[x0, . . . , xn]. To give a concrete example, consider the case of
P2, and the curve
D0 = V (x3 + y3 + z3) ⊆ P2k
This is clearly integral, since the defining equation is irreducible. Similarly, we
can consider the subscheme
D1 = V (xyz)
This subscheme is reduced, but not irreducible: D has three irreducible compo-
nents V (x), V (y), V (z).
The main feature of divisors to talk about sums and differences of such
subschemes, thereby turning them into a group. This is illustrated in the
example above, by writing D1 = V (x)+V (y)+V (z). The sum here is completely
formal – it is an element in the free group on integral subschemes of codimension
one. Such a sum is by definition, a Weil divisor.
225
There is an equivalence relation defined on such objects, designed to capture
when two divisors belong to the same family. In the example D and D′ belong
to the same family, namely
D[s:t] = V (sxyz + t(x3 + y3 + z3))
More precisely, there is a closed subscheme D ⊆ P1 × P2 defined by the above
equation, so that the fiber over a closed point [s : t] ∈ P1 is exactly the curve
D[s:t] of P2k. Geometrically, we have a family of ‘moving divisors’, parameterized
over the projective line. We say the two divisors are linearly equivalent. The key
feature is that there is this morphism f : D → P1, and f−1([0 : 1]) = D0 and
f−1([1 : 0]) = D1. Moreover, the quotient
g =x3 + y3 + z3
xyz
defines a rational function on P2. This is the pullback of the rational function
s/t on P1: g = f ∗(s/t).
There is a second approach to divisors, which is perhaps more algebraic in
nature, namely Cartier divisors. The definition is motivated by the fact that
integral subschemes of codimension one are typically defined by a single equation
locally. Note that in the above example, the special properties of projective space
imply that D0 and D1 are defined globally by a single equation. It is not hard to
come up with examples of schemes with subschemes of codimension one that are
not. In fact, for most schemes, the concept of a ‘globally defined equation’ does
not make sense, since we do not have global coordinates to work with. However,
locally this concept makes sense: We can consider subschemes Y ⊆ X so that
the ideal sheaf IY is locally generated by a single element fi ∈ Ai, on some affine
covering X =⋃
SpecAi. In other words, the ideal sheaf IY is an invertible sheaf.
While Weil divisors are conceptual and more geometric, Cartier divisors do
have some advantages. For instance, they are very closely related to invertible
sheaves and line bundles, which in turn makes computations with them easier.
For instance, given a morphism f : X → Y , we would like to define a ‘pullback’
of a divisor on Y to a divisor on X – this turns out to only be possible for Cartier
divisors. There are also other settings where the special properties of Cartier
divisors are essential, for instance defining intersection products.
That being said, for a smooth scheme we will see that the various notions of
a divisor are equivalent, and there are natural ways to switch between the four
226
Chapter 14. Divisors and linear systems
categories in the diagram
Weil divisors line bundles
Cartier divisors invertible sheaves
The main interest in these concepts is of course that they give us tools to classify
varieties and schemes. So given X, perhaps as an abstract scheme, we can first
look for a divisor D on it. If the corresponding invertible sheaf L = OX(D) is
globally generated, we have a morphism to projective space f : X → Pn and
we can use this to study the geometry of X: We can ask about the fibers of f ,
whether it is an closed immersion, and if so, what the equations of the image is.
We will in this chapter assume that
All schemes are integral and noetherian.
Noetherianness is somewhat important, since we want to talk about the decom-
position of a closed subschemes into its irreducible components. The assumption
of integrality, especially irreducibility, is not essential for most of the statements,
but it makes the definitions more transparent, and more importantly, the proofs
considerably simpler (e.g., not having to worry about nilpotent elements allows
us to work with meromorphic functions as elements in a fraction field, rather
some more obscure localization). Still, we remark that the theory of Cartier
divisors work in a completely general setting, although there are some subtle
points one has to take into account. If you are curious about general background
on meromorphic functions on arbitrary schemes, see EGA IV4, section 20.
14.1 Weil divisors
In this section X will denote an integral, normal noetherian scheme. Recall that
this means that each local ring OX,x is an integral domain, which is integrally
closed in its function field K = K(X).
Definition 14.1. A prime divisor on X is a closed integral subscheme Y of
codimension 1. A Weil divisor on X is a finite formal sum
D =∑
niYi (14.1.1)
where ni ∈ Z and Yi are prime divisors.
We say D is effective if all the ni are non-negative in (14.1.1).
We call⋃i Yi the support of D.
We denote by Div(X) the group of Weil divisors; this is the free abelian
group on prime divisors.
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14.1. Weil divisors
If Y is a prime divisor on X and U ⊆ X is an open set, then Y ∩ U is
naturally a prime divisor on U . It follows that we obtain a presheaf U 7→ Div(U).
This is in fact a sheaf; it coincides with⊕x∈X,codim x=1
jx∗(Zx)
where jx : x → X is the inclusion of a point x of codimension 1 in X. We will
denote this sheaf by Div.
14.1.1 The divisor of a rational function
The main reason for the normality assumption is that if X is normal, then it
is regular in codimension one. This implies that for each point η ∈ X with
codim η = 1, each local ring OX,η (which has Krull dimension 1) is a discrete
valuation ring, with a corresponding valuation vY : K×Z. The concept of a
valuation is a generalization of the ‘order’ of a zero or a pole of a meromorphic
function in complex analysis. Intuitively, an element f ∈ K× has positive
valuation N if it vanishes to order N along Y , and negative valuation −N if it
has a pole of order N there.
To say what vY is explicitly, we define for an element f ∈ OX,η,
vY (f) = n
where n is the integer so that f ∈ mn −mn−1. In the function field, an element
f is represented by a fraction g/h and we define vY (f) = vY (g)− vY (h). (Check
that this is independent of the chosen representative). This means that OX,η =
v−1Y (Z≥0), and the maximal ideal m = v−1
Y (Z≥1).
Definition 14.2. If f ∈ K×, we define its divisor as
div (f) =∑
vY (f)Y
Divisors of the form div (f) are called principal divisors, and they form a subgroup
Div0(X) ⊆ Div(X).
This is well defined by the following lemma:
Lemma 14.3. Suppose that X is an integral normal noetherian scheme with
fraction field K and let f ∈ K. Then vY (f) = 0 for all but finitely many prime
divisors Y .
Proof. We first reduce to the case when X is affine. Let U = SpecA be an open
affine subset such that f |U ∈ Γ(U,OX). Since X is noetherian, the complement
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Chapter 14. Divisors and linear systems
Z = X−U is a closed subset of X which has finitely many irreducible components;
in particular, only finitely many prime divisors Y are supported in Z. So we
reduce to the affine case X = SpecA and f ∈ Γ(X,OX), by ignoring these
finitely many components. Then vY (f) ≥ 0 and vY (f) > 0 if and only if Y is
contained in V (f); and since V (f) has only finitely many irreducible components
of codimension 1, we are done.
Proposition 14.4. Let f ∈ K×, and let Y ⊆ X be a prime divisor with generic
point η. Then
1. vY (f) ≥ 0 if and only if f ∈ OX,η
2. vY (f) = 0 if and only if f ∈ O×X,η.
If X is projective, then vY (f) ≥ 0 for all Y if and only if vY (f) = 0 for all Y if
and only if f ∈ k∗.
Example 14.5. Consider X = A1k = Spec k[t] and let K = k(t). Here prime
divisors in X correspond to closed points p ∈ X. Let f = x2(x−1)x+1
. Then vp(f) = 0
for all Y except when p = 0,±1, where we have
v0(f) = 2, v1(f) = 1, v−1(f) = −1
Hence the divisor of f is 2V (t) + V (t− 1)− V (t+ 1).
Example 14.6. Consider X = P1k = Proj k[u, v] and let K be the fraction field.
Let f = u2−v2(u−2v)2
. On D(u) we use the coordinate t = v/u so that K = k(t). Here
f = 1−t2(1−2t)2
, and the only non-zero valuations are
v1(f) = 1, v−1(f) = 1, v1/2(f) = −2
The rational function f does not have any poles or zeroes at the point [0 : 1], so
the divisor is [1 : 1] + [1 : −1]− 2[2 : 1].
14.1.2 The divisor of a section of an invertible sheaf
Let L be an invertible sheaf and let s be a rational section (a section defined over
an open set). We can define a divisor div s as follows. For Y a prime divisor, let
y be the generic point. Let U ⊆ X be a neighbourhood of y such that there is a
trivialization φ : L|U → OX |U . On U , φ(s) defines a rational section of OX(U),
and hence an element of K. We define vY (s) = vY (f). It is not hard to see that
this is independent of U .
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14.1. Weil divisors
Example 14.7. Let L = O(2) on X = P1k = Proj k[u, v] and let K be the
fraction field. Let f = v2
u+v. On D(u) we use the coordinate t = v/u so that
K = k(t). Here f = t2
1+t, and the only non-zero valuations are
v0(f) = 2, v−1(f) = −1.
So on D(u) we get the divisor 2[1 : 0] − [1 : −1]. On D(v), we can use the
coordinate t = u/v, where f becomes f = tt+1
. The non-zero valuations are the
following:
vt=0(f) = 1, vt=−1(f) = −1.
So we get [0 : 1]− [−1 : 1] = [0 : 1]− [1 : −1]. On X = D(u) ∪D(v), we then
find the divisor is
2[1 : 0]− [1 : −1] + [0 : 1].
14.1.3 The class group
Definition 14.8. We define the class group of X as
Cl(X) = Div(X)/Div0(X)
Two Weil divisors D,D′ are linearly equivalent (written D ∼ D′) if they have
the same image in Cl(X), or equivalently, that D −D′ is principal.
Example 14.9. In fact, any divisor on A1k for an algebraically closed field k is
principal. Indeed, if D =∑n
i=1 ni[pi] where ni ∈ Z and pi ∈ k, then
f =n∏i=1
(t− pi)ni
is an element of k(t)× with div (f) = D. It follows that Cl(A1k) = 0 in this case.
14.1.4 Projective space
Proposition 14.10. Cl(Pnk) = Z.
Write Pnk = ProjR, R = k[x0, . . . , xn]. Prime divisors on Pnk correspond to
height one prime ideals in R, i.e., p = (g) where g is a homogeneous irreducible
polynomial. We can use this to define the degree of a divisor, by taking the
degrees of the corresponding polynomials:
Div(Pn)→ Z∑niV (gi) 7→
∑ni deg gi
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Chapter 14. Divisors and linear systems
If f ∈ K(Pnk) is a rational function, then f can be written as a quotient of two
homogeneous polynomials of the same degree. (Indeed, K = K(Pn) = K(U) =
K(R(x0)) and, so every rational non-zero function on Pn gives rise to an element
of F of degree 0, via the inclusion K(R(x0))× → K. Conversely, any fraction
in K of degree 0 gives an element in R(x0).) Now, if f ∈ K(Pnk) is a rational
function, we can write it as f =∏
i fnii where the fi are irreducible coprime
polynomials in R and ni ∈ Z. Let us first show that
div f =∑
ni[V (fi)]
If Y ⊆ X is a prime divisor, let y ∈ X be the generic point. Y = V (g) for
some irreducible polynomial g of degree d. For any other polynomial h of degree
d, the fraction g/h is a generator of myOX,y. We can write f = (g/h)rf ′ with
r = ni if fi divides g (and 0 if no fi divides g) and f ′ a rational function not
containing g in the numerator or the denominator. Hence vY (f) = r, and hence
(f) =∑ni[V (fi)]. Note that deg div f =
∑ni deg fi = 0, because f is a rational
function. It follows that the degree map descends to a map
deg : Cl(Pn)→ Z∑niV (gi) 7→
∑ni deg gi
We claim that this is an isomorphism. Note that degH = 1, where H = V (x0)
is a hyperplane, so deg is surjective. Now, any Z =∑ni[V (fi)] in the kernel
of deg, must have∑ni deg fi = 0. Consider the element f =
∏i f
nii , which
now defines an element of K. Using the formula for div f above, we see that
Z = div f , and hence Z is a principal divisor, and so deg is injective.
Example 14.11. Consider the curve X as in Figure , given by X = V (y2x−x3 − xz2) ⊆ P2. By the above discussion any two lines L,L′ in P2 are linearly
equivalent. Since X is integral, then there are well-defined restrictions L|X and
L′|X , which are linearly equivalent divisors on X. The figure below shows one
example where L|X = P +Q+R and L′ = 2S + T .
14.1.5 Weil divisors and invertible sheaves
Let D be a Weil divisor on X. We define the sheaf OX(D) as follows:
OX(D)(U) = f ∈ K|(div f +D)|U ≥ 0 ∪ 0= f ∈ K| vZ(f) ≥ −vZ(D), ∀Z ∈ U a point of codim 1 ∪ 0
This is a quasi-coherent sheaf on X. (We will see soon that it is invertible if and
only if D is a Cartier divisor).
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14.1. Weil divisors
Example 14.12. Let X be the projective line P1k = Proj k[x0, x1] over k and let
D = V (x0) = [1 : 0]. Let U0 = Spec k[x1/x0] = Spec k[t], U1 = Spec k[x0/x1] =
Spec k[s] be the standard covering of P1k (so s = t−1 on U0 ∩ U1). Note that the
point [1 : 0] does not lie in U0 = D+(x0). This means that a rational function
f ∈ K such that f +D is effective on U0 must be regular on U0, i.e.,
Γ(U0,OX(D)) = k[t]
On U1, we are looking at elements f ∈ k(s) having valuation ≥ −1 at s = 0.
This implies that
Γ(U1,OX(D)) = k[s]⊕ k[s]s−1
Now we think of elements in Γ(X,OX(D)) as pairs (f, g) with f, g ∈ K sections
of OX(D) over U0 and U1 respectively, so that f = g on U0 ∩ U1. Here f = f(t)
is a polynomial in t, and
g(s) = p(s) + q(s)s−1 = p(t−1) + q(t−1)t.
If f = g in k[t, t−1] we must deg p, q ≤ 1. This implies that
Γ(X,OX(D)) = k ⊕ kt.
14.1.6 The class group and unique factorization domains
The term ‘class group’ comes from algebraic number theory and its origins be
traced back Kummer’s work on Fermat’s last theorem. If A is a Dedekind domain
then Cl(SpecA) coincides with the class group of A, which measures how far A
is from being a unique factorization domain.
Proposition 14.13. Let A be a noetherian integral domain and let X = SpecA.
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Chapter 14. Divisors and linear systems
Then A is a UFD if and only if Cl(SpecA) = 0 and X is normal.
Proof. If A is a UFD, then it is integrally closed, and hence normal. Moreover,
A is a UFD if and only if every prime ideal p of height 1 is principal. We want
to show that this condition is equivalent to Cl(X) = 0.
⇒: If Y ⊆ X is a prime divisor, then Y = V (p) for p of height 1. Hence
Y = V (f) = div f for some f , and so Cl(X) = 0.
⇐: If Cl(X) = 0, let p be a prime of height 1, and let Y = V (p) ⊆ X. By
assumption, there is a f ∈ K× such that div (f) = Y . We want to show that
f ∈ A and that p = (f).
vY (f) = 1, so f ∈ Ap and f generates pAp. If q ⊆ A is any prime ideal not
equal to p, then Y ′ = V (q) is a prime divisor not equal to Y . vY ′(f) = 0, since
(f) = Y , and so f ∈ Aq. So in all, we have
f ∈⋂
q∈SpecA
Aq = A
where the equality comes from the fact that A is an integrally closed domain and
the ‘algebraic Hartog’s theorem’: A function which is regular in codimension 1
is regular (see Matsumura p. 124). Hence f ∈ A, and in fact f ∈ A ∩ pAp = p.
We show that f generates p: Take any f ∈ p. Then vY (g) ≥ 1 amd vY ′(g) ≥ 0
for all Y ′ 6= Y . It follows that vY ′(g/f) ≥ 0 for all prime divisors Y ′. Hence
g/f ∈ Aq for all q prime of height 1, and hence g/f ∈ A, by the above. It follows
that g ∈ fA and so p = fA is principal.
14.1.7 A useful exact sequence
Given a scheme X and an open subset U , the restriction of a prime divisor on
X is a prime divisor on U , so it is natural to ask how the two class groups are
related. The answer is given by the following theorem:
Theorem 14.14. Let X be a normal, integral scheme, let Z ⊆ X be a closed
subscheme and let U = X−Z. If Z1, . . . , Zr are the prime divisors corresponding
to the codimension 1 components of Z, then there is an exact sequence
r⊕i=1
ZZi → Cl(X)→ Cl(U)→ 0 (14.1.2)
where the map Cl(X)→ Cl(U) is defined by [Y ] 7→ [Y ∩ U ].
Here we regard each Zi as a subscheme of X with the reduced scheme structure
(cf. Proposition 8.37)
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14.2. Cartier divisors
Proof. If Y is a prime divisor on U , the closure in X is a prime divisor in X, so
the map is surjective.
We just need to check exactness in the middle. Suppose D is a prime divisor
which is principal on U . Then D|U = div f for some f ∈ K(U) = K = K(X).
Now div f is a divisor on X such that D|U = div (f)|U . Hence D − div (f) is
supported in X − U , and hence it is a linear combination of the Zi.
As a special case, we see that removing a codimension 2 subset does not
change the group of Weil divisors. So for instance Cl(A2 − 0) = Cl(A2).
Example 14.15. Consider the projective line P1k over a field k, and let p be a
point. We have the exact sequence
Z[p]→ Cl(P1)→ Cl(A1)→ 0
We saw that Cl(A1) = 0, so the map Z→ Cl(P1) is surjective. It is also injective:
If [nP ] = 0 in Cl(P1) for some n. Then nP = div f for some f ∈ k(P1), so if
div f |A1 = 0, we have f ∈ Γ(A1,O) = k∗. Hence f is constant, and so n = 0. It
follows that Cl(P1) = Z.
14.2 Cartier divisors
Let X be a noetherian integral scheme with function field K. Let KX denote
the constant sheaf with value K. The constant sheaf K ×X with value K× (the
group of non-zero elements of K) is a subsheaf, and contains O×X , the sheaf of
units in O×X .
Definition 14.16. A Cartier divisor D on X is an element of Γ(X,K ×X /O
×X).
This definition of a Cartier divisor is pretty obscure, and one rarely thinks of
a divisor in this way. We can shed some light on the definition by considering an
open cover Ui of X. The sheaf axiom sequence takes the following form
0→ Γ(X,K ×X /O
×X)→
∏i
Γ(Ui,K×X /O
×X)→
∏i,j
Γ(Uij,K×X /O
×X)
From this, we see that a Cartier divisor is therefore given as a set of sections of
K ×X /O
×X over the open sets Ui that agree on the overlaps Uij.
If we shrink the Ui we may assume that si are induced by a section fi of
K ×X over Ui, that is, an element of K×. To see what it means that two such
data Ui, fi agree on Uij we keep in mind that they are sections of K ×X /O
×X .
The condition si|Uij = sj|Uij translates into the statement that fi|Uijf−1j |Uij is a
section of O×X over Uij. In other words, there are units cij ∈ OX |Uij such that
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Chapter 14. Divisors and linear systems
fi = cijfj over Uij. In K× we have cij = fifj
, which implies that the cij satisfy
the ‘cocyle conditions’
cik = cijcjk; cji = c−1ij ; cii = 1
The idea here is that the fi give the local equations for a subscheme of X. On
an affine open set U where fi is regular, the zero-set V (fi) defines a codimension
1 closed subset of U . If both fj and fi are regular on U , their zero-sets must be
the same since they differ by a unit.
Definition 14.17. The pairs (Ui, fi) are called the local defining data or the
local equations for the divisor D (with respect to the covering Ui).
Such defining data are not unique: (Ui, fi)i∈I and (Vj, gj)i∈I give the
same Cartier divisor if fig−1j ∈ Γ(Ui ∩ Vj,O×X) for all i, j.
Now, the set of Cartier divisors naturally forms an abelian group: Given
D and D′ represented by the data (Ui, fi)i∈I and (Vi, gi)i∈I , we can define
D +D′ as the Cartier divisor associated to the data
(Ui ∩ Vj, figj)i,j
Moreover, the inverse −D will be defined as (Ui, f−1i )i∈I , and the identity is
defined by the data (Ui, fi)i∈I where fi ∈ Γ(Ui,O×X). We denote the group of
Cartier divisors by CaDiv(X) = Γ(X,KX/O×X).
Definition 14.18. We say that a Cartier divisor is principal if it is equal (as an
element of CaDiv(X)) to (X, f) where f ∈ K×.
The set of principal Cartier divisors form a subgroup of CaDiv(X), which
is typically much smaller than CaDiv(X). However, by definition, any Cartier
divisor is ‘locally principal’.
We define CaCl(X) to be the group of Cartier divisors modulo principal
divisors:
CaCl(X) = CaDiv(X)/(X, f)|f ∈ K×
We say that two Cartier divisors D,D′ are linearly equivalent, if D−D′ = (X, f)
for some principal divisor (X, f), or equivalently, [D] = [D′] in CaCl(X).
14.2.1 Cartier divisors and invertible sheaves
Let D be a Cartier divisor on a scheme X given by the data (Ui, fi). We can
associate to it an invertible sheaf, which we denote by OX(D). On Ui we take
the the subsheaf f−1i OUi ⊆ KUi of KX . This subsheaf is isomorphic to OUi and
has f−1i as a local generator. So over an affine subset U = SpecA ⊆ Ui, the
sheaf is the sheaf associated to f−1i A ⊆ K(A).
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14.2. Cartier divisors
On the intersection, Ui ∩ Uj, we have fi = cijfh where cij is an invertible
section of OUij . This means that f−1i OUij = f−1
j OUij as subsheaves of KUij . We
have therefore, constructed a sheaf on each Ui, and the elements coincide on
the intersections Uij. In order to be able to glue to a sheaf, there is a cocycle
condition that has to be satisfied. But since these sheaves are all subsheaves of
a fixed sheaf K , the gluing maps are actually identity maps, and the cocycle
condition is automatically satisfied. It follows that the sheaves f−1i OUi glue to
a sheaf OX(D) defined on all of X. It is by construction invertible, since it is
invertible on each Ui.
Explicitly, OX(D) is defined as the subsheaf of KX given by
Γ(V,OX(D)) = f ∈ K|fif ∈ Γ(Ui ∩ V,OX)∀i
Two different data (Ui, fi) and (Vj, gj) for the same divisor D give rise to the
same invertible sheaf. This is because over Ui ∩ Vj, we have fi = dijgj for some
sections dij ∈ OX(Ui ∩ Vj)×. This means that f−1i OUi∩Vj = g−1
i OUi∩Vj , and so
the sheaf is uniquely determined.
Moreover, by how we defined the sum D +D′ we have
Proposition 14.19. Let X be an integral noetherian scheme and let D and D′
be two Cartier divisors.
(i) OX(D +D′) ' OX(D)⊗ OX(D′)
(ii) OX(D) = OX(D′) if and only if D and D′ are linearly equivalent.
Proof. We can pick a covering Ui so that both D and D′ are both represented
by data (Ui, fi), (Ui, f′i). Then D +D′ is defined by (Ui, fif
′i). Locally, over Ui
the sheaf OX(D +D′) is defined as the subsheaf of K given by (fif′i)−1OUi =
f−1i f ′−1
i OUi . The tensor product is locally(!) given as f−1OUi ⊗ f−1OUi , which
is clearly isomorphic to f−1i f ′−1
i OUi via af−1i ⊗ bf ′i
−1 7→ abfi−1fj
−1.
For the second claim, it suffices (by point (i)) to show that OX(D) ' OX if
and only if D is a principal Cartier divisor. So suppose that OX(D) ⊆ KX is a
sub OX-module which is isomorphic to OX . Then the image of 1 in OX will be a
section of OX(D) which generates OX(D) everywhere. This means that (X, f)
is the local defining data, and D is a principal Cartier divisor.
Conversely, if D = (X, f), then OX(D) = f−1OX , and multiplication by f
gives an isomorphism OX(D) ' OX .
14.2.2 CaCl(X) and Pic(X)
By the item (ii) in Proposition 14.19. We see that the map CaDiv(X)→ Pic(X),
which sends D to the class of OX(D) in Pic(X) is additive and has the subgroup
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Chapter 14. Divisors and linear systems
of principal divisors CaDiv0(X) as its kernel. This means that the induced map
ρ : CaCl(X)→ Pic(X) is injective. In this section we will show that this map is
also surjective, so that CaCl(X) = Pic(X).
Proposition 14.20. When X is integral, the map ρ : CaCl(X)→ Pic(X) is an
isomorphism.
Proof. We need to show that ρ is surjective. It suffices to show that any invertible
OX-module L is a submodule of KX : If L ⊆ K , let Ui be a trivializing cover of
L and let gi be its local generators. Then we have L|Ui = giOUi ⊆ KUi and the giare rational functions on Ui. On Uij = Ui ∩ Uj, we have giOUij = L|Uij = gjOUij ,
and it follows that gi = cijgj for units cij ∈ O×Uij . Consequently, (Ui, g−1i ) forms
a set of local defining data for a Cartier divisor D, and of course we have
L = OX(D).
Let L be an invertible sheaf and consider the sheaf L⊗OX KX . Let Ui ⊆ X
be a set of open subsets such that L|Ui = OX |Ui . Note that the restriction of
L ⊗OX KX ' KX to each Ui is a constant sheaf. Since X is irreducible, any
sheaf whose restriction to opens in a covering is constant, is in fact a constant
sheaf, and therefore L⊗OX KX ' KX as sheaves on X. Now we can regard L
as a rank 1 subsheaf of KX using the composition L→ L⊗KX ' KX . Hence
we get a Cartier divisor D on X such that L = OX(D).
14.2.3 From Cartier to Weil
Let D be a Cartier divisor given by the data (Ui, fi). If Y is a prime divisor on
X, with generic point η, then since Ui is a cover, η lies in some Ui. We can then
define
vY (D) = vY (fi)
This is independent of the choice of Ui: If η ∈ Ui ∩ Uj, then fif−1j is an element
of O×X(Ui ∩ Uj), and so vY (fif−1j ) = 0, and hence vY (fi) = v(fj).
This gives a map φ : CaDiv(X)→ Div(X) defined by
φ(D) =∑Y
vY (D)Y
Proposition 14.21. φ is injective.
Indeed, if φ(D) = 0, then all vY (fi) = 0 for all i (so that ηY ∈ Ui). Then the
Hartogs’ lemma implies that fi ∈ O×(Ui), so that D = 0 (as a Cartier divisor).
Note that by the definition of (f), a principal Cartier divisor (X, f) gives rise
to a principal Weil divisor. It follows that we get an induced map CaCl(X)→Cl(X)
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14.2. Cartier divisors
Proposition 14.22. CaCl(X) = Div(X) if and only if X is locally factorial
(all the local rings OX,x are UFDs).
Proof. The inclusion CaDiv(X) ⊆ Div(X) is an isomorphism if and only if the
map of sheaves K ×X → Z1 is surjective, i.e., the map is surjective for every stalk.
However, at the stalk at x ∈ X, this map is simply
K× → Div(Spec OX,x)
which sends f to the sum∑vp(f)V (p) where p is a height 1 prime ideal of OX,x.
This is surjective if and only if every p ⊆ OX,x is a principal ideal. This happens
if and only if OX,x is a UFD.
Corollary 14.23. On a non-singular variety X, there is a bijective correspon-
dence between
(i) Line bundles
(ii) Invertible sheaves
(iii) Cartier divisors
(iv) Weil divisors
up to isomorphism.
From this, we get the following theorem:
Theorem 14.24. Let k be a field. Then Pic(An) = Cl(An) = CaCl(An) = 0.
Proof. The polynomial ring is a UFD.
14.2.4 Projective space
Let us take a closer look at the projective space Pnk over a field k. Write Pnk =
ProjR where R = k[x0, . . . , xn]. Consider the standard covering Ui = D+(xi) of
Pnk .
Let F (x0, . . . , xn) ∈ Rd denote a homogeneous polynomial of degree d. Then
the function F (x/xi) = F (x0xi, . . . , xi−1
xi, 1, xi+1
xi, . . . , xn
xi) defines a non-zero regular
function on Ui, and the collection
(Ui, F (xj/xi))
forms a Cartier divisor on X. Indeed, on the overlap Ui∩Uj we have the relation
F (x/xi) = (xj/xi)n F (x/xj)
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Chapter 14. Divisors and linear systems
and xj/xi is a regular, and invertible function. The corresponding invertible sheaf
is OPnk (d). Two homogeneous polynomials F,G of the same degree d give linearly
equivalent divisors, because the quotient F (x)/G(x) is a global rational function
on Pnk . In particular, this applies to F and xd0, and we get OPnk (n) = OPk(1)⊗n.
Corollary 14.25. On Pnk any invertible sheaf is isomorphic to some O(m).
In the case of P1 over a field k, the above result is not so difficult to prove
directly. Indeed, recall that P1 is obtained via gluing U0 = Spec k[s] and
U1 = Spec k[t] along U01 = Spec k[s, s−1] = Spec k[t, t−1] using the identification
t = s−1. So given an invertible sheaf L on P1, the restriction of it to each open
must be trivial (since Pic(A1k) = 0), we must have isomorphisms φi : L|Ui → OUi .
In particular, over U012 we obtain two isomorphisms φi|U01 : L|U01 → OU01 . In
particular, the map ψ : φ1φ−10 : OU01 → OU01 is an isomorphism. This is induced
by a map k[s, s−1]→ k[s, s−1] which is just multiplication by some polynomial
p(s, s−1). Note that p(s, s−1) cannot vanish at any point on U01 (otherwise ψ
would not be an isomorphism over the stalk at that point). Hence p(s, s−1) = sn
for some n ∈ Z. But that implies that L ' OP1(n).
14.3 Effective divisors
We say that a Cartier divisor D is effective (and write D ≥ 0) if D can be
represented by local data (Ui, fi) where the rational functions (which a priori
are just assumed to be sections of KX over Ui) actually lie in OX(Ui). It is
straightforward to verify that if this is true for one set of data, then it holds for
any other set as well.
We will write D ≥ D′ if D −D′ ≥ 0, i.e., the difference D −D′ is effective.
Note also that if D and D′ are both effective, then so is D + D′. This means
that Pic(X) is an ordered group.
Effective divisors are of paramount importance in algebraic geometry; they
carry lots of essential geometric information. With the set up above, D being
effective is equivalent to the statement that OX(−D) (regarded as a subsheaf of
KX is contained in OX . The OX-module OX(−D) is a coherent sheaf of ideals
which is locally generated by one element (in other words, it is locally principal).
We will usually denote the corresponding closed subscheme also by D (this is a
somewhat bad example of abusing the notation, but it has its advantages).
The inclusion OX(−D) ⊆ OX induces an exact sequence
0→ OX(−D)→ OX → OD → 0
where the right hand side can be interpreted both as the cokernel of the left-most
map and as the structure sheaf of the subscheme associated to D. Moreover, the
239
14.3. Effective divisors
support of the OD (i.e., the set of points x ∈ X, such that OD,x 6= 0) coincides
with underlying topological space of D.
So effective divisors give rise to closed subschemes of codimension one. The
converse is also true: If D is a closed subscheme which is locally defined by
one equation (which is not a zero-divisor), then we can find, for any x ∈ X,
an open affine neighbourhood Ux and an element fx ∈ A = Γ(Ux,OX) so
that D = V (fx) = SpecA/(fxA). Two such elements fx, f′x generate the same
principal ideal, so there must be a relation fx = cxf′x for some cx ∈ A×. From this,
we see that (Ux, fx) form the defining data for a Cartier divisor, which we will
also denote by D. Indeed, on an overlap Uxy = Ux∩Uy we have fx|Uxy = cxyfy|Uxyfor a section cxy of O×Uxy . We have therefore proved the following theorem:
Theorem 14.26. Let X be an noetherian, integral scheme. Then there is a
one-to-one correspondence between closed subschemes D ⊆ X locally defined by
a principal ideal, and effective Cartier divisors on X.
The inclusion OX(−D) ⊆ OX can be dualized, i.e., we apply the functor
HomOX (−,OX), and obtain a map α : OX → OX(D). Such a map is uniquely
determined by its value on 1, i.e., the global section σ = α(1) ∈ Γ(X,OX(D)).
Conversely, given a global section Γ(X,OX(D)), we dually a map OX(−D)→OX . When X is integral, then this gives us a divisor, which is of course the
effective divisor D. We have therefore proved:
Theorem 14.27. Let X be an noetherian, integral scheme and let D be a Cartier
divisor on X. Then D is effective if and only if OX(D) has a non-zero global
section. For each section σ, we get a divisor denoted by div σ. Two such sections
σ, σ′ give rise to the same divisor if and only if σ = cσ′ where c ∈ Γ(X,OX)×.
14.3.1 Linear systems
Definition 14.28. The set of effective divisors D′ linearly equivalent to D is
denoted by |D|. This is called the complete linear system of D.
The name ‘linear system’ comes from the special case when X is a projective
variety X over a field k (thus X is integral, separated of finite type over k). In
this case, we have Γ(X,OX)× = k×, and the previous discussion shows that the
linear system |D| is given by
|D| = D′|D′ ≥ 0 and D′ ∼ D= (Γ(X,OX(D))− 0) /k×
= PΓ(X,OX(D))
So the linear system |D| is (as a set) a projective space. When X is projective,
the cohomology groups H0(X,OX(D)) is a finite dimensional (we will prove
240
Chapter 14. Divisors and linear systems
this fact in Chapter 17), so the set of divisors D′ linearly equivalent to D is
parameterized by a projective space Pnk .
Definition 14.29. A linear system of divisors is a linear subspace of a complete
linear system |D|.
Example 14.30. Consider the case X = Pnk and D = dH, where H is the
hyperplane divisor (so H is a Cartier divisor with OX(H) ' OX(1). In this case
the linear system of D associated to OX(dH) is given by
|D| = PRd ' PN
where N =(n+dd
)− 1. The points of this projective space correspond to degree
d hypersurfaces.
14.4 Examples
14.4.1 The quadric surface
Let k be a field, and let Q = P1k×P1
k. Recall that Q embeds as a quadric surface
in P3k via the Segre embedding:
So we can view Q both as a fiber product P1×P1 and the quadric V (xy−zw) ⊆ P3.
Weil divisors on Q
Since Q is a product of two P1s there are natural ways of constructing Weil
divisors on Q from those on P1. For instance, we can let
L1 = [0 : 1]× P1 ⊆ Q,
which is a prime divisor on Q corresponding to the ‘vertical fiber’ of Q. Similarly,
L2 = P1 × [0 : 1] is a Weil divisor on Q.
ZL1 ⊕ ZL2 → Cl(Q)→ Cl(Q− L1 − L2)→ 0
241
14.4. Examples
We claim that the first map is injective, and in fact that
Cl(Q) = ZL1 ⊕ ZL2.
For this, we need to analyse the two divisors L1, L2 a bit closer.
Cartier divisors on Q
To study Cartier divisors on Q, we first need a covering. Q is covered by four
affines
U00 = Spec k[x, y] U10 = Spec k[x−1, y]
U01 = Spec k[x, y−1] U11 = Spec k[x−1, y−1]
Consider P1k = W0 ∩ W1, where W0 = Spec k[t],W1 = Spec k[t−1]. The first
projection p1 : Q→ P1k is induced by the ring maps
k[t]→ k[x, y] k[t−1]→ k[x−1, y]
t 7→ x t−1 7→ x−1
k[t−1]→ k[x, y−1]; k[t−1]→ k[x−1, y−1];
t 7→ x t−1 7→ x−1
Let p = [0 : 1] be the Weil divisor on P1. The Cartier data is given by
(W0, t), (W1, 1). The pullback D = p∗1(p) is a Cartier divisor on Q, corresponding
to the Weil divisor [0 : 1]×P1. Writing K = k(x, y), the Cartier data is given by
(U00, x), (U10, 1)
(U01, x), (U11, 1)
Let L1 = [0 : 1] × P1k and L2 = P1
k × [0 : 1]. Consider the restriction of D to
L2. L2 is covered by the two open subsets V0 = U00 ∩ L2 = Spec k[x, y]/y, V1 =
U10 ∩ L2 = Spec k[x−1, y]/y. In terms of these opens, D|L2 has Cartier data
(V0, x), (V1, 1)
obtained by restricting the data above. In particular, identifying L2 ' P1, we
see that OQ(D)|L1 ' OP1(1). In particular, since Cl(P1) = Z, no multiple nD is
equivalent to 0 in Cl(Q) – if that were the case, we would have OQ(nD) ' OQ,
and hence OQ(nD)|L2 ' OQ|L2 ' OP1 , a contradiction.
In all, this shows that the first map in the exact sequence (14.4.1) is injec-
tive. We also have that Cl(Q − L1 − L2) = 0: Indeed Q − L1 − L2 = U11 =
242
Chapter 14. Divisors and linear systems
Spec k[x−1, y−1] which is the Spec of a UFD. Hence
Cl(Q) ' ZL1 ⊕ ZL2
If D is a divisor on Q, D ∼ aL1 + bL2 and we call (a, b) the ‘type’ of D.
The canonical divisor of Q
We saw in the previous section that Cl(Q) ' Z2. Thus it makes sense to ask how
to represent the canonical divisor of Q in terms of L1, L2. Since Q = P1 × P1
and KP1 = −2p, it should perhaps not come as a big surprise that
KQ = −2L1 − 2L2
In fact, ΩQ = p∗1ΩP1 ⊕ p∗2ΩP1 . So taking∧2, we get
ωQ =2∧
(p∗1ΩP1 ⊕ p∗2ΩP1)
= p∗1ΩP1 ⊗ p∗2ΩP1
= OQ(−2, 0)⊗ OQ(0,−2) = OQ(−2,−2)
A second way of seeing this, is to use the fact that Q is embedded by the Segre
embedding Q ⊆ P3 as a smooth quadric surface. Then the Adjunction formula
of Proposition 11.43 on page 194 tells us that
ΩQ = ωP3 ⊗ OP3(Q)|Q = OP3(−4)⊗ OP3(2)|Q = OQ(−2)
This gives the same answer as before, since OP3(1)|Q = OQ(L1 + L2).
14.4.2 The quadric cone
Let X = SpecR where R = k[x, y, z]/(xy − z2), and k has characteristic 6= 2.
Let Z = V (x, z) be the closed subscheme corresponding to the line x = z = 0.Note that Z ' Spec k[x, y, z]/(xy−z2, x, z) = Spec k[y] is integral of codimension
1.
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14.4. Examples
A singular quadric surface
Note that X −Z = Spec k[x, y, y−1, z]/(xy− z2) = Spec k[y]y[t, u]/(t− u2) =
Spec k[y]y[u] which is a a UFD. It follows that Cl(X − Z) = 0. Recall now the
sequence
Z→ Cl(X)→ Cl(X − Z)→ 0
where the first map sends 1 to [Z]. Hence Cl(X) is generated by [Z].
We first show that 2Z = 0 in Cl(X). This is because we can consider the
divisor of x. This can only be non-zero along Z. The valuation here is 2: The
local ring is
(k[x, y, z]/(xy − z2))(x,z)
since y is invertible here, we see that x ∈ (z2) and that z is the uniformizer.
Now we show that Z is not a principal divisor. It suffices to prove that this
is not principal in Spec OX,x where x ∈ X is the singular point of X. The local
ring here is
(k[x, y, z]/(xy − z2))(x,y,z)
In this ring p = (x, z) is a height 1 prime ideal, but it is not principal: Let
m ⊆ OX,x be the maximal ideal. Note that x, y ∈ m, since x, y are not units.
However, we can compute that the vector space m/m2 (which is the Zariski
cotangent space at x) is 3-dimensional, spanned by x, y, z. Then I = (y, z) ⊆ m,
and x, y give a 2-dimensional subspace I/m2 of m/m2. Hence, since x and y
are linearly independent here, there couldn’t be an element f ∈ OX,x for which
x = af, y = bf . Hence
Cl(X) = Z/2.
Note that X − 0, 0, 0 is factorial. Hence removing a codimension 2 subset
has an effect on CaCl(X). Recall however, that the class group of Weil divisors
Cl(X) stays unchanged under removing a codimension 2 subset.
Projective quadric cone
Let X = ProjR where R = k[x, y, z, w]/(xy − z2). Let H = V (w) be the
hyperplane determined by w. We have
0→ ZH → Cl(X)→ Cl(X −H)→ 0
(Here H is an ample divisor, hence non-torsion in Cl(X), so the first map is
injective). X −H is isomorphic to the affine quadric cone from before, hence
Cl(X −H) = Z/2. Using this sequence, we see that Cl(X) = Z, generated by12H.
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Chapter 14. Divisors and linear systems
14.4.3 Quadric hypersurfaces in higher dimension
Theorem 14.31 (Nagata’s lemma). Let A be a noetherian integral domain, and
let x ∈ A− 0. Suppose that (x) is prime, and that Ax is a UFD. Then A is a
UFD.
Proof. We first show that A is normal. Of course Ax is normal, being a UFD.
So if t ∈ K(A) is integral in A, it lies in Ax. We need to check that if a/xn ∈ Axis integral over A and x 6 |a, then n = 0. If we have an integral relation
(a/xn)N + b1(a/xn)N−1 + · · ·+ bN = 0
Multiplying by xnN give get aN ∈ xA, so x|a, because A is an integral domain.
This shows that the divisor D = div x is an effective divisor and so there is an
exact sequence
ZD → Cl(SpecA)→ Cl(SpecAx) = 0→ 0
The image of the left-most map is 0, so Cl(A) = 0, and so A is a UFD.
Let A = k[x1, . . . , xn, y, z]/(x21 + · · · + x2
m − yz). We will prove that A is a
UFD for m ≥ 3. A is a domain, since the defining ideal is prime. Apply Nagata’s
lemma with the element y:
Ay = k[x1, . . . , xn, y, y−1, z]/(y−1(x2
1 + · · ·+ x2m)− z) ' k[x1, . . . , xn, y, y
−1, z]
which is a UFD. We show that y is prime: Taking the quotient we get
A/y = k[x1, . . . , xn, x]/(x21 + · · ·+ x2
m)
which is an integral domain, because x21 + . . . , x2
m is irreducible (for m ≥ 3).
Note that for m = 2, we get the quadric cone, which we have seen is not a
UFD.
Applying a change of variables, we find the following:
Proposition 14.32. Let k be a field containing√−1 and let X = V (x2
0 + · · ·+x2m) ⊆ An+1
k = Spec k[x0, . . . , xn].
1. m = 2, Cl(X) = Z/2
2. m = 3, Cl(X) = Z
3. m ≥ 4, Cl(X) = 0
Proposition 14.33. Let X = V (x20 + · · ·+ x2
m) ⊆ Pn = Proj k[x0, . . . , xn].
1. m = 2, Cl(X) = Z
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14.4. Examples
2. m = 3, Cl(X) = Z2
3. m ≥ 4, Cl(X) = Z
14.4.4 Hirzebruch surfaces
Let r ≥ 0 be an integer and consider the scheme X which is glued together by
the four affine scheme charts
U00 = Spec k[x, y]
U01 = Spec k[x, y−1]
U10 = Spec k[x−1, xry]
U11 = Spec k[x−1, x−ry−1]
This is a non-singular, integral 2-dimensional scheme over k. When k = C, this
are the so-called r-th Hirzebruch surface. In many ways, these behave as the
’mobius strips’ in algebraic geometry.
Note in particular, when r = 0, we get P1k × P1
k.
Divisors
Let us define two divisors D1, D2 by the following Cartier divisors:
Writing K = k(x, y), the Cartier data is given by
D1 =
[(U00, x), (U01, x)
(U10, 1), (U11, 1)
], D2 =
[(U00, y), (U01, 1)
(U10, y), (U11, 1)
]We will show that D1, D2 generate Pic(X). Note that both of these divisors are
effective, since they are defined by rational functions which are regular on the
Uij . In fact D1 ' P1, since it is glued together by V0 = U00∩x = 0 = Spec k[y]
and V1 = U01 ∩ x = 0 = Spec k[y−1]. Moreover, D2 restricted to D1 is given
by the Cartier divisor (V0, y), (V1, 1), which corresponds to a point on D1. Hence
OD1(D2) ' OP1(1). This shows that D2 is non-torsion in Cl(X). A similar
argument shows that D1 is non-torsion in Cl(X).
Let D′1 be the Cartier divisor
D′1 =
[(U00, 1), (U01, 1)
(U10, x−1), (U11, x
−1)
],
We can compute that D′1|D1 the divisor (V0, 1), (V1, 1) which is principal. In fact,
div x = D1 −D′1
246
Chapter 14. Divisors and linear systems
So, D1 = D′1 in Cl(X). This shows that D1 and D2 are independent in Cl(X) –
D′1 restricts to 0 on D1 by O(1) on D2.
Now let U = X −D1 −D2. This is isomorphic to U11 which is the spectrum
of k[x−1, x−ry−1] which is a UFD. The exact sequence
ZD1 ⊕ ZD2 → Cl(X)→ 0
and the previous analysis shows that Cl(X) = ZD1 ⊕ ZD2.
Sheaf cohomology
We want to compute H i(X,OX(D)) for a divisor D = aD1 + bD2. As usual, we
utilize a Cech complex.
We want to mimick the computation for Pn. In that proof, the polynomial
ring k[x0, . . . , xn] played an important role – the groups in the Cech complex
corresponded to degree 0 localizations of it. For X there is no such Z-graded
ring lying around, but we can get by by introducing the bigraded ring
R = k[x0, x1, y0, y1]
where the degrees of the variables are defined by
deg x0 = (1, 0), deg x1 = (1, 0), deg x2 = (0, 1), deg x3 = (−r, 1).
By identifying x = x1x0
, y =xr0y1y0
, we find that the Cech complex of Uij can be
writtenR[x0x1] R[x0x1y0]
⊕ ⊕R[x0x1] R[x0x1y0]
⊕ → ⊕ → R[x0x1y0y1]
R[x0x1] R[x0x1y0]
⊕ ⊕R[x0x1] R[x0x1y0]
where the bracket means that we take the (0, 0)-part of the localization. So for
instance,
R[x0y0] = R
[x1
x0
,xr0y1
y0
]As in the Pn case, we now have a bigraded isomorphism⊕
a,b∈Z
H0(X,OX(aD1 + bD2)) '⋂i,j
R[xiyj ] = R
In particular, H0(X,OX(aD1 + bD2)) can be identified with polynomials of
247
14.4. Examples
bidegree (a, b) in R. So for example H0(X,O(D1)) corresponds to degree (1, 0)-
polynomials, e.g., polynomials in x0, x1. These two sections corresponds to the
sections 1, x above. Similarly, H0(X,O(D2)) is 1-dimensional.
Perhaps the most interesting divisor is E = D2 − rD1,which is effective.
This has H0(X,OX(nE)) = k for every n ≥ 0, by E is not the trivial divisor.
Therefore E couldn’t possibly by globally generated. In fact, E ' P1 and
OX(E)|E ' OP1(−r)
Maps to projective space
Note that OX(D1) is globally generated by the sections x0, x1. These define a
map
f : X → P1
This is an example of a P1-fibration – the closed fibers here are all isomorphic to
P1. In fact, one can show that any such fibration over P1 is isomorphic to this
example for some choice of r.
248
Chapter 15
Two applications of cohomology
15.1 Vector bundles on the projective line
As an instructive and not too complicated example of how cohomology groups
can give important results, we give a proof that all locally free sheaves on the
projective line P1k decompose as direct sums of invertible sheaves, and as all
invertible sheaves are of the form OP1k(n) for some integer n, we have the following
theorem:
Theorem 15.1. Assume that k is a field and that E is a locally free sheaf of
rank r on the projective line P1k. Then there is an isomorphism
E ' OP1k(α1)⊕ · · · ⊕ OP1
k(αr),
where the αi are integers, uniquely defined up to ordering.
We need some preperatory results about zeros of sections of locally free
sheaves om curves. Assume for a moment that C is a non-singular curve and
that E is locally free sheaf of finite rank on C. Let σ : OC → E be an injective
map of sheaves on C; such maps correspond bijectively to sections of E, that we
as well denote by σ (the correspondence is σ ↔ σ(1)). The dualized the map σ∨,
sits in an exact sequence
E∗ // OC// OZ(σ)
// 0 (15.1.1)
where Z(σ) is a closed subscheme of C. Generically σ is injective and therefore
σ∨ is generically surjective, and hence Z is not equal to the whole curve C.
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15.1. Vector bundles on the projective line
It follows that Z(σ) is finite C being a curve. The scheme Z(σ) is called the
zero-scheme of φ.
As C is non-singular, any non-zero coherent ideal I ⊆ OC is an invertible
sheaf. The ideal sheaf of Z(σ) – or the image of σ∨, if you want – is therefore an
invertible sheaf Lσ, and we may tensor the sequence (15.1.1) by Lσ−1 to obtain
the following factorization of σ∨
E∗⊗ Lσ−1 σZ // // OC // Lσ
−1.
For later references we formulate this as a proposition:
Proposition 15.2. Let C be a non-singular curve over a field k and E a locally
free sheaf of finite rank on C. Assume that σ is a non-zero section of E. Then
there is a finite subscheme Zσ in C and a surjective map E∗⊗ Lσ−1 σZ // // OC
where Ls denotes the invertible sheaf being the ideal of Z(σ), and a factorization
σ∨ = (ι⊗ Lσ−1) σZ where ι denotes the inclusion of the ideal Lσ in OC.
Proof of the theorem. The first step in the proof relies on two fundamental
facts about locally free sheaves of finite rank on projective spaces. Firstly,
sufficiently high positive twists have global sections; stated slightly differently
Γ(P1k, E(n)) 6= 0 if n 0 (this property locally free sheaves of finite rank share
with all coherent sheaves).
Secondly, sufficiently negative twists do not have global sections, i.e., it holds
true that Γ(P1k, E(−n)) = 0 for n >> 0. The third salient point is that any
invertible sheaf on the projective line is isomorphic to OP1k(α) for some integer α,
in other words, the theorem is true in rank one. The proof goes by induction on
the rank of E, and this will be the start of the induction.
We now let n0 be the greatest integer such that Γ(P1k, E(−n0)) 6= 0, and
after having replaced E by E(−n0), we may thus assume that Γ(P1k, E) 6= 0, but
Γ(P1k, E(−α)) = 0 for all α > 0. Let σ be a global section of E (it is called a
minimal section).
Lemma 15.3. The minimal section σ does not vanish anywhere.
Proof. Assume that Zσ 6= 0; then its sheaf of ideals equals OP1k(−α) for some
α > 0, and by (15.2) there is a surjective map E∗(α) → OP1k, whose dual is a
section of E(−α). This contradicts the fact that σ is a minimal section.
To prove the theorem we proceed as announced by induction on the rank r
of E, the r = 1 being taken care of by the fact that every invertible sheaf on
P1k is isomorphic to OP1
k(α) for some α ∈ Z. Choose a minimal section σ of E.
Then there is an exact sequence
0 // F // E∗ // OP1k
// 0 (15.1.2)
250
Chapter 15. Two applications of cohomology
where the kernel F is locally free of rank r−1. By induction F '⊕
1≤i≤r−1 OP1k(αi).
Lemma 15.4. H1(P1k, F ) = 0
Proof. Taking the dual of (15.1.2) and twisting by O(−1) gives:
0 // OP1k(−1) // E(−1) // F ∗(−1) // 0
Then the long exact sequence of cohomology shows H0(F ∗(−1)) = 0. But then
also H0(F ∗(−2)) = 0 = H0(F ∗ΩP1), so H1(F ) = 0 by Serre duality. (Note that
we only need Serre duality for a direct sum of invertible sheaves of the form
OP1k(αi) in this proof).
Since H1(P1k, F ) = 0, the long exact sequence associated to 15.1.2 shows that
we have a surjection
Γ(P1k, E
∗) // Γ(P1k,OP1
k) // 0
and the section 1 can be lifted to a section τ of E∗. This is translates into the
diagram
0 // F // E∗ // OP1k
// 0
OP1k
τ
OO
=
==
It follows that the sequence (15.1.2) is split, and as a consequence we have
E∗ ' OP1k⊕ F,
that is E ' OP1k⊕⊕
1≤i≤r−1 OP1k(−αi) and the proof is complete.
15.2 Non-split vector bundles on Pn
A locally free sheaf is said to be split if it is isomorphic to a direct sum of
invertible sheaves. For instance, we showed before that any locally free sheaf
on on P1k is split. Let us give one example of a non-split locally free sheaf. We
consider again the sheaf E of rank n on Pnk sititing in the exact sequence
0→ E → On+1Pnk→ OPnk (1)→ 0.
Let us show that E is not split, i.e., E is not isomorphic to a direct sum of
invertible sheaves. Since Pic(Pnk) is generated by the class of O(1), this would
mean that E ' OPnk (a1)⊕ · · ·OPnk (an) for some integers a1, . . . an.
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15.2. Non-split vector bundles on Pn
Recall that for n ≥ 2, we have H1(Pnk ,O(m)) = 0 for any m ∈ Z. So if
we could show that H1(Pnk ,E ) 6= 0, we would have a contradiction. Actually,
H1(Pnk ,E ) = 0, but we can instead consider E (−1), which fits into
0→ E (−1)→ On+1Pnk
(−1)→ OPnk → 0.
Taking the long exact sequence in cohomology, we get
· · · → H0(O(−1)n+1)→ H0(OPnk )→ H1(E )→ H1(O(−1)n+1)→ · · ·
Here the two outer cohomology groups are zero, since O(−1) does not have any
global sections, and H1(Pnk ,OPnk (−1)) = 0. Hence, by exactness, we find that
H1(E (−1)) ' H0(OPnk ) = k. This implies that E (−1), and hence E cannot be a
sum of invertible sheaves, and we are done.
However, coming up with examples of non-split vector bundles of low rank
is a notoriously difficult problem, even for projective space. In fact, a famous
conjecture of Hartshorne says that any rank 2 vector bundle on Pn for n ≥ 5 is
split. (On P4 this statement does not hold, as shown by the so-called Horrocks–
Mumford bundle). This is related to the conjecture that any smooth codimension
2 subvariety of Pn is a complete intersection.
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Maps to projective space
Given a scheme X it is natural to ask when there is a morphism to a projective
space
f : X → Pn,
or when there is a closed immersion X → Pn. Given such a morphism, we get
geometric information about X using this map, e.g., using the fibers f−1(y) or
pulling back sheaves from Pn.
The corresponding question for An has already been answered. Morphisms
X → An are in one-to-one correspondence with elements of Γ(X,OX)n. In fancy
terms, An represents the functor
AffSch → Sets
X 7→ Γ(X,OX)
So which functor does projective space represent? Intuitively we would like
to associate to each morphism f : X → Pn to a set of data on X. As we have
seen, there is not so much information Γ(X,OX) in the context of projective
schemes. However, we do have something canonical associated to Pn, namely
sections of OPn(1). Taking the usual global sections x0, . . . , xn on OPn(1), we
can pull back via the map f ∗ and get n+ 1 sections si = f ∗xi of the invertible
sheaf f ∗O(1). Note that there is no point of Pn where the xi simultaneously
vanish. More precisely, for every y ∈ Pn, the stalk OPn(1)y is generated by one
of the xi (as an OPn-module). So my the properties of the pullback, we see that
the same statement holds for the si and X.
In this chapter we will se that there is a way to reverse this process, i.e., that
from a given invertible sheaf L and n+ 1 global sections si ∈ Γ(X,L) with the
253
16.1. Globally generated sheaves
above property, we can uniquely reconstruct a morphism f : X → Pn so that
f ∗OPn(1) = L and f ∗xi = si.
16.1 Globally generated sheaves
Definition 16.1. Let X be a scheme and let F be an OX-module. We say F is
globally generated (or generated by global sections) if there is a family of sections
si ∈ F(x), i ∈ I, such that the images of sx generate Fx as an OX,x-module.
Equivalently, F is globally generated if there is a surjection
OIX → F → 0
for some index set I.
Let us consider a few examples:
Example 16.2. On an affine scheme any quasi-coherent sheaf is globally gener-
ated. Indeed, if X = SpecA, F = M , for some A-module M , then picking any
presentation AI →M → 0 for M shows that F is globally generated.
Example 16.3. X = ProjR. Then F = O(1) is globally generated if R
is generated in degree 1. Indeed, if this is the case, we have a surjection
R ⊗ R1 → R(1). Taking ∼, we get OX ⊗ R1 → OX(1) → 0, which shows that
O(1) is globally generated.
Example 16.4. The tangent bundle of Pn is globally generated. To see why, we
recall the Euler sequence on Pn:
0→ O → O(1)n+1 → TPn → 0
Here the leftmost map is given by multiplication by the vector (x0, . . . , xn), and
the rightmost sends xi to ∂∂xi
. So in fact, even TPn(−1) is globally generated.
16.1.1 Pullbacks and pushforwards
Let us recall a few things about pulling back sheaves and sections. If f : X → Y
is a morphism of schemes, and G is an OY -module, we define f ∗G as the tensor
product f−1G ⊗f−1OY OX . The stalk of f ∗G at a point x ∈ X is isomorphic to
Gf(x)⊗OY,f(x)OX,x .
If G is an invertible sheaf, the pullback f ∗G is also invertible. Indeed, if
U ⊆ X and V =⊆ Y are open affine subsets such that f(U) ⊆ V , and L|U ' OU ,
then f ∗L|V ' OV (recall that a morphism pulls back the structure sheaf of the
target gives to the structure sheaf of the source).
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Chapter 16. Maps to projective space
We can pull back sections s ∈ Γ(Y,G) as well. Let us consider the affine
case first, where U = SpecB and V = SpecA. In this case, G = M for some
A-module, and f ∗G = M ⊗A B. So finally, if s ∈ Γ(U,G) = M , then the pullback
is given by f ∗(s) = s⊗ 1 ∈M ⊗A B.
Proposition 16.5. Let f : X → Y be a morphism of schemes. If F is an
OY -module which is globally generated by sections sii∈I , then the pullbacks
ti = f ∗(si) generate f ∗F .
Proof. The si determine a surjection OIY → F → 0, and the pullback OI
X →f ∗F → 0 (which is surjective since f ∗ is right-exact) is the map determined by
the ti. Hence f ∗F is globally generated by the ti.
Remark 16.6. For the pushforward, f∗F is typically not globally generated even
when F is the structure sheaf. For example, if f : P1 → P1 is the map induced
by k[u2, v2] ⊆ k[u, v], then f∗OP1 ' OP1 ⊕ OP1(−1). The latter is not globally
generated: If it were, we would in particular get a surjection OI → O(−1)→ 0,
however O(−1) is clearly not globally generated, since it has no global sections
at all.
16.2 Morphisms to projective space
16.2.1 Example: The twisted cubic
Consider the ring R = k[u3, u2v, uv2, v3] with the grading such that the mono-
mials have degree 1. We have seen that ProjR = P1k (since R is the Veronese
subring k[u, v](3) of k[u, v]). R is isomorphic to the ring
k[x0, x1, x2, x3]/(x21 − x0x2, x0x3 − x1x2, x
22 − x1x3).
This shows that ProjR embeds as a cubic curve in P3k.
Intuitively, we would like to say that this morphism f : X → P3 is given by
something like
[u, v] 7→ [u3, u2v, uv2, v3].
However, here u, v are not regular functions (they however define rational func-
tions), so to define an actual morphism of schemes, we need to be a little bit more
rigorous. Note, however, that on D+(u) = SpecR(u) = Spec k[x1x0, x2x0, x3x0
], the
ratio t = vu
is a regular section Γ(D+(u),O). Moreover, the ring homomorphism
φ : k[x1x0, x2x0, x3x0
] → k[t]x1x0
7→ tx2x0
7→ t2
x3x0
7→ t2
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16.2. Morphisms to projective space
gives a morphism of affine schemes
D+(u)→ D+(x0) ⊆ P3k
Note that this corresponds to the usual affine twisted cubic A1 → A3, t 7→(t, t2, t3). Similarly, on D+(v), the ratio s = u
vdefines a morphism D+(x0) →
D+(x3). On the overlaps, these maps are compatible with the standard gluing
construction of P1k, and so we get finally a morphism P1
k → P3k.
16.2.2 Morphisms and globally generated invertible sheaves
The morale of the previous example is the morphism is not specified using a set
of regular functions, but rather, sections of an invertible sheaf. Using the same
type of gluing argument, we will prove the following general result:
Theorem 16.7. Let X be a scheme over a ring A.
(i) If f : X → PnA is a morphism over A, then f ∗(O(1)) is an invertible sheaf
which is globally generated by the sections si = f ∗(xi) for i = 0, . . . , n.
(ii) If L is an invertible sheaf on X which is globally generated by n+ 1 sections
s0, . . . , sn, then there exist a unique morphism f : X → PnA such that
L ' f ∗(O(1)) and si = f ∗(xi) under this isomorphism.
Proof. (i) This is a consequence of Proposition 16.5.
(ii) Given n+ 1 sections s0, . . . , sn of an invertible sheaf L, we will construct
a morphism to PnA, such that si is the pullback of xi. Let
Xi = Xsi = x ∈ X|si(x) 6= 0.
(Recall that s(x) is the residue class of s in Lx/mxLx.) As we have seen, this is
the complement of the support of the section si, and is therefore an open set.
By the assumption that these sections generate L, we see that X is covered by
the sets Xi.
Now, we have given local isomorphisms φi : L|Ui → OX |Ui , which we can use
to view the fractionsjsi
can be regarded as a regular section on Xi, that is, an
element of Γ(Xi,OX).
Write R = A[x0, . . . , xn] and consider the ring homomorphism given by
R(xi) → Γ(Xi,OXi)xjxi7→ sj
si
By the correspondence between ring homomorphisms and maps into affine
schemes, we obtain a morphism of schemes fi : Xi → D+(xi). It is not so hard
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Chapter 16. Maps to projective space
to see that on the overlaps Xi ∩Xj, two maps are given by the same ratios of
sections, and so they glue to a morphism f : X → Pn. By construction of the
map f , we have f ∗xi = si.
Abusing notation, we will refer to a morphism φ : X → PnA as ‘given by the
data (L, s0, . . . , sn) and write
X → PnAx 7→ [s0(x) : · · · : sn(x)]
One should still keep in mind that the sections si are sections of L, rather than
regular functions.
Given a scheme X with s0, . . . , sn of a line bundle L, there is a maximal open
subset U such that the sections generate L for all points in U . We then get a
rational map φ : X 99K PnA, that is φ is a morphism on the smaller open set U .
Example 16.8. Let X = P1k = Proj k[s, t] and L = O(2). Then L is globally
generated by s2, st, t2 and the corresponding morphism
X → P2k
[s : t] 7→ [s2 : st : t2]
has image V (x0x2 − x21) which is a smooth conic.
Example 16.9 (Cuspidal cubic). Let X = A1k and L = OX . Then, Γ(X,L) =
k[t] is infinite dimensional over k. Choosing the three sections 1, t2, t3, we get a
map of schemes
X → P2k
t 7→ [1 : t2 : t3]
whose image in P2 is the cuspidal cubic minus the point at ∞.
Example 16.10 (P1 as a quotient space). Let X = A2, and L = OX . Then,
Γ(X,L) = k[x, y]. If we take the sections x, y, then they generate L outside
V (x, y). Hence we get a morphism of schemes
A2k − V (x, y)→ P1
k
(x, y) 7→ [x : y]
which we is the quotient map used to define P1.
Example 16.11 (Projection from a point). Consider the projective space X =
PnA and sections x1, . . . , xn of O(1), then these sections generate O(1) outside
257
16.3. Application: Automorphisms of Pn
the point p corresponding to I = (x1, . . . , xn) (that is, what we would classically
write as the point p = (1 : 0 : · · · : 0)). The induced morphism PnA−V (I)→ Pn−1A
is the projection from p.
Example 16.12 (Cremona transformation). Consider the projective space
X = P2A and sections x0, x1, x2 of O(1), then the sections x0x1, x0x2, x1x2
generate O(2) outside V (x0x1, x0x2, x1x2) corresponding to the three points
(1, 0, 0), (0, 1, 0), (0, 0, 1). The induced rational map P2A 99K P2
A is the Cremona
transformation.
Example 16.13 (The Veronese surface). Consider X = P2, and L = OP2(2). If
x0, x1, x2 are projective coordinates on X, then the quadratic monomials
x20, x
21, x
22, x0x1, x0x2, x1x2
form a basis for H0(X,L), and generate L at every point. The corresponding
map X → P5 is an embedding; the image is the Veronese surface. It is a classical
fact that the image is defined by the 2× 2 minors of the matrixu0 u1 u2
u1 u3 u4
u2 u4 u5
Example 16.14. Let us consider again the case Q = P1 × P1. Keeping the
notation from Section 14.4.1, we have two divisors, L1 = [0 : 1] × P1, L2 =
P1 × [0 : 1]. Note that eacj Li is globally generated (being the pullback of a base
point free divisor on P1). The corresponding map is of course the i-th projection
map pi : Q→ P1.
If x0, x1 is a basis for Γ(X,L1), and y0, y1 is a basis for Γ(X,L2), we find that
Γ(X,L1 + L2) is spanned by the sections
s0 = x0y0, s1 = x0y1, s2 = x1y0, s3 = x1y1
Moreover, these sections generate OQ(D) everywhere, and so we get a map
Q→ P3
This is of course nothing but the Segre embedding; note the quadratic relation
between the four sections s0s3 − s1s2 = 0.
16.3 Application: Automorphisms of Pn
If k is a field, then any invertible (n+1)×(n+1) matrix m gives an automorphism
Pnk → Pnk (since it induces an automorphism of k[x0, . . . , xn]). Moreover, two
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Chapter 16. Maps to projective space
matrices m and m′ determine the same automorphism if and only if m = λm′
for some non-zero scalar λ ∈ k∗ (. So we are led to consider the projective linear
group
PGLn(k) = GLn(k)/k∗
Theorem 16.15. Autk(Pn) = PGLn(k).
Proof. The above shows that there is a containment ⊇. To show the reverse
inclusion, let φ : Pnk → Pnk be any automorphism. Then we get an induced map
φ∗ : Pic(Pn)→ Pic(Pn)
which is an isomorphism. Since Pic(Pn) = Z, we must have either φ∗OPn(1) =
OPn(1) or φ∗OPn(1) = O(−1). The latter case is impossible, since φ∗(OPn(1))
has a lot of global sections, whereas OPn(−1) has none. So φ∗(OPn(1)) = OPn(1).
In particular, φ∗ gives a map
Γ(Pn,OPn(1))→ Γ(Pn,OPn(1)),
which is a isomorphism of k-vector spaces. However, these vector spaces can be
identified with 〈x0, . . . , xn〉, and so φ∗ gives rise to an invertible (n+ 1)× (n+ 1)-
matrix m. By construction m induces the map φ, and so φ comes from an
element of PGLn(k).
16.4 Projective embeddings
We have seen how morphisms X → Pn corresponds to invertible sheaves L plus
n+ 1 global sections s0, . . . , sn that generate it. Given this it is natural to ask
which data (L, si) that corresponds to special types of morphisms, e.g., closed
immersions. The following criterion tells us how to check this locally:
Proposition 16.16. Let φ : X → PnA be a morphism corresponding to the data
(L, s0, . . . , sn). Then φ is a closed immersion if and only if for each i = 0, . . . , n,
Xi = Xsi is affine, and the homomorphism A[y0, . . . , yn]→ Γ(Xi,OX) yj 7→ sjsi
is surjective.
Proof. ⇒: If φ is a closed immersion, we may view X as a closed subscheme
of PnA (so it is given by some ideal in A[y0, . . . , yn]). Note that in this case
Xi = X ∩ D+(xi) is a closed subscheme of the affine scheme D+(xi). By our
results on closed subschemes of affine schemes, Xi corresponds to an ideal in
A[y0, . . . , yn](x0), and the map A[y0, . . . , yn]→ Γ(Xi,OX) yj 7→ sjsi
is surjective.
⇐: Since Xi is affine, and the map to the right is surjective, we see that Xi
corresponds to a closed subscheme of D+(xi). As X is glued together by the Xi
we see that X is a closed subscheme of PnA as well.
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16.4. Projective embeddings
In practice however, this condition is not so hard to verify (that is, not much
easier than checking directly that the induced map is an embedding). To get a
better criterion we first introduce some more notation.
Definition 16.17. Let X be a scheme over a field k.
1. A linear series V is a subspace V ⊆ Γ(X,L) where L is an invertible sheaf
on X.
2. A linear series V separates points if for any two points x, y ∈ X there is a
section s ∈ V such that s(x) = 0 but s(y) 6= 0
3. A linear series V separates tangent vectors if for any x ∈ X the set
s ∈ V |s(x) 6= 0 spans mxLx/m2x.
The definitions here come from the following theorem
Theorem 16.18. Let X be a projective scheme over an algebraically closed field
k and let φ : X → Pnk be a morphism over k corresponding to (L, s0, . . . , sn).
Then φ is a closed immersion if and only if V = spans0, . . . , sn separates points
and tangent vectors.
We will not prove this theorem here, but let us at least say a few words about
it.
If φ is a closed immersion, we can consider X ⊆ Pnk as closed subscheme, and
the si are simply restrictions of th xi to X. Now the theorem is essentially a
consequence of the easy geometric fact that the x0, . . . , xn separate points and
tangent vectors on Pn. Indeed, we can pick a linear form l = a0x0 + · · ·+ anxnsuch that the hyperplane H = V (l) meets x but not not y (here we are using
that k is algebraically closed!). l|X is a linear combination of the si, and so V
separates points. Similarly, linear forms on Pn separate tangent vectors, so they
also separate tangent vectors on the subscheme X.
The hard part is showing that the converse holds. If V is a linear series
separating points and tangent vectors, then the morphism φ is injective (since
you can use linear forms for distinguish the images of two points). The main
ingredient we need is that φ is proper, which implies that φ is closed, and
furthermore that φ gives a homeomorphism onto a closed set in Pn. Given this,
the rest of the proof is relatively straightforward: We need only check that
OPn → φ∗OX is surjective, which can be done on stalks.
What makes this theorem more powerful than the previous is that the
conditions on sections to separate points and tangent vectors can be turned in
to statements about restriction maps of certain sheaves being surjective. The
latter task is something we can approach using the machinery of cohomology.
And indeed, using the long exact sequence in cohomology we can turn this into
simple, computable numerical criteria that guarantee that the corresponding
map is an embedding.
260
Chapter 16. Maps to projective space
16.5 Projective space as a functor
Recall that we defined a functor F : Schop → Sets to be representable if there is
a scheme X and an isomorphism of functors Φ : hX ' F ; i.e., for each S ∈ Schop
a bijection Φ(S) : Hom(S,X)→ F (S).
We saw in Chapter 6 that affine space An = SpecZ[x1, . . . , xn] represented
the functor F taking a scheme S to Γ(X,OX)n; this just amounts to saying
that a morphism X → An is determined by n regular functions. More generally
Corollary 3.7 says that an affine scheme SpecA is represented by the functor
F (S) = HomRings(A,Γ(X,OX)).
So which functor does projective space represent? Intuitively we would like
to associate to each morphism f : X → Pn to a set of data on X. As we have
seen, there is not so much information Γ(X,OX) in the context of projective
schemes. However, we do have something canonical associated to Pn, namely
sections of O(1).
As a first question, we should ask what F (Spec k) should correspond to.
Motivated by the very definition of projective space over a field, we would like
to say that the value of F on closed points should correspond to lines in a vector
space. More precisely, morphisms Spec k → Pn should be in correspondence with
lines l ⊆ kn+1 through the origin. This assignment should also be continuous,
e.g., for each open U ⊆ Pn, we should have a sub-line bundle LU of the trivial
bundle Pnk ×An+1k . By the correspondence between vector bundles and invertible
sheaves, this is equivalent to a subsheaf L ⊆ On+1U which is an invertible sheaf.
This motivates the following definition:
Definition 16.19. Define a functor F : Schop → Sets by defining for a scheme
X
F (X) =
invertible subsheaves L ⊆ On+1X
=
invertible sheaf quotients On+1X → L→ 0
/ ∼
where the ∼ says that quotients On+1X → L→ 0, On+1
X →M → 0 are equivalent
if have the same kernel.
Note that on projective space Pn = ProjZ[x0, . . . , xn], we do have such a
quotient
On+1Pn → OPn(1)→ 0.
Indeed, on a D+(f) we can define it by sending an element (t0, . . . , tm) ∈ R(f)
to x0t0 + · · · + xntn ∈ R(1)(f). Of course this quotient is not canonical, as it
depends on the choice x0, . . . , xn.
261
16.5. Projective space as a functor
Theorem 16.20. The functor F is represented by the scheme Pn.
Proof. We need to define natural transformations of the two functors hPn to F .
For a scheme X, we define
Φ(X) : Hom(X,Pn)→ F (X)
by sending a morphism f : X → Pn to the (equivalence class of the) quotient
On+1X → f ∗O(1)→ 0
which is the pullback of On+1 → O(1)→ 0 on Pn. It is clear that this assignment
is functorial, i.e., it gives a natural transformation. Moreover, by definition Φ
sends the identity map idPn to the equivalence class of the sequence above.
To prove the theorem, we need to construct an inverse Ψ to Φ. In other
words, to each quotient On+1X → L → 0, we need to produce a morphism of
schemes f : X → Pn so that On+1Pn → OPn(1)→ 0 pulls back to On+1
X → L→ 0.
However, from the quotient On+1X → L → we obtain n + 1 sections s0, . . . , sn
by taking the n + 1 maps OX → On+1X → L (where the first map corresponds
to a ‘coordinate inclusion’). We then define Ψ by associating On+1X → L→ to
the morphism f : X → Pn ∈ Hom(S,Pn). One can check that two choices of
quotients induce the same map f : X → Pn, so this is well-defined.
Finally, in the construction of the morphisms in Theorem 16.7, we have
L = f ∗O(1) and si = f ∗xi, and so Ψ is the inverse to Φ.
Why should one care about such a statement? There are several good reasons,
but perhaps the most basic is that it tells us how to think about points of the
scheme projective space as corresponding to something geometric. Indeed, a
priori, the only thing we know about PnZ is that it is glued together by n + 1
coordinate affine n-spaces. The theorem above shows that in fact does what it is
supposed to do; the k-points Pn(k) correspond to lines in the vector space kn+1
for any field k.
The case of P1 is particularly vivid. The functor of points of A1 shows that a
morphism X → A1 is equivalent to giving an element of Γ(X,OX). In less precise
terms this is what we think of as a ‘regular function’ on X. The corresponding
statement for P1 is the following. A map X → P1 corresponds to a section s of
a line bundle L on X, or in other words a ‘meromorphic function’, which is only
required to be regular on Xs. .
16.5.1 Generalizations
If we modify of the functor of points of Pn, we obtain other interesting examples
of schemes parameterizing geometric objects. The following examples are only
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Chapter 16. Maps to projective space
meant as basic illustrations of this point - they will not appear later in the notes.
Example 16.21 (The Grassmannian). Consider the functor
F (S) = rank r locally free quotients On → Q→ 0 / ∼
where two quotients again are defined to be equivalent if they have the same
kernel. Then F is represented by a scheme, known as the Grassmannian Gr(r, n).
This is scheme can realized as the projective scheme in P(n+rr )−1
Z = ProjZ[pi1,...,ir ]
where 0 ≤ i1 < i2 < . . . ir ≤ n. Explicitly, Gr(r, n) is given by the Plucker
equationsr+1∑k=0
(−1)kpi1,i2,...,ir−1,jk · pi1,...,jk,...,jr+1= 0
(over all sequences of indices i1, . . . , ir and j1, . . . , jr+1). We can define the map
Gr(k, n)→ P(n+rr )−1
Z using the Yoneda lemma, by giving a natural transformation
between the functors. If S is a scheme, this transformation takes an S-valued
point of Gr(k, n), that is, a quotient On → Q→ 0, to ∧rOn → ∧rQ→ 0, which
since ∧rQ has rank 1, defines a point in P(n+rr )−1
Z (S).
Proving that this scheme actually represents Gr(k, n) is similar to what we
did for projective n-space, although it is slightly more involved, due to the
Plucker equations.
Example 16.22 (Projective bundles). Let E be a locally free sheaf on a scheme
X. Consider the functor on Sch/X given by
F (h : Y → X) = invertible sheaf quotients h∗E → L→ 0
Then F is represented by a scheme P(E )→ X over X called the projectivization
of E . One can think of closed points of this scheme as hyperplanes in the fibers
of E .
Example 16.23 (Proj of an OX-module). Let X be a scheme and let F be a
quasi-coherent sheaf of OX-modules. Consider the functor on Sch/X given by
F (h : Y → X) = invertible sheaf quotients h∗F → L→ 0
Then F is represented by a scheme Proj(F). Many geometric constructions can
be formulated as such schemes. For instance, if I is a sheaf of ideals on X, then
Proj(F) can be identified with the blow-up of X along I (see Hartshorne II.7).
263
Chapter 17
More on projective schemes
17.1 Ample invertible sheaves and Serre’s the-
orems
The prototype of an invertible sheaf is the sheaf O(1) on X = ProjR, which is
globally generated if R is generated in degree 1. As we have seen, this corresponds
to a morphism f : X → Pn, with the property that f ∗OPnZ (1) = OProjR(1). In
the case where f is a closed immersion, we can regard X as a subscheme of
Pn, and O(1) is simply the restriction OPnZ (1)|X . This restricted invertible sheaf
has a special property - R and hence X is completely recovered by the global
sections of it and the polynomial relations between them. This motivates the
following definition:
Definition 17.1. Let X be a scheme over S. A invertible sheaf L on X is said
to be very ample over S if there is a closed immersion i : X → PnS such that
L ' i∗OPnS(1). L is ample if L⊗N is very ample for some N > 0.
Theorem 17.2 (Serre). Let X ⊆ PnA be a projective scheme over a noetherian
ring A and let L be an ample invertible sheaf. Let F be a coherent sheaf on X.
Then there is an integer m0 such that F ⊗ Lm is globally generated (by a finite
set of global sections) for all m ≥ m0.
Proof. By replacing L with a multiple, we may assume that L is very ample
and even that L = i∗O(1) for a projective embedding i : X → Pn. Since X is
noetherian, i is finite and F is coherent, we have that i∗F is coherent on Pn.
Moreover, (i∗F)(m) = i∗(F ⊗ Lm), so F ⊗ Lm is globally generated if and only
if (i∗F)(m) is. Hence we may assume that X = PnA.
265
17.1. Ample invertible sheaves and Serre’s theorems
Write PnA = ProjR where R = ProjA[x0, . . . , xn]. We can cover X by open
sets D+(xi) where i = 0, . . . , n, such that F|D(xi) = Mi some finitely generated
R(xi)-module. For each i we can choose finitely many elements sij generating Mi.
Regarding sij as sections of the sheaf Mi, we see that there are integers mij such
that xmiji sij extend to global sections of F(mij). Take m = maxijmij, then for
each i, j we have a section tij ∈ Γ(X,F(m)).
We claim that the global sections tij generate F(m) locally. It is sufficient to
prove this on the opens D+(xi) over which F(m) is isomorphic to the R(xi)-module
Mi(m) 'Mi ⊗R R(m), which is isomorphic to xmi Mi as a graded R(xi)-module.
Since sij generates Mi, we see that xmi sij generate F(m)|D+(xi).
This shows that F(m0) is globally generated for the chosen m0 = m above.
Then also F(m) is globally generated for m ≥ m0, by multiplying the generating
sections of F(m0) by the various monomials in the xi.
In particular, we see that any coherent sheaf can be written as a quotient of
a direct sum of invertible sheaves of the form O(−m): Take any F(m) which is
globally generated; then there is a surjection OI → F(m)→ 0 (with I finite!),
which becomes
O(−m)I → F → 0 (17.1.1)
after tensoring by O(−n).
Theorem 17.3 (Serre). Let X = ProjR be a noetherian projective scheme over
SpecA, where A is a noetherian ring, and let F be a coherence sheaf on X.
Then:
(i) The cohomology groups
H i(X,F)
are finitely generated A-modules for every i ≥ 0.
(ii) There exists an n0 > 0 such that
H i(X,F (n)) = 0.
for all n ≥ n0.
Proof. As in the above proof, we immediately reduce to the case where X = PnA.
Note that both of the conclusions hold for the twisting sheaves OX(n). To
prove it for any coherent F , take a quotient of the form (17.1.1) and let K be
the kernel, so that we have an exact sequence
0→ K → E → F → 0
where E = O(−n)I , with I finite. Note that K is again coherent.
266
Chapter 17. More on projective schemes
(i): Take the l.e.s of cohomology to get
H i(X,E )→ H i(X,F)→ H i+1(X,K )
We can now prove the theorem by downwards induction on i: H i+1(X,K )
and H i(X,E ) are both finitely generated, and hence so is H i(X,F), since A is
noetherian.
(ii): Twist the above sequence by OX(m) and take the long exact sequence
in cohomology to get
H i(X,E (m))→ H i(X,F(m))→ H i+1(X,K (m))
By downward induction on i, and the fact that H i(X,E (m)) for any m, we find
that H i(X,F(m)) = 0.
17.1.1 Euler characteristic
Let X be a noetherian scheme over a field k and let F be a coherent sheaf on
X. By Serre’s theorem, the cohomology groups H i(X,F) are finite dimensional
k-vector spaces, so it makes sense to ask about their dimensions. It turns out
that the alternating sum of these dimensions is the right thing to look at:
Definition 17.4. Let X be a projective scheme of dimension n. We define the
Euler characteristic of F as
χ(F) =n∑k=0
(−1)k dimHk(X,F)
Proposition 17.5. Let X be a projective scheme and let O(1) be the very ample
invertible sheaf. Then the function
PF(m) = χ(F(m))
is a polynomial in m
This polynomial is called the Hilbert polynomial of F . When F = M , this
coincides with the Hilbert polynomial of M as defined in commutative algebra.
17.2 Proper and projective schemes
Let S be a scheme and let X be a scheme over S. We say call X is projective
over S (or that the structure morphism f : X → S is projective) if f : X → S
267
17.2. Proper and projective schemes
factors as f = π i where i : X → PnS is a closed immersion and π : PnS → S is
the projection.
The primary examples are of course X = ProjR where R is a graded R0-
algebra generated in degree 1 and S = SpecR0. In this case, we can define the
projective immersion i by taking a surjection R0[x0, . . . , xn]→ R, which upon
taking Proj, a closed immersion X → PnR0.
Theorem 17.6. A projective morphism X → S is proper.
268
Chapter 18
The Riemann–Roch theorem and
Serre duality
18.1 Curves
We will in this chapter assume that k is an algebraically closed field. Recall that
a variety over k is a separated, integral scheme of finite type over k. Throughout
this section, a curve will mean a 1-dimensional variety over k.
18.2 Divisors on Curves
Let X be a non-singular curve. Since X has dimension 1, a Weil divisor D on X
is a finite formal combination of points on X,
D =∑
nipi
where pi ∈ X. We say that D is effective if each ni ≥ 0.
The degree of D is defined as the sum degD =∑ni. Equivalently, if we view
D as a Cartier divisor given by the data (Ui, fi), then
degD =∑x∈X
vx(D)
where vx(D) = vx(fi) as before.
Lemma 18.1. Let D be a non-zero effective divisor on X. Then degD =
dimkH0(D,OD).
269
18.2. Divisors on Curves
Proof. Note that D is a finite scheme, hence finite. Let A = H0(D,OD). Then
A =⊕x∈D
OD(x) =⊕x∈X
OD,x
Hence dimk A =∑
x∈D dim OD,x. Now, vx(D) equals the length of OD,x. Since
k is algebraically closed, this length coincides with dimk OD,x, which gives the
claim.
Recall that each Weil divisor determines an invertible sheaf OX(D), which
over an open set U takes the value
OX(D)(U) = f ∈ K|(div f +D)|U ≥ 0
Then D is effective if and only if Γ(OX(D)) 6= 0. In particular, if Γ(X,OX(D))
has dimension at least 2, there is a second effective divisor D′ =∑miqi such
that D and D′ are linearly equivalent.
18.2.1 The genus of a curve
Definition 18.2. The arithmetic genus of X is the defined as
pa(X) = dimkH1(X,OX)
The geometric genus of X is defined as
pg(X) = dimkH0(X,ΩX)
Note that χ(OX) = dimH0(X,OX) − dimH1(X,OX), so we get pa(X) =
1− χ(OX).
At first sight, these numbers are defined using different sheaves, and there
is no a priori reason to expect that they should have anything to do with each
other. However, we shall see later in the chapter that there is a strong relation
between them: pa = pg. For the time being we will still refer to pa as the genus
of X.
Example 18.3. When X = P1, we have H1(P1,O) = 0 and H0(P1,ΩP1) =
H0(P1,OP1(−2)) = 0, so both genera are zero.
Example 18.4. Let X ⊆ P2 be a plane curve, defined by a homogeneous poly-
nomial f(x0,1 , x2) of degree d. In Chapter 12, we computed that H1(X,OX) 'k(d−1
2 ). Hence the arithmetic genus of X is (d−1)(d−2)2
.
270
Chapter 18. The Riemann–Roch theorem and Serre duality
18.3 The canonical divisor
Let X be a non-singular curve. Then the sheaf of differentials ΩX is a locally
free sheaf of rank 1, i.e., an invertible sheaf. Thus ΩX gives rise to a divisor,
which we denote by KX . We can describe KX as a Cartier divisor as follows.
Over an open subset U ⊆ X a local section of ΩX is given by fUdx. We can
then define a Cartier divisor with the data
(U, fU)
This is well-defined, because on the overlaps U ∩ V , two sections fUdx, fV dy are
related by fU = dydxfV , and dy
dxis a unit in OX(U ∩ V ).
Any other section ω′ of ΩX has the form fω for some f ∈ K×. Therefore,
div (ω′) = div (ω) + div (f)
so the divisor class [KX ] := [div (ω)] ∈ Cl(X) associated to ω is independent of
the chosen ω. We call this the canonical class of X. Also, any divisor D so that
[D] = [KX ] is called a canonical divisor and it is denoted by KX .
Definition 18.5. A canonical divisor is a divisor KX , so that OX(KX) ' ΩX .
Example 18.6. Let X = P1k, and consider the open set U0 = D+(x0). On U0 we
have a local coordinate t = x1x0
, and we consider the differential form dt = d(x1x0
).
On the overlap U0 ∩ U1, we have
d
(x0
x1
)= d(t−1) = −t−2dt = −
(x0
x1
)2
d
(x1
x0
)It follows that ΩX ' OP1(−2).
18.4 Hyperelliptic curves
Let us recall the hyperelliptic curves defined in Chapter 3. For an integer g ≥ 1
we consider the scheme X glued together by the affine schemes U = SpecA and
V = SpecB, where
A =k[x, y]
(−y2 + a2g+1x2g+1 + · · ·+ a1x)and B =
k[u, v]
(−v2 + a2g+1u+ · · ·+ a1u2g+1)
As before, we glue D(x) to D(u) using the identifications u = x−1 and v = x−g−1y.
271
18.4. Hyperelliptic curves
There is a morphism f : X → P1 coming from the inclusions k[x] ⊆ A and
k[u] ⊆ B; note that u = x−1 gives the standard gluing of P1 by the two affine
schemes Spec k[x] and Spec k[u]. The corresponding morphism X → P1 is finite,
of degree 2, as we are adjoining a square root of an element of k[x].
Let us compute the genus of X using Cech cohomology. We will use the
covering U = U, V above. Viewing the first ring as a k[x]-module, we can write
k[x, y]
(−y2 + a2g+1x2g+1 + · · ·+ a1x)= k[x]⊕ k[x]y
and similarly k[u]⊕ k[u]v as a k[u]-module for the ring on the right.
The first map in the Cech complex of OX has the following form:
d0 : (k[x]⊕ k[x]y)⊕(k[x−1]⊕ k[x−1]x−g−1y
)→ k[x±1]⊕ k[x±1]y
(p(x) + q(x)y, r(x−1) + s(x−1)x−g−1y) 7→ p(x)− r(x−1) + (q(x)− s(x−1)x−g−1)y
From this we can read off H0(X,O) = ker d0 = k and H1(X,O) ' kg, with
basis yx−1, yx−2, . . . , yx−g. In particular, g is exactly the arithmetic genus of the
curve.
For g = 2, we get a particularly interesting curve – a smooth projective curve
which cannot be embedded in P2. Indeed, any smooth curve in P2 has a degree d
and corresponding arithmetic genus 12(d− 1)(d− 2). However, there is no integer
solution to 12(d− 1)(d− 2) = 2!
Proposition 18.7. There exist non-singular projective curves which cannot be
embedded in P2.
To prove this, we need to verify that the curve is in fact projective. That
is, we need to find some projective embedding of it. To do this, we will need to
work out the cohomology groups H0(X,OX(nP )) for a point p ∈ X.
Let us for simplicity assume that a2g+1 = 1. Let p be the unique point in
u = v = 0 in X. In the local ring at p, we have
u = v2(1 + a2gu+ · · ·+ a1u2g)−1 = v2(unit),
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Chapter 18. The Riemann–Roch theorem and Serre duality
and hence vOp generates mp. We compute the some valuations of elements in
Op:
νp(v) = 1, νp(u) = 2, νp(x) = −2, νp(y) = 1 + (g + 1)(−2) = −(2g + 1)
We’ve seen that Γ(X,OX) = k, which agrees with our expectation that there
are no non-constant regular function on a projective curve. Let us consider the
case where the functions are allowed to have poles at p (and only at p). In other
words, we are interested in elements s ∈ Γ(X,OX(p)). Note that the point p
does not lie in U ; this means that s is regular there, and hence is a polynomial
in x, y. Now,
k[x, y]
−y2 + a2g+1x2g+1 + · · ·+ a1x= k[x] + k[x]y (as a k[x]-module)
So, any element s can be expressed as f(x) + h(x)y. We then can calculate
νp(f(x) + h(x)y) = minνp(f(x)), νp(h(x))νp(y)= min−2 deg f,−(2 deg h+ 2g + 1)
Thus, even if we allow a pole or order 1 at p, we still have Γ(X,OX(p)) = k.
On the other hand for 2p we gain an extra section, corresponding to f(x) = x:
Γ(X,OX(2p)) = k1, x
Note that O(2p)p = Op · x. The section x ∈ Γ(X,OX(2p)) is nonvanishing at
p, while the section 1 ∈ Γ(X,OX(2p)) is vanishing at p, since 1 = u · x and
u ∈ m ⊆ Op. Note that the linear series generated by 1, x generates OX(2p)
everywhere, inducing the morphism
Xϕ→ P1
(x, y) 7→ [1 : x]
which is the same morphism as the double cover above.
We can allow even higher order poles at p. The computation above shows
that
Γ(X,O(3p)) =
k1, x, y if g = 1
k1, x if g > 1
273
18.5. The Riemann Roch theorem for curves
Using the embedding criterion of Chapter 10, one can show that in the case
g = 1, the sections x0 = 1, x1 = x, x2 = y give an embedding
X → P2k
(x, y) 7→ [1 : x : y]
The image is even seen to be a non-singular cubic curve: One computes that
Γ(X,O(6p)) is 6-dimensional, but we have 7 global sections: 1, x, y, x2, xy, x3, y2.
That means that there must be some relation between them. But of course this
is the relation
y2 = a3x3 + a2x
2 + a1x
giving the following relation in P2:
x22x0 = a3x
31 + a2x0x
21 + a1x
20x1
For g = 2, this map is not an embedding, since the image is P1. However, the
map given by 5p is: We obtain
Γ(X,OX(5p)) = k1, x, x2, y
These sections generate OX(5p), so we obtain a morphism
φ : X → P3
given by u0 = 1, u1 = x, u2 = x2, u3 = y. Notice that u0u2− u21 = 0, so X lies on
a quadric surface. In fact, the image of φ is precisely the relations between the
sections:
C = V(u2
1 − u0u2, u20u2 + u3
2 − u1u23, u
20u1 + u1u
22 − u0u
23
)⊆ P3
k
This shows that X is projective.
18.5 The Riemann Roch theorem for curves
When X is a curve over a field k, the cohomology groups H i(X,F ) are k-vector
spaces and we define will write
hi(X,F) := dimkHi(X,F)
Note that in this case, hi(X,F) = 0 for all i ≥ 2, so we have two invariants
h0(X,F) and h1(X,F) to work with.
Recall, that we defined for a sheaf F , the Euler characteristic χ(F) as the
274
Chapter 18. The Riemann–Roch theorem and Serre duality
alternating sum of the hi(X,F). One useful property of χ(X,−) is that it is
additive on exact sequences:
Lemma 18.8. Let 0 → F ′ → F → F ′′ → 0 be an exact sequence of sheaves.
Then
χ(F) = χ(F ′) + χ(F ′′)
This follows because if 0→ V0 → V1 → · · · → Vn → 0 is an exact sequence
of k-vector spaces, then∑
i(−1)i dimk V = 0. Applying this to the long exact
sequence in cohomology gives the claim.
The most important sequence we will encounter is the ideal sheaf sequence
of a point p ∈, which takes the form
0→ OX(−p)→ OX → k(p)→ 0 (18.5.1)
where the first map is the inclusion and the second is evaluation at p. Here we
have identified the ideal sheaf mp ⊆ OX by the invertible sheaf OX(−p), and the
sheaf i∗Op with the skyscraper sheaf with value k(p) at p. If L is an invertible
sheaf, we can tensor (18.5.1) by L and get
0→ L(−p)→ L→ k(p)→ 0 (18.5.2)
where L(−p) is the invertible sheaf of sections of L vanishing at p. Taking χ, we
get
χ(L(−p)) = χ(L)− χ(k(−p)) = χ(L)− 1.
Theorem 18.9 (Easy Riemann–Roch). Let X be a smooth projective curve of
genus g and let D be a Cartier divisor on X. Then
χ(OX(D)) = h0(OX(D))− h1(X,OX(D)) = degD + 1− g
Proof. Let p ∈ X be a point and consider the sequence (18.5.2) with L =
OX(D + p). Then, as we just saw, χ(OX(D + p)) = χ(OX(D)) + 1. Also the
right-hand side of the equation above increases by 1 by adding p to D (since
deg(D + p) = degD + 1). This means that the theorem holds for a divisor D if
and only if it holds for D+p for any closed point p. So by adding and subtracting
points, we can reduce to the case when D = 0. But in that case, the left hand
side of the formula is by definition dimkH0(X,OX)− dimkH
1(X,OX) = 1− g,
which equals the right hand side.
The formula above is extremely useful because the right hand side is so easy to
compute. The number we are really after is the number h0(X,OX(D)), since this
is the dimension of global sections of OX(D). This in turn would help us to study
X geometrically, since we could use sections of OX(D) to define rational maps
275
18.5. The Riemann Roch theorem for curves
X 99K Pn. So if we, for some reason, could argue that say, H1(X,OX(D)) = 0,
we would have a formula for the dimension of the space of global sections of
OX(D).
In any case, we can certainly say that h1(X,OX(D)) ≥ 0, so we get the
following bound on h0(X,OX(D)):
Corollary 18.10. h0(X,OX(D)) ≥ degD + 1− g
Example 18.11. A typical feature is that H1(X,OX(D)) = 0 provided that
the degree degD is large enough. This is essentially a consequence of Serre’s
theorem. To give an example, consider again the case where X is a hyperelliptic
curve of genus 2, as in the introduction. We have the following table of the
various cohomology groups H i(X,OX(np)) for the point p = (u, v):
D 0 1p 2p 3p 4p 5p 6p 7p
χ(OX(D)) -1 0 1 2 3 4 5 6
H0(X,OX(D)) 1 1 2 2 3 4 5 6
H1(X,OX(D)) 2 1 1 0 0 0 0 0
and it is not so hard to prove directly using the Cech complex thatH1(X,OX(np)) =
0 for all n ≥ 4.
Fortunately, there are more general results which tell us whenH1(X,OX(D)) =
0. This is due to the following fundamental theorem:
Theorem 18.12 (Serre duality). Let X be a smooth projective variety of dimen-
sion n and let D be a Cartier divisor on X. Then for each 0 ≤ p ≤ n,
dimkHp(X,OX(D)) = dimkH
n−p(X,OX(KX −D))
So if X is a curve, we get that H1(X,OX(D)) = H0(X,OX(KX −D)) and
the Riemann–Roch theorem takes the following form:
Theorem 18.13 (Riemann–Roch). Let X be a projective curve of genus g and
let D ∈ Div(X) be a divisor. Then
h0(X,OX(D))− h0(X,OX(KX −D)) = degD + 1− g
This is a much stronger statement than the Riemann–Roch formula we had
before, since group H0(X,OX(KX −D)) is easier to interpret: it is the space of
global sections of the sheaf associated to the divisor KX −D, or equivalently
ΩX(−D). It is usually easier to argue that there can be no such global sections
of this divisor. For instance, in the case degD < dimKX then KX −D cannot
be effective: if s ∈ H0(X,OX(KX −D)), then the divisor of s defines an effective
276
Chapter 18. The Riemann–Roch theorem and Serre duality
divisor on X, that is, a sum div (s) =∑nipi where ni ≥ 0, contradicting our
assumption degKX < dimD.
So what is this degree of the canonical divisor KX? From Serre duality,
we get that H0(X,OX(KX)) and H1(X,OX) have the same dimension, so the
geometric genus and arithmetic genus agree:
pg = pa = g.
Then applying the Riemann–Roch formula to D = KX , we get
g − 1 = dimkH0(X,OX(K))− dimkH
0(X,OX(KX −K)) = degK + 1− g
and so degKX = 2g − 2. This observation gives us
Corollary 18.14. Suppose that D is a Cartier divisor of degree > 2g− 2. Then
H1(X,OX(D)) = 0, and
dimkH0(X,OX(D)) = degD + 1− g
Moreover, if degD = 2g − 2, then H1(X,OX(D)) 6= 0 only if D ∼ KX .
18.6 Proof of Serre duality for curves
Riemann–Roch and Serre duality are two of the very deep theorems in math-
ematics. The statement holds true in any dimension and they have numerous
consequences.
In this section we will outline the proof of Theorem 18.12 for curves. We
follow the proof in of Serre’s Groupes algebriques et corps de classes, with some
changes to make the proof as self-contained as possible.
So let X be a smooth projective curve over an algebraically closed field k
and let K denote the function field of X.
The product∏
x∈X K will play an important role in the following proof. It has
a natural structure of a (big!) k-vector space. An element r = (rx) ∈∏
x∈X K is
a rational function rx ∈ K for each closed point x ∈ X. These rational functions
can a priori be chosen in a completely arbitrary way, but or course for special
elements r, the rx can be linked together more closely. For instance, the function
field K is ‘diagonally embedded’ into∏
x∈X K: For each f ∈ K we get the
element r(f) by rx = f for all x ∈ X.
We follow Serre and call an element r ∈∏
x∈X K a repartition if rx ∈ OX,x
for all but finitely many x. Thus r consists of infinitely many rational functions
rx, indexed by the closed points in X, but all but finitely many of them are
regular near x for each x. We denote the set of repartitions by R. This is clearly
277
18.6. Proof of Serre duality for curves
a subring of∏
x∈X K, as rxsx is regular near x if both rx are sx are. Also R
contains the ‘diagonally embedded’ K ⊆∏
x∈X K as a subring: If f ∈ K then f
lies in OX,x for almost all x and hence r(f) is a repartition.
Consider now a divisor D =∑npp on X. This defines an OX-module
OX(D) ⊆ K (where K is the constant sheaf with value K on X). This is similar
to the definition of OX(D), but we treat each point and each rational function
rx separately. The stalk of OX(D) at a point x consists of all elements f ∈ Ksuch that vx(f) ≥ −vx(D).
Let R(D) ⊆ R denote the additive subgroup of repartitions r ∈ R so that
vx(rx) ≥ −vx(D). Equivalently, we pick the r = (rx) so that each rx lies in the
stalk OX(D)x ⊆ K. Then R(D) has the structure of a sub-k-vector space of R.
If D ≤ D′ (or in other words, D′−D is effective), then clearly R(D) ⊆ R(D′),
as −vx(D′) ≤ −vx(D). Note also that R(0) coincides with the repartitions
r = (rx) so that rx ∈ OX,x is regular for every x ∈ X.
The quotient R/R(D) is naturally contained in the product∏
x∈X R/OX(D)x,
and it can be identified with the direct sum⊕
x∈X R/OX(D)x since the elements
of R are contained in OX,x for almost all x.
Proposition 18.15. We have a canonical isomorphism
H1(X,OX(D)) = R/(R(D) +K)
We will from now on define I(D) := R/(R(D) +K).
Proof. We start out with the short exact sequence
0→ OX(D)→ K → S → 0
where S = K/OX(D) is the quotient sheaf. The sheaf K is a constant sheaf, and
hence all of its higher cohomology groups vanish; in partcular H1(X,K) = 0.
Taking the long exact sequence in cohomology, we get
K → H0(X,S)→ H1(X,OX(D))→ 0
So it remains to identify the global sections of S.
For this, we need a simpler description of S. At a point x ∈ X, we have
Sx = (K/OX(D))x = f ∈ K|vx(f) + vx(D) < 0
and hence R/R(D) =⊕
x∈X Sx.We would like to show that S equals the the direct sum its skyscraper sheaves,
that is, S =⊕Sx and so sections of S coincides with a selection of elements
of Sx for x ∈ X, almost all of which are zero. Given this, the space of global
sections of S is indeed⊕
x Sx = R/R(D), which gives the proposition.
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Chapter 18. The Riemann–Roch theorem and Serre duality
In any case, we recall that an element of Sx is represented by an element
(r, U), where r ∈ S(U) and U ⊆ X is an open set containing x. Since K → S is
surjective, we may assume that r is represented by an element of K.
The element r may or not satisfy vx(r) < −vx(D) (i.e., r 6∈ OX(D)x): This
happens if the class [r] ∈ Sx is non-zero. Note that vy(r) = 0 for y in a
neighbourhood V of U − x: This is because the rational function is regular in a
neighbourhood of x (here we use that X is a curve), and we can assume that
it is has no zero there. We also have that vx(D) = 0 in a neighbourhood W of
U − x. Hence [r] vanishes in the stalks Sy for y ∈ V ∩W .
Now we conclude the proof using the following lemma:
Lemma 18.16. Let F be a sheaf on a quasicompact topological space X so that
the closed points are dense in x. Suppose that for each point x ∈ X and each
element s ∈ Fx there exists an open set U 3 x and a section σ ∈ F(U) which
induces s in Fx and vanishes on U − x. Then
F(X) =⊕x∈X
Fx
where the sum is over the closed point of X.
Proof. We have an injective map
F(X)→⊕x∈X
Fx
which sends a section σ ∈ F (X) to σx in each stalk. We will show that the image
is the whole direct sum of the stalks Fx (i.e., that a global section of F is zero
for all but a finite number of points x, and there are no conditions on the points
x and the ‘values’ sx).
Let σ ∈ F(X), and choose an open set Ux of each point x ∈ X so that σ|Ux−xvanishes. Since X is quasicompact, X can be covered by finitely many of the
sets Ux, say Ux1 , . . . , UXr . So σ is zero outside of the points x1, . . . , xr.
Conversely, let x1, . . . , xr be points in X and suppose we are given an element
si ∈ Fxi for each i = 1, . . . , r. Each si can be extended to a section σi of F in
some neighbourhood Ui of xi which vanishes on Ui−xi. If we let Vi = Ui−xjj 6=i,then the Vi form an open cover of X, which is such that Vi ∩ Vj contains none
of the points x1, . . . , xr. Therefore, σ|Vi∩Vj = 0 = σj|Vi∩Vj and the σi glue to a
global section of F with the desired property.
The take-away from this is that we have an explicit description ofH1(X,OX(D))
in terms of rational functions. This is already interesting in the case D = 0. In
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18.6. Proof of Serre duality for curves
this case, H1(X,OX) = 0 means that for any collection of rational functions rxwhich are allowed to have poles at a finite number of points in X, there is a
rational function f ∈ K so that rx − f is regular for every x ∈ X.
18.6.1 Laurent tails
In this section A will be a discrete valuation ring, which is an algebra over its
residue field k = A/m, where of course m is the maximal ideal of A. Our main
example will be when A = OX,x where X is a curve over an algebraically closed
field and x is a closed point, or more generally A = ΩX,η where η is a point of
codimension 1.
Let K be the fraction field of A and let v : K → Z denote the normalized
valuation. The field K has a decreasing filtration (infinite in both directions) of
A-modules Ki given by
Ki = f ∈ K|v(f) ≥ i
Note that K0 = A, and Ki = mi for i ≥ 0. Moreover, the quotient Ki/Ki+1 has
the structure of a k-vector space of dimension 1: it is spanned by [ti] where t is
a uniformizing parameter of A.
Lemma 18.17. Any element f ∈ K can be written as
f =∑i>0
a−it−i + g
where g ∈ A.
Proof. This is obvious if f ∈ A. More generally, take f ∈ F and suppose
n = v(f) < 0. Then f ∈ Kn and the class [f ] in Kn/Kn+1 is a scalar multiple of
the generator [tn], say [f ] = α[tn] where α ∈ k. In particular, f − αrn maps to
zero in Kn/Kn+1, and hence v(f − αtn) ≥ n+ 1. So by induction on |v(f)| we
can write f =∑
i>0 α−it−i + g and we are done.
18.6.2 Residues on A
Let t be a uniformizing parameter in A, and consider a differential ω = fdt which
lies inM = ΩK|k. We define the residue of f (relative to t) as the coefficient a−1
in the above decomposition of f . This is denoted by Rest f , Rest fdt or Rest ω.
We have
fdt = a−nt−ndt+ · · ·+ a−2t
−2dt+ Rest f · t−1dt+ g dt
An important point, which we will address below, is that this definition does not
depend on the choice of uniformizing parameter t.
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Chapter 18. The Riemann–Roch theorem and Serre duality
Here are some standard properties of residues
Lemma 18.18. (i) Rest is a k-linear map Rest : ΩA|k → k.
(ii) Rest(fdt) = 0 if f ∈ A
(iii) Rest(df) = 0 for f ∈ K.
(iv) Rest(g−1dg) = v(g) for all g ∈ K − k.
Proof. These properties are easy to verify. (i) is clear. (ii) follows since all
elements of A have zero residues by definition. For (iii), we write
f =∑i>0
a−it−i + g
where g ∈ A. This gives that df =∑
i>0(−i)a−it−i−1dt + dg where dg ∈ ΩA|k.
The coefficient of t−1 vanishes.
For (iv), write g = tnα, where n = v(g) and α ∈ A×. Differentiation gives
that dg = ntn−aαdr + tndα, which gives that
g−1dg = nt−1dt+ α−1dα
As dα ∈ ΩA|k and α is invertible in A, so α−1dα ∈ ΩA|k which gives the claim.
18.6.3 Residues on X
Let now X be a smooth projective curve over k, and let ΩX = ΩX|k denote
the sheaf of differentials. For each divisor D we also have the invertible sheaf
ΩX(D) = ΩX ⊗ OX(D) of differentials with poles at worst order vx(D) on D for
each x ∈ X.
We let M = ΩK|k denote the space of meromorphic differentials on X. Note
that M ' Kdt is a 1-dimensional K-vector space. Given an element ω ∈ M,
and a point x ∈ X, K is the fraction field of OX,x so we can use the residue map
from the previous section. In particular, we get a map
Resx :M→ k.
Proposition 18.19. Let X be a smooth projective curve over k. Then
(i) (Independence of uniformizer) Resx(ω) does not depend on the choice of t.
(ii) (Residue theorem) For any ω ∈ ΩK|k, Resx(ω) = 0 for all but finitely many
x. Moreover,∑
x∈X Resx ω = 0.
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18.6. Proof of Serre duality for curves
Over the field k = C, these are indeed the same residues you know from
complex analysis. Here the theorem is easy to prove: (i) follows since the residue
is uniquely determined by
Resx(ω) =
∫σ
ω
for a small circle σ around x. The point (ii) follows just as easily, by the Residue
theorem. Indeed, if Di are small disks around each pole of ω, ω is holomorphic
in U = X −⋃Di, and so
∫∂Uω = 0, which means that the residues sum to zero.
One can prove the above properties (i) and (ii) over any field, completely
algebraically, but it is a bit more involved (especially in positive characteristic),
so we will not do so here.
We now return to the proof of Serre duality. We recall the set of repartitions
R, which form a vector space over K. (If f ∈ K and r is a repartition, the
product fr is given by (fr)x = frx: Again this makes sense since a rational
function is regular outside a finite number of points.)
Definition 18.20. Let I(D) = R/(R(D) +K) and let
J(D) = Homk(I(D), k)
denote the space of k-linear functionals on I(D).
We can (and will!) think of the elements of J(D) as the linear functionals on
the huge ring of repartitions R that vanish on the subspace R(D) +K. So we
have an inclusion
J(D) ⊆ Homk(R, k).
If D′ ≥ D, we saw before that R(D) ⊆ R(D′), which implies that J(D′) ⊆ J(D).
We define
J =⋃D
J(D)
where the sum is over all divisors on D. This is yet another horribly big vector
space over k.
Actually, J also has the structure of a vector space over K, via the product
fα(r) = α(fr). Note that α(fr) is a k-linear functional that vanishes on K.
If α vanishes on R(D), and div f = D′, then fα vanishes on R(D − D′): If
r ∈ R(D −D′), then fr ∈ R(D) because
vx(fr) = vx(f) + vx(r) ≥ vx(f)− (vx(D) + vx(f)) = −vx(D)
and hence α(fr) = 0. Hence fα belongs to J(D −D′) ⊆ J .
Proposition 18.21. We have dimK J ≤ 1
282
Chapter 18. The Riemann–Roch theorem and Serre duality
Proof. Assume the contrary and let α, β be two linearly independent elements
of J . Let D be a divisor, such that α and β both lie in J(D). Let d = degD.
For each n ∈ Z we pick a divisor Dn so that degDn = n (for instance
Dn = nP for a point P will do). If f ∈ H0(X,OX(Dn)), then fα ∈ J(D −Dn),
by the argument above.
Define the map
H0(X,OX(Dn))⊗H0(X,OX(Dn)) → J(D −Dn)
(f, g) 7→ fα + gβ
This is injective since α, β are assumed to be linearly independent over K. In
particular,
dimk J(D −Dn) ≥ 2h0(X,OX(Dn))
We will show that this gives a contradiction by estimating these dimensions using
Easy Riemann–Roch.
On the left-hand side we get
dimk J(D −Dn) = dimk I(D −Dn)
= h1(X,OX(D −Dn))
= h0(X,OX(D −Dn)− (d− n) +O(1)
= n+O(1)
for n 0. Here O(1) means a constant independent of n and Dn.
On the right-hand side we use the inqueality h0(D) ≥ degD + 1− g:
2h0(X,OX(Dn)) ≥ 2 deg(Dn) +O(1) = 2n+ O(1)
So if n 0 we get a contradiction as the two sides cannot be equal. Hence α, β
cannot be linearly independent, which finishes the proof.
18.6.4 The final part of the proof
The statement of Serre duality is that two vectors are dual to each other. We
have to define the pairing explicitly and check that it is indeed perfect1.
Let ω ∈ M be a meromorphic differential on X. This defines a canonical
divisor
div (ω) :=∑x∈X
vx(ω)x
1Recall that a bilinear map M × N → A is a perfect pairing if the induced map M 7→HomA(N,A) is an isomorphism
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18.6. Proof of Serre duality for curves
Note that this means that the invertible sheaf ΩX(−D) is the sheaf of differentials
so that divω −D ≥ 0.
Definition 18.22. We define the Serre duality pairing as follows:
〈 , 〉 :M×R → k
(ω, r) 7→ 〈ω, r〉 =∑x∈X
Resx(rxω)
This has the following properties:
(1) 〈ω, r〉 = 0 if r ∈ K(by the Residue theorem)
(2) 〈ω, r〉 = 0 if r ∈ R(D) and ω ∈ ΩX(−D)(X).
(The product rxω cannot have a pole for any x ∈ X, because the zeroes must
at least cancel the poles by the assumptions on r and ω.
(3) 〈fω, r〉 = 〈ω, fr〉 if f ∈ K(Since both pairings evaluate to the same sum of the residues over fωr.)
Now, fixing a meromorphic differential ω ∈ ΩX(−D)(X), gives a linear
functional θ(ω) on R. By properties (1) and (2), this descends to a functional
on R/(R(D) +K). Hence we get a map
θ : ΩX(−D)→ J(D)
(Recall that J(D) is the dual of R/(R(D) +K)).
Lemma 18.23. Let ω ∈M such that θ(ω) ∈ J(D). Then ω ∈ ΩX(−D)(X).
Proof. Suppose this is not true, so that ω 6∈ ΩX(−D)(X). Hence there is a
point x ∈ X so that ω has a pole in x with a higher order than allowed by D:
vx(ω) < vx(−D).
Pick the repartition r ∈ R(D) by defining
rq =
0 for q 6= x
t−vx(ω)−1 for q = x
Then vx(rxω) = −1, and so
〈ω, r〉 =∑q∈X
Res(rqω) = Resx(rω) 6= 0
This means that θ(ω) 6= 0 on R(D). However, θ(ω) is supposed to vanish on all
of R(D) +K, so we have a contradiction.
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Chapter 18. The Riemann–Roch theorem and Serre duality
Theorem 18.24 (Serre duality). The map θ induces an isomorphism
θ : H0(X,ΩX(−D))→ H1(X,OX(D))∨
Proof. θ is injective. Let ω ∈ H0(X,ΩX(−D)) be so that θ(ω) = 0. Then by
the previous lemma, we have that ω ∈ ΩX(−D′)(X) for every divisor D′. This
implies that ω = 0, since it is impossible to have a non-zero ω ∈ M so that
divω −D′ ≥ 0 for every divisor D′ (just take D′ to have very high degree).
θ is surjective. θ is K-linear, and we can view it as a map from M to J .
By definition, dimKM = 1, and we also have dimK J ≤ 1. An injection of
finite dimensional vector spaces into a smaller dimensional vector space must be
surjective. Hence if α ∈ J(D), we get a meromorphic differential ω ∈M so that
θ(ω) = α, and so the previous lemma shows that ω ∈ Ω(−D).
285
Chapter 19
Applications of the
Riemann–Roch theorem
In this chapter we give a few of the (many) consequences of the Riemann–Roch
formula. We start by translating the results of Chapter 16 into concrete numerical
criteria for a divisor D to be base-point free or very ample. Then we use these
results to classify all curves of all genus ≤ 4.
19.1 Morphisms of curves
Let f : X → Y be a morphism of schemes. Recall that we say that f is said to
be finite if the following two conditions are satisfied:
(i) f is affine, i.e., if U ⊆ Y is open and affine, then f−1(U) ⊆ X is afiine.
(ii) If U = SpecA and V = f−1(U) = SpecB, then B is a finite A-algebra (via
f#).
For such morphisms, the pushforward f∗OX is a coherent OY -module.
If y ∈ Y is a closed point, corresponding to the sheaf of ideals my ⊆ OY , then
the scheme theoretic fibre Xy is defined in X by the ideal sheaf f ∗my ·OX ⊆ OX .
Note that in the case X = SpecB, and Y = SpecA, the scheme theoretic fibre
over y is naturally isomorphic to Spec(B/myB) where my is the maximal ideal
in the point y. If k(y) = OY,y/myOY,y = A/m denotes the residue field, then we
have f−1(x) ' Spec(B⊗A k(y)). Note that B⊗A k(y) is a finite algebra over the
field k(y). Such an algebra has only finitely many prime ideals (and all of these
287
19.1. Morphisms of curves
are maximal). In particular, a finite morphism can only have a finite number of
preimages over a closed point y ∈ Y .
The converse of this statement is of course not true. If f : U = SpecAf →X = SpecA is an open immersion, e.g., f : Spec k[x, y]/(xy − 1) → Spec k[x],
the number of preimages are of course finite (0 or 1 in fact). However, Af is
usually not finite as an A-algebra.
However if f : X → Y is a projective morphisms (or more and its fibers are
finite, then f is finite. We will prove this in the next section for curves.
We can be a little bit more precise when talking about the ‘number of
preimages’ of f . Let Y = SpecA be an integral noetherian scheme; so that A is
a noetherian integral domain. Let f : X → Y be a finite morphism. Then X is
also affine, say X = SpecB for some finite A-algebra B. We finally suppose that
f is dominant. This means that the image of each component of X is dense in X.
As A is an integral domain, this is equivalent to A ⊆ B, and q ∩ A = 0 for each
associated primeideal q ⊆ B. It follows that we can extend the inclusion A ⊆ B
to an inclusion K(A) ⊆ K(B). Since B is a finite A-algebra, this extension is
finite. We define the degree of f as the degree of the field extension
[K(B) : K(A)] = dimK(A) K(B).
Example 19.1. Let Y = Spec k[x], X = Spec k[x, y]/(x−y2) and let f : X → Y
denote the projection map (induced by k[x] ⊆ k[x, y]/(x−y2). This is a morphism
of degree two, since K(X) = k(y) is a degree two extension of K(Y ) = k(x)
(since x = y2). Note that the fiber of f over the point 0 ∈ Y has set-theoretically
only one preimage, namely the point (0, 0) ∈ Y corresponding to (0, 0). However,
the scheme theoretic fiber here has a non-reduced structure that reflects the
number 2:
Xy = Spec k[x, y]/(x− y2)⊗k[x] k(x) ' Spec k[y]/y2
19.1.1 Morphisms of curves are finite
Let us now specialize to the case of curves. For a closed point x ∈ X, the local
ring OX,x is a discrete valuation ring and there is an element t ∈ OX,x generating
the maximal ideal m. We say that t is a local parameter or a uniformizing
parameter if vx(t) = 1. Note that we can always normalize the valuation so that
a generator of m has valuation 1.
Proposition 19.2. Let X be a projective non-singular curve over k and let Y
be any curve. Let f : X → Y be a morphism. Then either
(i) f(X) = a point in Y ; or
288
Chapter 19. Applications of the Riemann–Roch theorem
(ii) f(X) = Y ;
In the case (ii) K(X) ⊇ K(Y ) is a finite extension, f is a finite morphism and
Y is also projective.
Proof. Since X is projective, the image f(X) must be closed in Y (Add this
theorem). On the other hand X is irreducible, and hence so is f(X). Hence
f(X) is either a point or all of Y .
If f(X) = Y , then f is dominant, so we get an inclusion of function fields
K(Y ) ⊆ K(X). Both of these fields are finitely generated over k of the same
transcendence degree (i.e., 1), so the extension is finite algebraic.
Let V = SpecB ⊆ Y be an open set of Y and let A denote the integral
closure of B in K(X). Then by [Hartshorne I.6.7] we have SpecA = f−1(V ). So
since A is a finite B-module, we have that f is finite.
We also have the following theorem of [Hartshorne Ch. I. 6.12]:
Theorem 19.3. There is an equivalence of categories
1. Non-singular projective curves, and dominant morphisms
2. Function fields K of dimension 1 over k and k-homomorphisms.
19.1.2 Pullbacks of divisors
Given f : X → Y (non-constant) we have a well-defined pullback map
f ∗ : Pic(Y )→ Pic(X)
sending the class of an invertible sheaf L on X to that of f ∗L. If L = OX(D)
for some divisor, we can make this pullback a bit more explicit as follows.
Let t ∈ OY,y denote the uniformizing parameter. We consider t as an element
of K(X) via the field extension K(X) ⊇ K(Y ). In particular, we can talk about
the valuations of t in each local ring OX,x and define the Weil divisor
f ∗(y) =∑
x∈f−1(y)
vx(t)x
(Since f is finite, there are only finitely many preimages of y, and so the sum is
finite). The expression is also independent of the choice of t: If t′ is a different
uniformizing parameter, we can in any case write t′ = ut where u is a unit, so
that vx(t) = vx(t′).
Thus we get a well defined map
f ∗ : Div(Y )→ Div(X)
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19.1. Morphisms of curves
which descends to the above map f ∗ : Pic(Y )→ Pic(X).
Lemma 19.4. We have deg f ∗D = deg f · degD.
Proof. TODO.
Definition 19.5. For a morphism f : X → Y , we call the number ex = vx(t)
the ramification index of f at x. If ex > 1, we say that f is ramified at x.
Example 19.6. Let A = k[x] and let B = k[x, y]/(x− y2) ' k[y] where k is a
field. Let X = SpecB and let Y = SpecA. Let f : X → Y be the morphism
induced by the inclusion A → B (thus x 7→ y2).
The morphism f is ramified only at the origin (0, 0), and here the ramification
index is two. Indeed, x is a uniformizing parameter of OY,y = k[x](x), while y is
the uniformizer of OX,x = B(x,y). Then we have v(0,0)(x) = v(0,0)(y2) = 2.
The reader might notice a resemblance between the previous example and
Example 11.41, where ramification was defined in terms of the relative sheaf of
differentials ΩX|Y . In that example, ΩX|Y was a torsion sheaf supported on the
single point (0, 0). This correspondence between the two notions of ramification
is a general fact (at least in characteristic 0), and we have the useful formula for
the ramification indexes of curves:
ep = length(ΩX|Y )p + 1
19.1.3 Morphisms to P1
Let X be a non-singular curve, and let f ∈ K. Then f induces a morphism
φ : X → P1 in the following way. Let U = X−Supp f−1(∞) and V = X−f−1(0),
so that f ∈ OX(U) and 1/f ∈ OX(V ).
Write P1 = Proj k[x0, x1]. On D+(x0), the map k[x1/x0]→ OX(U) sending
x1/x0 7→ f , induces a map φU : U → D+(x0) ⊆ P1. Similarly, on D+(x1), we
get a map φV : V → P1. These maps coincide on U ∩ V , and therefore we get a
morphism
φ : X → P1
This morphism is non-constant, hence finite.
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Chapter 19. Applications of the Riemann–Roch theorem
19.2 Very ampleness criteria
Recall the criterion of Theorem 16.18, that an invertible sheaf L is very ample if
and only if its linear system separates points and tangent vectors. Using Riemann–
Roch we can translate that result into a very simple, numerical criterion for very
ampleness on a curve:
Theorem 19.7. Let X be a non-singular projective curve and let D be a divisor
on X. Then
(i) |D| is base-point free if and only if
h0(D − P ) = h0(D)− 1 for every point P ∈ X.
(ii) |D| is very ample if and only if
h0(D − P −Q) = h0(D)− 2 for every two points P,Q ∈ X
(including the case P = Q)
(iii) A divisor D is ample iff degD > 0
Proof. (i) We take the cohomology of the following exact sequence
0→ OX(D − P )→ OX(D)→ k(P )→ 0
and get
0→ H0(X,OX(D − P ))→ H0(X,OX(D))→ k
From this sequence, we get h0(D)− 1 ≤ h0(D − P ) ≤ h0(D).
The right-most map takes a global section of OX(D) and evaluates it at P . To
prove that |D| is base point free, we must prove that there is a section s ∈ OX(D)
which does not vanish at P , or equivalently, that the map H0(X,OX(D))→ k
is surjective. But this happens if and only if h0(D − P ) = h0(D)− 1.
(ii) If the above inequality is satisfied, we see in particular that |D| is base-
point free. So it determines a morphism φ : X → Pn. We can use Theorem
16.18 that ensure that φ is an embedding. That is, φ separates (a) points and
(b) tangent vectors.
For (a), we are assuming that h0(D − P − Q) = h0(D) − 2, so the divisor
D − P is effective and does not have Q as a base point (by (i)). But this means
that there is a section of H0(X,OX(D − P )) which doesn’t vanish at Q. We
have H0(X,D − P ) ⊆ H0(X,D), so we get a section of OX(D) which vanishes
at P , but not at Q. Hence |D| separates points.
For (b), we need to show that |D| separates tangent vectors, i.e., the elements
of H0(X,OX(D)) should generate the k-vector space mPOX(D)/m2POX(D) at
291
19.3. Curves on P1 × P1
every point P ∈ X. This condition is equivalent to saying that there is a divisor
D′ ∈ |D| with multiplicity 1 at P : Note that dim TP (X) = 1, dim TPD′ = 0 if
P has multiplicity 1 in D′ and dim Tp(D′) = 1 if P has higher multiplicity. But
this is equivalent to P not being a base-point of D − P . Applying (i) again, we
see that h0(D − 2P ) = h0(D)− 2, so we are done.
(iii) By definition, D is ample if mD is very ample for m 0. So the result
follows by the next result, since any divisor of degree ≥ 2g+ 1 is very ample.
Proposition 19.8. Let X be a non-singular projective curve and let D be a
divisor on X. Then
1. If degD ≥ 2g, then |D| is base-point free.
2. If degD ≥ 2g + 1, then |D| is very ample.
Proof. By Serre duality, h1(D) = h0(K−D) = 0 because degD > degK = 2g−2.
Similarly, h1(D − P ) = 0.
(i) Applying Riemann–Roch, we find that h0(D − p) = h0(D) − 1 for any
P ∈ X, so we are done by the above theorem.
(ii) In this case we also get h1(D − P − Q) = 0, so Riemann–Roch shows
that h0(D − P −Q) = h0(D)− 2, which is the conclusion we want.
19.3 Curves on P1 × P1
Let us consider one central example, namely curves on Q = P1×P1. Recall, that
Cl(Q) = ZL1 ⊕ ZL2 where L1 = [0 : 1]× P1 and L2 = P1 × [0 : 1].
We can use this to prove that Q contains non-singular curves of any genus
g ≥ 0. (This is in contrast with the case of P2, where only genera of the form(d−1
2
)were allowed).
To prove this, consider the divisor D = aL1 + bL2 where a, b ≥ 1. D is
effective, so let C ∈ |D| be a generic element.
Lemma 19.9. C is non-singular.
To compute the genus of C, we use the formula 2g− 2 = deg ΩC . So we need
to find ΩC and compute its degree. This is best computed using the Adjunction
formula of Proposition 11.43:
ΩC = ωQ ⊗ OQ(C)|X= OQ(−2L1 − 2L2)⊗ O(aL1 + bL2)|C= OC((a− 2)L1 + (b− 2)L2)
To compute the degree of this, we consider the degrees of L1|C and L2|C separately.
Note that the degree degL1|C is invariant under linear equivalence, so we can
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Chapter 19. Applications of the Riemann–Roch theorem
compute the degree of any [s : t]× P1 for a general point [s : t]× P1. The point
is that as a Weil divisor, L1|X is obtained by intersecting [s : t] × P1 with X.
When [s : t] ∈ P1 is a general point, the intersection X ∩ [s : t]× P1 is a reduced
subscheme of X, consisting of b points (as C ⊆ Q = P1 × P1 is a divisor of type
aL1 + bL2). Hence degL1|C = b and degL2|C = a. It follows that
2g − 2 = deg ΩC = (a− 2)b+ (b− 2)a = 2ab− 2a− 2b
Solving for g gives us the following theorem:
Theorem 19.10. Let Q = P1 × P1. Then a generic divisor C in |aL1 + bL2| is
a smooth projective curve of genus
g = (a− 1)(b− 1).
In particular, Q contains non-singular curves of any genus g ≥ 0.
19.4 Curves of genus 0
The results of the previous results are particularly strong when the genus is small.
For instance, when g = 0, any divisor of positive degree is very ample! We can
use this to classify all curves of genus 0.
Lemma 19.11. Let X be a non-singular curve. Then X ' P1 if and only if
there exists a Cartier divisor D such that degD = 1 and h0(X,OX(D)) ≥ 2. In
this case, the divisor D is even very ample.
Proof. This follows directly from Proposition 19.8. Still, let us give an alternative
proof, which is perhaps a little bit more enlightening.
Let g ∈ H0(X,OX(D)). Then D′ ∼ div g +D ≥ 0, so we may assume that
D is effective. Since degD = 1, we must have D = p for some point p ∈ X. Now
take f ∈ H0(X,OX(D))− k. As above, f induces a morphism φ : X → P1. This
morphism has degree equal to 1, so it is birational, and hence X is isomorphic
to P1.
Proposition 19.12. A non-singular curve X is isomorphic to P1 if and only if
Pic(X) ' Z.
Proof. We have seen that the Picard group of any Pnk is isomorphic to Z.
Conversely, suppose X is a curve with Pic(X) ' Z. Let p, q be two distinct
points on X. By assumption, OX(p) ' OX(q), so the linear system |p| has at
least two sections. Then X ' P1k by the previous lemma.
Theorem 19.13. Any curve of genus 0 over an algebraically closed field is
isomorphic to P1.
293
19.5. Curves of genus 1
Proof. Let p ∈ X be a point and consider the divisor D = p. If X has genus 0,
then 1 = degD > 2g − 2 = −2, so H1(X,OX(D)) = 0. Then Riemann-Roch
tells us that
dimH0(X,OX(D)) = 1 + 1− 0 = 2
Hence X ' P1k by Lemma 19.11.
We conclude by yet another characterisation of P1:
Lemma 19.14. Let C be a non-singular projective curve and D any divisor of
degree d > 0. Then
dim |D| ≤ degD
with equality if and only if C ' P1.
Proof. This is (IV, Ex. 1.5) in Hartshorne. Although one might guess that this
lemma follows from Riemann-Roch this is not the case. Riemann-Roch gives a
different sort of relationship between the dimension and degree of a divisor. We
induct on d.
First suppose d = 1. There is an exact sequence
0→ OC → OC(P )→ k(P )→ 0.
Now h0(OC) = 1 and h0(k(P )) = 1 therefore h0(OC(P )) ≤ 2 so dim |P | ≤ 1. If
dim |P | = 1 then |P | has no base points so we obtain a morphism C → P1 of
degree degP = 1 which must be an isomorphism so C is rational.
Next suppose D = P1 + · · ·+Pd. Let D′ = P1 + · · ·+Pd−1. There is an exact
sequence
0→ OC(D′)→ OC(D)→ k(Pd)→ 0.
Now h0(OC(D′)) ≤ d by induction and h0(k(Pd)) = 1 so h0(OC(D)) ≤ d+1, there-
fore dim |D| ≤ d with equality iff h0(OC(D′)) = d. By induction h0(OC(D′)) = d
iff C is rational.
19.5 Curves of genus 1
A curve in P2k of degree three has genus 1. This follows from our earlier work on
the canonical bundle, which showed ωX ' OP2k(d− 3)|X ' OX . In this section,
we note that in fact every curve of genus 1 arises this way:
Theorem 19.15. Any projective curve of genus 1 can be embedded as a plane
cubic curve in P2k.
Proof. Pick a point P ∈ X and consider the divisor D = 3P . D has degree
3 ≥ 2g + 1, so it is very ample. Let φ : X → P2k denote the embedding. The
image φ(X) is a smooth curve of degree equal to deg φ∗OP2(1) = degD = 3.
294
Chapter 19. Applications of the Riemann–Roch theorem
In contrast to the g = 0 case, there is however many non-isomorphic genus 1
curves. For instance, in the Ledengre family of curves in Xλ ⊆ P2 given by
y2z = x(x− z)(x− λz)
where λ ∈ k, each Xλ is isomorphic to at most a finite number of other Xλ′ ’s.
19.5.1 Divisors on X
Let us study the divisors on X. To make the discussion a bit more concrete, let
X ⊆ P2 be the curve given by y2z = x3 − xz2. We claim that there is an exact
sequence
0→ X(k)→ Cl(X)deg−−→ Z→ 0
This means that the Class group Cl(X) is very big – its elements are in bijection
with the k-points of X, of which there might be uncountably many. In particular
X cannot be isomorphic to P1.
If L ⊆ P2 is a line, we get a divisor L|X . That is, we take a section s ∈ OP2(1)
defining L and restrict it to X. The divisor of s ∈ OX(1) consists of three points
P,Q,R (counted with multiplicity). In particular, since any two lines are linearly
equivalent on P2, we get for every pair of lines L,L′ and corresponding triples
P,Q,R, a relation
P +Q+R ∼ P ′ +Q′ +R′
(where ∼ denotes linear equivalence).
Let us consider the point O = [0, 1, 0] on X. This is a special point on X: it
is an inflection point, in the sense that there is a line L = V (z) ⊆ P2 which has
multiplicity three at O, so that L restricts to 3O on X. This has the property
that any three collinear points P,Q,R in X satisfy
P +Q+R ∼ 3O
We will use these observations to define a group structure on the set of closed
points X(k), using the point O as the identity. The group structure will be
induced from that in Cl(X).
Consider the subgroup Cl0(X) ⊆ Cl(X) consisting of degree 0. This fits into
the exact sequence
0→ Cl0(X)→ Cl(X)deg−−→ Z→ 0
295
19.5. Curves of genus 1
We now define a map
ξ : X(k) → Cl0(X)
P 7→ [P −O]
Lemma 19.16. ξ is a bijection.
Proof. ξ is injective: ξ(P ) = ξ(Q) implies that P ∼ Q. Then P = Q (otherwise
X would be rational, by Proposition 19.12). (Alternatively, it follows because
h0(X,OX(P )) = 1).
ξ is surjective: Take a divisor D =∑niPi ∈ Div(X) of degree 0. Then
D′ = D+O has degree 1, so by Riemann–Roch, H0(X,OC(D′)) is 1-dimensional.
Hence there exists an effective divisor of degree 1 in |D′|, which must then be
of the form D′ = Q. But that means that D + O ∼ Q, or, D ∼ Q − O, as
desired.
Using this bijection, we can put a group structure on the set X(k):
Theorem 19.17. The set of k-points X(k) on a genus 1 form a group.
The group law has the following famous geometric interpretation. Given two
points p1, p2 ∈ X, we draw the line L they span (see the figure below). This
intersects X in one more point, say p3. In the group Cl0(X) we have
p1 + p2 + p3 = 3O
To define the ‘sum’ p1 + p2 (which should be a new k-point of X), we then look
for a point p4 so that
p4 −O = (p1 −O) + (p2 −O)
or in other words, p4 +O = p1 +p2. By the above, this becomes p4 +O = 3O−p3
or, p3 + p4 + O = 3O. This tells us that we should define p4 as follows: We
draw the line L′ from O to p3 (shown as the dotted line in the figure), and define
p4 to be the third intersection point of L′ with X. By construction, we get
(p1 −O) + (p2 −O) = (p4 −O) in Cl0(X).
296
Chapter 19. Applications of the Riemann–Roch theorem
Given the equation of X in P2, and coordinates for the points p1, p2, we can
of course write down explicit formulas for the coordinates of p4, and they are
rational functions in the coordinates of p1, p2. This is almost enough to justify
that X is a group variety, i.e., it is an algebraic variety equipped with morphisms
m : X ×X → X satisfying the usual group axioms.
19.6 Curves of genus 2
Let X be a non-singular projective curve of genus 2. We saw one example of
such a curve earlier in this chapter, namely the curve obtained by gluing two
copies of the affine curve y2 = p(x) where p(x) is a polynomial of degree five.
The condition that X is non-singular implies that p has distinct roots.
We already saw in Chapter XX that a genus 2 curve cannot be embedded in
the projective plane P2k (since 2 is not a triagonal number). However, we show
the following:
Theorem 19.18. Any curve of genus 2 is isomorphic to a hyperelliptic curve
Here, a curve C is said to be hyperelliptic if there is a base point free linear
system of degree 2 and dimension 1. Equivalently, there exists points P,Q ∈ Xso that the invertible sheaf L = OX(P + Q) is globally generated and by two
global sections.
It is classical notation that a base point free linear system of degree d and
dimension r is called a grd. So to say that a curve is hyperelliptic is to say that it
has a g12.
Example 19.19. If g = 0, then X ' P1. Let D = 2P , then H0(D) =
kx20 +kx0x1 +x2
1, so |D| ' P2 is identified with the space of quadratic polynomials
up to scaling. If we take two quadratic polynomials q0, q1 with no common zeroes,
we get a base-point free linear system g12 ⊆ |D|.
297
19.7. Curves of genus 3
Example 19.20. If g = 1 any divisor of degree 2 gives a g12 by Riemann-Roch.
Indeed, if D has degree 2 then
h0(D)− h0(K −D) = 2 + 1− g = 2
and deg(K − D) = −2 so h0(D) = 2 and hence dim |D| = 1. This D is
base-point free, since D − p has degree 1, and hence since X is not rational,
h0(D − p) = 1 = h0(D)− 1.
Example 19.21. Let X ⊆ P1 × P1 be a smooth divisor of bidegree (2, g + 1).
Then KX ' OP1×P1(0, g − 1) and X has genus g. Moreover, the projection
p2 : X → P1 is finite of degree 2, which shows that X is hyperelliptic.
The projections p1, p2 : Q → P1 give rise to a degree 2 and a degree g + 1
morphism of X to P1. Thus there exists a 2:1 morphism f : X → P1. f
corresponds to a base point free linear system on X of degree 2 and dimension 1.
Thus X is hyperelliptic.
In this example, ΩX = OQ(X) ⊗ ωQ|X = OQ(2, g + 1) ⊗ OQ(−2,−2) =
OX(0, g − 1). The latter invertible sheaf has g independent global sections so
X has genus g. Moreover KX is base point free, but not very ample, since the
corresponding morphism X → P2 is not an embedding (it maps X onto a conic).
To prove the theorem, we must produce a degree two map φ : X → P1. We
have a natural candidate: the canonical divisor KX , which has degree 2g− 2 = 2.
We claim that KX is base point free.
Note that we cannot apply Proposition 19.8 directly to prove this, since the
degree is too small. However, we can use Riemman–Roch to check directly that
the conditions in Theorem 19.7 apply. That is, we need to show that for every
point P ∈ X, we have
h0(X,KX − P ) = h0(X,KX)− 1 = 2− 1 = 1
Applying Riemann–Roch to the divisor D = P , we also get h0(P )−h0(KX−P ) =
1 + 1− 2 = 0. As P is effective, and X is not rational, we have h0(P ) = 1, and
so also h0(X,KX − P ) = 1, as we want.
19.7 Curves of genus 3
The case of curves of genus 3 is especially interesting. We have seen two examples
of curves of genus 3 so far:
Example 19.22. A plane curve X ⊆ P2 of degree d = 4 has genus 12(d− 1)(d−
2) = 3.
298
Chapter 19. Applications of the Riemann–Roch theorem
Notice that
ΩX = OP2(d− 3)|X = OX(1)
so ΩX is very ample, since it is the restriction of the very ample invertible sheaf
OP2(1) on P2. Hence KX is very ample, and the corresponding morphism is
exactly the given embedding X → P2.
Example 19.23. The curves in Section 18.4 on page 271 can be chosen to have
genus g = 3. In this case, X admits a 2:1 map to P1.
Example 19.24. A curve X on the quadric surface Q ' P1 × P1 in P3 of type
(2,4) is hyperelliptic. It is a curve of degree 6 and genus 3.
Thus these examples are a bit different. The curves in the first example are
‘canonical’ whereas the curves in the other two are ‘hyperelliptic’. We show that
this distinction is a general phenomenon for curves of genus three:
Proposition 19.25. Let X be a curve of genus 3. Then there are two possibili-
ties:
(i) KX is very ample. Then X embeds as a plane curve of degree 4.
(ii) KX is not very ample. Then X is a hyperelliptic curve, and it embeds as a
(2, 4) divisor in P1 × P1. Moreover, KX ∼ 2F , where F = L1|X .
We willl deduce this from a more general result:
Theorem 19.26. Let X be a curve of genus ≥ 2. Then K is very ample if and
only if X is not hyperelliptic.
Proof. K is very ample if and only if h0(K − P −Q) = h0(K)− 2 = g − 2 for
every P,Q ∈ X. By Riemann–Roch, we compute
h0(P +Q)− h0(K − P −Q) = 2 + 1− g = 3− g
Hence K is very ample if and only if h0(P +Q) = 1 for every P,Q.
If X is hyperelliptic, then there is a map φ : X → P1, so that φ∗([1 : 0]) =
P +Q for some points P,Q ∈ X (possibly equal). Here the linear system |P +Q|is 1-dimensional, so h0(X,P +Q) = 2, and hence KX is not very ample.
If X is not hyperelliptic, we have h0(X,P +Q) = 1 for any P,Q (otherwise
it is ≥ 2, and P + Q induces a map X → P1 of degree two), and hence KX is
very ample.
We still need to check the last part of the above theorem, namely that every
hyperelliptic curve arises as a curve of type (2,4) on Q ⊆ P3.
299
19.7. Curves of genus 3
We proceed as follows. Let D = P1 + · · · + P4 denote a generic degree 4
divisor on X (so P1, . . . , P4 are general points of X). By Riemann–Roch, we get
h0(D)− h0(K −D) = 4 + 1− 3 = 2
We claim that h0(K −D) = 0, so that h0(D) = 2. Note that K −D has degree
2g − 2− 4 = 0, so K −D is a divisor of degree 0. This is effective if and only
if K ∼ D. However, there is a 4-dimensional family of divisors of the form
P1 + · · · + P4, wheras the space of effective canonical divisors has dimension
dim |K| = 2. Hence if the points Pi are chosen generically, K −D will not be
effective, and hence the claim holds.
From this, we obtain a linear system |D| of dimension 1. We claim that D is
base point free. We need to show that
h0(D − P ) = h0(D)− 1 = degD + 1− 3)− 1 = 1
for every point P . Suppose not, and let P be a base point of D. Since D =
P1 + P2 + P3 + P4 we may suppose that P = P4.
By Riemann–Roch, we are done if we can show h0(K−D+P ) = 0. However,
K − D + P = K − P1 − P2 − P3. There is a 3-dimensional space of effective
divisors of the form P1 + P2 + P3 for points Pi ∈ X, but only a 2-dimensional
linear system of effective canonical divisors |K|. Hence K−D+P is not effective.
We therefore have two morphisms from our hyperelliptic curve X; f : X → P1
(induced by the g12) and g : X → P1 (induced by D). By the universal property
of the fiber product, this gives a morphism
φ = (f × g) : X → P1 ×k P1
We claim that this is a closed immersion. Let F = P +Q ∈ |g12|. The map D+F
induces the map F : X → P3, which coincides with j φ where j : P1 × P1 is the
Segre embedding. To prove the claim, it suffices to show that F is an embedding,
or equivalently that D + F is very ample.
First claim that K ∼ 2F . Since both of these divisors have degree 4 it
suffices to show that K − 2F is effective. Note that in any case h0(X, 2F ) ≥ 3,
since if H0(X,F ) = 〈x, y〉, then x2, xy, y2 are linearly independent in H0(X, 2F )
(understand why!). Now applying Riemann–Roch to D = 2F , we get
h0(2F )− h0(K − 2F ) = 4 + 1− 3 = 2
so h0(K − 2F ) ≥ 1, and K ∼ 2F as we want.
Now, to show that D + F is very ample, we need to show that
h0(X,D + F − p− q) = h0(D + F )− 2
300
Chapter 19. Applications of the Riemann–Roch theorem
for any pair of points p, q ∈ X. By Riemann–Roch again, we can conclude if we
know that h0(K −D − F + p+ q) = 0. But since K ∼ 2F , we have
K −D − F + p+ q ∼ F −D + p+ q
These are divisors of degree 0, so if this is effective, we must have D ∼ F + p+ q.
However, the space of effective divisors of the form D′ + p + q with D′ ∼ F
is 3-dimensional (since |F | has dimension 1, and p and q can be chosen freely
on X). On the other hand, as we have seen, the space of divisors of the form
D = P1 + · · · + P4 is of dimension 4, so choosing D generically means that
thisF −D+ p+ q is not effective. It follows that h1(D− p− q) = h0(D+F )− 2
and hence D is very ample.
19.8 Curves of Genus 4
Recall that curves of genus g ≥ 2 split up into two disjoint classes.
(I) Hyperelliptic curves: X admits a 2:1 to P1
(II) Canonical curves: KX is very ample
Here’s an example of a genus 4 curve in P1 × P1:
Example 19.27. Consider a type (2, 5) curve C on Q ⊆ P3. Then C has degree
7 = 2 + 5 and C is hyperelliptic (because of the degree 2 map coming from
projection onto the first fact p1 : Q→ P1). A type (3, 3) curve on Q is also of
genus 4. It is a degree 6 complete intersection of Q and a cubic surface. Curves
of type (3, 3) have at least two g13’s.
In fact, using the same strategy as for g = 3, one can show that any
hyperelliptic curve of genus 4 arises this way.
19.8.1 Classifying curves of genus 4
We start with an abstract curve X of genus 4. We may assume that X is not
hyperelliptic (since in that case it embeds as a (2, 5)-divisor on P1 × P1). So
we assume that KX is very ample. Therefore we have the canonical embedding
X → Pg−1 = P3. The degree of the embedded curve is degωX = 2g − 2 = 6.
Thus we can view X as a degree 6 genus 4 curve in P3.
What are the equations of X in P3? To answer this question we use a very
powerful technique in curve theory, namely we combine Riemann–Roch with the
301
19.8. Curves of Genus 4
sheaf cohomology on Pn. Twisting the ideal sheaf sequence of X by OP3(2) and
taking cohomology gives the exact sequence
0→ H0(P3, IX(2))→ H0(P3,OP3(2))→ H0(X,OX(2))→ · · ·
Note that OP3(1)|X = KX . Applying Riemann-Roch states to the divisor
D = 2KX , we get
h0(OX(2)) = deg 2KX + 1− g + h1(OX(D)) = 12 + 1− 4 + 0 = 9.
(Note that h1(OX(2)) = h0(KX − 2KX) = h0(−KX) = 0 since KX is effective).
Since h0(P3,OP3(2)) = 10 it follows that the map H0(OP3(2)) → H0(OX(2))
must have a nontrivial kernel so h0(P3, IX(2)) > 0.
The upshot of this is that we now know that X lies in some surface of degree
2. Since X is integral, this surface cannot be a union of hyperplanes. So X lies
on either a singular quadric cone Q0 = V (xy − z2) or the nonsingular quadric
surface Q = V (xy − zw).
If C lies on Q then it must have a type (a, b) which must satisfy a+ b = 6 and
(a− 1)(b− 1) = 4. The only solution is a = b = 3. Since OQ(3, 3) ' OP3(3)|Q,
this implies that C is the restriction of a divisor on P3, that is, C = Q ∩ S for a
degree 3 surface S.
The other possibility is that C lies on Q0. Computing as before, we obtain
0→ H0(OX(3))→ H0(OP3(3))→ H0(OX(3))→ · · ·
As before one sees that h0(OX(3)) = 15 and h0(OP3(3)) = 20. Thus h0(OC(3)) ≥5. Let q ∈ H0(OC(2)) be the defining equation of Q0. Then xq, yq, zq, wq ∈H0(OC(3)). But h0(OC(3)) ≥ 5 so there exists an f ∈ H0(OC(3)) so that
the global sections xq, yq, zq, wq, f are independent. Thus there is an f not in
(q). Since f 6∈ (q) we see that S = Z(f) 6⊇ Q so C ′ = S ∩ Q is a degree 6
not necessarily nonsingular or irreducible curve. Since C ⊆ S and C ⊆ Q it
follows that C ⊆ C ′. Since these are both integral curves of the same degree
degC = 6 = degC ′, we must have C = C ′. Thus in the case that C lies on Q0
we see that C is also a complete intersection C = Q0 ∩ S for some cubic surface
S.
This proves the following theorem:
Theorem 19.28. Let X be a non-singular curve of genus 4. Then either
(i) X is hyperelliptic (in which case X embeds as a (2, 5)-divisor in P1 × P1);
or
(ii) X = Q ∩ S is the intersection of a quadric surface and a cubic surface in
P3.
302
Appendix A
To be deleted
A.1 L.E.S.
Theorem A.1. Let 0 → F1 → F2 → F3 → 0 be a short exact sequence of
sheaves. Then there is a long exact sequence
0→ H0(X,F1)→ H0(X,F2)→ H0(X,F3)δ−→ H1(X,F1)→ H1(X,F2)→ H1(X,F3)→ · · ·
Proof. This follows essentially by repeated use of the snake lemma. To start, we
have a commutative diagram
0 0 0
0 F1(X) F2(X) F3(X)
0 C 01 (X) C 0
2 (X) C 03 (X) 0
0 Z11 (X) Z1
2 (X) Z13 (X)
A subtle but key point is that the middle row is exact on the right, since the
short exact sequence is exact on every stalk. (In fact this observation is one of
main reasons why we use the Godement sheaves in the definition of cohomology
in the first place.)
Note that H1(X,Fi) is by definition the cokernel of the vertical map in
303
A.1. L.E.S.
column i. Applying the snake lemma, we obtain an exact sequence
0→ H0(X,F1)→ H0(X,F2)→ H0(X,F3)δ−→ H1(X,F1)→ H1(X,F2)→ H1(X,F3)
We will show how to continue this sequence inductively. Applying the above
reasoning to the exact sequence
0→ Z11 → Z1
2 → Z13 → 0
we get a diagram
0 0 0
0 Z11 (X) Z1
2 (X) Z13 (X)
0 C 11 (X) C 1
2 (X) C 13 (X) 0
0 Z21 (X) Z2
2 (X) Z23 (X) 0
(where the vertical maps Z1 → C1 are induced by κ’s, as before) and a new
6-term exact sequence
H0(X,Z1)→ H0(X,Z02 )→ H0(X,Z0
3 )δ−→ H1(X,Z0
1 )→ H1(X,Z02 )→ H1(X,Z0
3 )
Note that by the recursive definition of cohomology, we have
H1(X,Z0i ) = H2(X,Fi)
for i = 1, 2, 3. We claim that this δ factors through a map
H1(X,F3) = H0(X,Z13 )/(imH0(X,C 0
3 )→ H1(F3))→ H1(X,Z01 ).
Again by the snake lemma, we get a diagram
H0(C 01 ) //
H0(C 02 ) //
H0(C 03 ) //
0
H0(Z1
1 ) //
H0(Z12 ) //
H0(Z03 ) δ //
H1(Z01 )
H1(F1) // H1(F2) // H1(F3)
δ99
304
Appendix A. To be deleted
We construct the dotted arrow δ in the diagram as follows: For t ∈ H1(F3) take
a lift s ∈ H0(Z03 ) and map it to δ(s). This is well-defined, since if s′ is another
lift, then s− s′ can be lifted to an element of H0(C 03 ), and the commutativity of
the upper right square shows that δs = δs′. Moreover, another diagram chase
shows that δ(s) = 0 if and only if s is in the image of H1(F2)→ H1(F3). Hence
by identifying H1(X,Z0i ) and H2(X,Fi), we get a sequence
· · · → H1(F2)→ H1(X,F3)δ−→ H2(X,F1)→ H2(X,F2)→ H2(X,F3)
which extends the previous sequence. If we now apply the same argument to
0→ Z i1 → Z i2 → Z i3 → 0
for each i ≥ 2, we see that we can define the remaining sequence inductively.
Write C•(U ,G) for the Cech complex of a sheaf G. From the Godement
resolution F → C 1 → C 2 → · · · , we can form a double complex of sheaves by
setting:
Kp,q = Cp(U ,C q)
We can picture this as a collection of sheaves on a 2D grid, with morphisms
going both in the upwards and the right direction:
......
......
0 C2(U ,F) C2(U ,C 0) C2(U ,C 1) C2(U ,C 2) · · ·
0 C1(U ,F) C1(U ,C 0) C1(U ,C 1) C1(U ,C 2) · · ·
0 C0(U ,F) C0(U ,C 0) C0(U ,C 1) C0(U ,C 2) · · ·
0 0 0 0
So if we set p = 0, the complex K0,• is simply Γ(X,C •), whose cohomology is,
by definition, Hq(X,F). If we set q = 0, we get a complex whose (p, 0)-entry is∏U1,...,Up∈U
Hq(U1 ∩ · · · ∩ Up,F |U1∩···∩Up).
In particular, we find the Cech cohomology as the cohomology of the q = 0
305
A.1. L.E.S.
column. Suppose that Hq(U1 ∩ · · · ∩ Up,F ) = 0 for all p and all q > 0. Then
the morphisms above are exact at Cp(U ,C q) with p, q ≥ 0, and we can conclude
by the following:
Proposition A.2. Suppose that K is a double complex in the first quadrant.
Assume that for all p > 0 we have Hp,•K = 0 and that for all q > 0 we have
H•,qK = 0. (In other words, the columns and rows are all exact, except in
degree 0.) Then are natural isomorphisms HpH0,•K ' HpH•,0K for p ≥ 0.
Proof. This is again a rather straightforward diagram chase. To make the chase
a little bit simpler, we can carry out the following process. We fix p ≥ 0 and
sequentially modify K, and keep track of how the cohomology of K changes if
we replace an entry by 0 and replace all arrows out of that entry by cokernels
and all arrows into that entry by kernels. That is, we replace a sequence
A B Cφ ψ
by either
A kerψ 0φ ψ
or 0 kerψ Cφ ψ
Doing this, we can modify K to get the following diagram:
0 0
K0,p K1,p 0
0 K1,p−1 K2,p−1 0
0. . . . . . Kp−1,2 Kp−1,2 0
0 Kp−1,1 Kp,1 0
0 Kp,0 0
All the arrows are isomorphisms, except possibly the left-most vertical maps, or
the lower horizontal maps. From this diagram it is clear that the cohomology
along the y-axis HpK0,∗ = K0,p is isomorphic to the cohomology along the x-axis,
HpK0,∗ = Kp,0.
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Appendix A. To be deleted
A.2 Pic(X) and H1(X,O×X)
So far, the Picard group has been defined as an abstract group defined in terms
of the tensor product of sheaves. But how do we compute it for a given X?
We can make the definition a little bit more down to earth as follows. The
essential data of L thus lies in how the OX |Ui are patched together: For each Uithere should be isomorphisms φi : LUi → OUi such that φi = φj on the overlaps
Ui ∩ Uj. Note that φj φ−1i is an isomorphism OUi∩Uj → OUi∩Uj . Now, we have
an isomorphism
HomOX (OUi∩Uj ,OUi∩Uj) ' Γ(Ui ∩ Uj,OX) (A.2.1)
(given by h 7→ h(1) and s ∈ Γ(Ui ∩ Uj,OX) gives a homomorphism given by
multiplication by s).
Recall the sheaf of units in O×X ; over an open set U ⊆ X, these where the
elements s having a multiplicative inverse s−1 ∈ Γ(U,OX). Equivalently, for each
x ∈ U , the germ sx does not lie in the maximal ideal of OX,x.
Via the correspondence (A.2.1), we find that the group of isomorphisms
OUi∩Uj → OUi∩Uj correspond exactly to the units in Γ(Ui ∩ Uj → OX). In other
words,
IsomOX (OUi∩Uj ,OUi∩Uj) ' Γ(Ui ∩ Uj,O×X)
To specify L, we must say how we glue OUi and OUj along Ui ∩ Uj – in other
words, we specify a unit sij ∈ Γ(Ui ∩ Uj,O×X) for each i, j. These elements sijcannot be chosen completely randomly, as we need to make sure that things are
compatible on the triple overlaps Ui ∩ Uj ∩ Uk. This means that they satisfy the
one constraint that on Ui ∩ Uj ∩ Uk: sijsjkski = 1, or in other words
sjks−1ik sij = 1
in Γ(Ui ∩ Uj ∩ Uk,OX). A restatement of this is that the collection sij ∈∏i,j Γ(Uij,O
×X) is a 1-cocyle! This means that we get an element in H1(X,O×X).
We can check that this is independent of the choice of cover Ui and that the map
Pic(X)→ H1(X,O×X) is in fact a group homomorphism.
Conversely, any element s ∈ H1(X,O×X) is represented by a cover U = Uiof X and cocycles sij ∈ C1(U ,O×X). The cocyle condition implies that the sijdefine isomorphisms OUi |Uij → OUi|Uij which glue the OUi together to form an
invertible sheaf. This provides an inverse to the above map. We have therefore
shown:
307
A.2. Pic(X) and H1(X,O×X)
Theorem A.3. Let X be a scheme. Then
Pic(X) ' H1(X,O×X).
308