equivariant algebraic geometry - university of...

Equivariant Algebraic Geometry

Dan Edidin

CHAPTER 1

Group Actions and Quotients

In this chapter we develop some basic facts about group actions and quotients.For the most part we work over an arbitrary (noetherian) base. In subsequentchapters, when we focus on equivariant intersection theory, our base will be thespectrum of a field.

1.1. Notation and apologia

Given a fixed ground scheme S an S-scheme is a scheme X together with amorphism X

p→ S. If X p→ S and Y q→ S are S-schemes then an S-morphism is a

morphism of schemes X φ→ Y such that q ◦ φ = p.If X is an S-scheme then its functor of points will be denoted HomS( , X) and

if T is an S-scheme then HomS(T, X) is denoted X(T). If X is an S-scheme then aT -valued point of X is an S-morphism T

x→ X; ie an element x ∈ X(T). If k is a fieldthen a morphism Speck x→ X will be call a point and if k is algebraically closedthen it will be called a geometric point. When X, Y are two S-schemes then the fiberproduct X×S Y will often be denoted X× Y.

Yoneda’s lemma implies that to give a morphism of S-schemes X → Y isequivalent to giving a morphism of functors (natural transformation) Hom( , X) →Hom( , Y).

1.1.1. Warning. Much of the time I will be implicitly assuming that S is a separatedNoetherian scheme and that all S-schemes are of finite type over S. However, alot of the Definitions and Propositions hold without this assumption, and as thesenotes mature I hope to clarify what hypothesis are necessary when.

1.2. Group schemes

1.2.1. Definition and Examples. We will start by giving the general definitionof a group scheme. As usual in algebraic geometry this is a relative notion.

1.2.2. Definition. Let S be a fixed ground scheme. An S-group is an S-schemeG

π→ S equipped with morphisms µ : G ×S G → G (composition), i : G → G (in-verse), e : S → G (identity) satisfying the following conditions.

(i) (Associativity) µ ◦ (1G ×S µ) = µ ◦ (µ×S 1G) as morphisms

G×S (G×S G) = (G×S G)×S G → G×S G → G

(here we use the convention that for any scheme X, 1X refers to the identity mor-phism).

3

4 Equivariant Algebraic Geometry

(ii) (Law of inverse) The compositions

G∆→ G×S G

1G×i→ Gµ→ G

and

G∆→ G×S G

i×1G→ Gµ→ G

both equal e ◦ π.(iii) (Law of identity) The compositions

G = S×S Ge×1G→ G×S G

µ→ G

G = G×S S1×e→ G×S G

µ→ G

are both equal to 1G.

1.2.3. Exercise. (i) (Trivial) Prove that i ◦ i = 1G.(ii) (Easy but very useful). Observe that if G is an S-group scheme and S ′ → S

is any morphism then S ′ ×S G has a canonical structure as an S ′-group scheme.Thus the notion of group scheme can be formulated categorically and so G shouldrepresent a group functor on the category of S-schemes. Make this precise.

1.2.4. Definition. If H and G are two S-groups than an S-morphism φ : H →G is a morphism of groups if φ commutes with the identity, composition andinverse homomorphism. (Equivalenty, if for all morphism T → S the map ofsets φ(T) : H(T) → H(T) is a group homomorphism. An S-group H is called anS-subgroup if φ is an immersion.

1.2.5. Definition. If k is a field and S = Speck then an S-group which is of finitetype is called an algebraic group.

1.2.6. Example. If φ : H → G is a morphism of S-groups then kerφ is defined asthe group representing the kernel of the map S-sheaves Hom( , H) → Hom( , G).The scheme structure is given by the cartesian diagram:

kerφ

��

// S

e

��H

φ // G

If G is separated then the section e : S → G is a closed emebedding [?, EGA4] sokerφ is a closed S-subgroup of H.

1.2.7. Example. The notion of the image of a morphism of S-groups is more subtle.However, in the case that G is a smooth algberaic group then a well known resultin theory of algebraic groups [?, Borel]tates that the image of a morphism (with itsreduced induced scheme structure) is a closed subgroup.

1.2.8. Example. Any finite group G determines an S-group structure on Spec⊕g∈GOS.

1.2.9. Definition. Of fundamental importance is the general linear group GLn(S) =

SpecRwhere R = OS[{Xij}i,j∈{1,...,n},D−1] where D is the determinant function in

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the variables Xij. Multiplication is induced by homomorphism of OS-algebras

µ] : R → R⊗OSR, Xij 7→ ∑

k

XikXkj

Inverse is induced by

i] : R → R, Xij 7→ (−1)i+jD−1Mji

where Mji is the the ji-ith minor of the matrix X = (Xij). (The right hand sideshould just be the adjoint formula for the inverse.) Finally, the identity is inducedby the morphism

R → OS, Xij 7→ δij

The group GL1(S) is denoted Gm(S) or simply Gm.If E is a locally free OS-module then we can define an S-group functor AutS(E)

by setting AutS(E)(S ′) to be the group of OS ′-module automorphisms of s∗E forany morphism S ′

s→ S.

1.2.10. Proposition. The functor AutS(E) is reprsented by and S-group AutS(E).

SKETCH OF PROOF. When E is free then a choice of a basis for E gives anisomorphism of functors AutS(E) → GLn(S) where n = rankE, so in this caseAutS(E) is representable.

Find an open cover {Ui}i∈I which trivializes E with trivializations ϕi : EUi→

GL(n,Ui). Then AutUi(EUi

) ' GL(n,Ui) by choice of basis, i.e., by the trivializa-tions ϕi. Then the isomorphism ϕ−1

j ◦ ϕi : EUi∩Uj→ EUi∩Uj

induce automor-phisms of AutUi∩Uj

(EUi∩Uj) which patch together to define a scheme AutS(E).

One can now show AutS(E) is naturally represented by AutS(E). �

1.2.11. Proposition. If S = Spec(A) is affine, E = P where P is a projective A-module of rank n, then AutS(E) is a closed subgroup of GL(N, S) for someN, i.e.,AutS(E) is linear.

KEY IDEA. Since P is projective we can write P ⊕ K ' F for some free A-module F. Then AutS(E) is a subgroup of AutS(F) ' GL(N, S) for some N. In factϕ ∈ AutS(F) is in AutS(E) if and only if ϕ fixes P and acts identically on K. �

1.2.12. Example.Consider A =

C[x,y](y2−(x3−x))

and look at the projective A-module (x, y). NotethatA is regular since its the coordinate ring of an affine curve of genus 1. The ideal(x, y) is projective of rank 1 since it is locally principal.

As an A-module I is the quotient of A2 via the map πwhich sends(

10

)7→ y

and(

01

)7→ x. Let K = kerπ.

Claim: K is the submodule generated by(

x−y

)and

(y

−(x2 − 1)

).

Proof of Claim. It is clear that these two elements lie in the kernel of π. To seethese two elements generate the kernel of π we suppose that

(r1r2

)7→ r1y+ r2x ∈

(y2−(x3−x)), i.e. there is a s ∈ C[x, y] such that r1y+r2x = sy2+s(x3−x). This isequivalent to (r1−sy)y+(r2−sx2+s)x = 0 in C[x, y]. As y, x is a regular sequencein C[x, y] there is a t ∈ C[x, y] such that r1 − sy = tx and r2 − sx2 + s = −ty. It


follows that(

r1r2

)= t(

x−y

)+ s(

y

−(x2 − 1)

), proving that kerπ is generated by(

x−y

)and

(y

−(x2 − 1)

).

Since I is projective, the projection π has a splitting f : I → A2, given by y 7→(1− x2

xy

)and x 7→ (

−y

x2

).

The section f also identifies I as the kernel of the map p : A2 → A,(

ab

) →ax2+by. Thus K is isomorphic as an A-module to the ideal (x2, y). Hence we havean isomorphism of A-modules A2 = (x2, y)⊕ (y, x)

We can now give equations for AutS(I) as a closed subgroup of AutS(A2) 'Spec

(A[x11, x12, x21, x22,D−1]

)where D = x11x22 − x12x21.

The subgroup of AutS(A2) which acts trivially on K = kerπ is given by twopairs A-linear equations in x11, x21, x21, x22(

x11 x12x21 x22

)(x

−y

)=

(x

−y

)(x11 x12x21 x22

)(y

−(x2 − 1)

)=

(y

−(x2 − 1)

)Note that in any localization of A, the two pairs of equations are equivalent, sinceK = kerπ is locally principal.

The subgroup of AutS(A2) that preserves I ⊂ A2 is the subgroup which pre-serves kerp. The corresponding equations are

(x2 y)

(x11 x12x21 x22

)(1− x2

xy

)=

(0

0

)

(x2 y)

(x11 x12x21 x22

)(−y

x2

)=

(0

0

)Again in any localization the two equations are equivalent, since I is locally

principal.

1.2.13. Exercise. Repeat the above example for the ring A = Z[√

−5] and I =

(3, 2+√

−5).

1.2.14. Remark So far all of the examples we have considered are affine groupschemes; ie the structure morphism G

π→ S is affine. However not all groupschemes, even nice ones (for example smooth and of finite type over a Noetherianbase) are affine. The basic example of a non-affine group is an abelian variety. Overa field, an abelian variety can be defined as complete connected algebraic group.Chevalley’s theorem states that any algebraic group is an extension of an affinealgebraic group by an abelian variety.

1.3. Group actions

1.3.1. Defintion. Let G be an S-group. A left action of G on an S-scheme X is amorphism G ×S X

σ→ X such that σ ◦ (1G × σ) = σ ◦ (µ × 1X). A right action isdefined analogously.


1.3.2. Observation. The axioms of a group imply that G acts on X by automor-phisms. Precisely if T is an S-scheme and T g→ G is a T -valued point of G then gdefines an automorphism of XT := T ×S X as the composition

XT = T ×T XTg×1X→ GT × XT

σT→ XT

where σT is the morphism obtained by base change from σ. The inverse to thisautomorphism is the compoisition

XT = T ×T XTg−1×1X→ GT × XT

σT→ XT

where g−1 = i ◦ g.

1.3.3. Example. (Conjugation). The action of a group on itself by conjugation canbe defined for an artibrary S-group. The action morphism σc : G × G → G is thecomposition

G×S G1G×∆→ G×G×S G×G

s12×i→ G×G×S G1G×µ→ G×G µ→ G

1.3.4. Terminology. Let G be and S-group and let X and Y be S-schemes equippedwith an action of G. We say that an S-morphism φ : X → Y is G-equivariantif φ commutes with the actions of G on X and Y respsectively. If Z is any S-scheme then we say that morphism ψ : X → Z is G-invariant if the diagram

G×S Xσ //

p2

��

X

ψ

��X

ψ // Z

commutes; ie ψ is G-equivariant where Z is given the trivial

G-action.

1.3.5. Exericse. If X → Z and Y → Z are G-equivariant morphisms then there is anaction ofG on X×Z Y such that the projections p1 : X×Z Y → X and p2 : X×Z Y → Y

are G-equivariant. Moreover, this action is unique up to G-equivariant isomor-phism. Hence, the category of S-schemes with a G-action and with morphismsG-equivariant morphisms is closed under fiber products.

1.4. Linear groups

1.4.1. Definition. An S-group G is linear if it is isomorphic to a closed subgroup ofGLn(S) for some n.

Obviously any linear group is affine but remarkably little is known about theconverse. The strongest result is the following.

1.4.2. Proposition. [?, Proposition 13.2]. Let R be a Noetherian, regular ring ofdimension at most two and let S = SpecR. Then any S-group which is flat, affineand of finite type is linear.

When S = Speckwe obtain the well known result.

1.4.3. Corollary. Any affine algebraic group is linear.

1.4.4. Remark. The restriction on the base is due to the method of proof. It usesthe fact that any finitely generated reflexive module over a regular Noetherian ring


of dimension at most 2 is automatically projective. The author does not known acounter-example for other affine schemes.

We give the proof in the case when dimR ≤ 1. It is based on the following localfiniteness result for group actions. Following Mumford we refer to this result asCartier’s lemma.

Observe that to give an action of an affine group G = Spec OG on an affineschemeX = Spec OX is equivalent to giving an OS-algebra homomorphism σ] : OX →OG ⊗OS

OX compatible with the comultiplication map µ] : OG → OG ⊗OSOG.

1.4.5. Lemma. (Cartier’s Lemma, [?, Proposition 11.8bis]) Let S be Noetherianscheme and let G be an S-group of finite type. LetM ⊂ OX be finite OS-submodule.Then there is a finite OS-submodule N ⊃M such that N is co-invariant; ie σ](N) ⊂OG ⊗N.

PROOF OF CARTIER’S LEMMA. Observe that σ](M) is a finitely generated sub-module of OG ⊗OS

OX so its contained in a submodule of the form OG ⊗M ′ withM ′ finitely generated as an OS-module. Explicitly ifM is generated by elementsmiand σ](mi) =

∑j aij ⊗mij then we can takeM ′ to be the OS-module generated by

themij.Now let

N = {x ∈ OX|σ](x) ∈ OG ⊗M ′}By definition M ⊂ N. Also, observe that N ⊂ M ′ because x = (1OS

⊗ e])(σ]x)where we identify OS ⊗OS

OX = OX. ThusN is necessarily finitely generated as wework over a Noetherian base. Now we must check that σ]N ⊂ OG ⊗N.

Let p : OX → OX/M ′ be the projection and let σ] : OX → OG ⊗OSOX/M ′ be the

composition (1OG⊗ p) ◦ σ]. By definition N is the kernel of σ].

Since σ] is given by an action of G = SpecOG on X = Spec OX we have that(1OG

⊗ σ]) ◦ σ] = (µ] ⊗ 1OX) ◦ σ].

Composing with the projection p : OX → OX/M ′ yields the equality (1OG⊗σ])◦

σ] = (µ]⊗p) ◦σ]. Hence (1OG⊗σ]) annihilates σ](N). Now OG is a flat OS-algebra

so ker(1OG⊗ σ]) = 1OG

⊗ (ker(σ]) = OG ⊗N. Thus σ](N) ⊂ OG ⊗N. �

1.4.6. Lemma.. If the module N in Cartier’s lemma is projective then there is amorphism of S-groups G → GLn(S) for some n.

PROOF. We will define a morphism G → AutS(N). By Proposition ?? AutS(N)is a closed subgroup of GL(n, S) for some n.

By adjunction, the map of OS-modulesN → OG ⊗OSN induced by the coaction

map is determined by an element θ ∈ EndOG(OG ⊗OS

N) which is necessarily anautomorphism. Thus given any morphism T

t→ Gwe obtain by pullback an elementt∗θ ∈ Aut(OT ⊗OS

N) and thus morphism of representable functors Hom( , G) →AutOs

(N) and thus a homomorphism of group schems G → AutS(N).Exercise. Verify this morphism of representable functors is in fact a homomorphismof group functors.

�

We can now complete the proof of Proposition 1.4.2 assuming that OS is regularof dimension at most 1.


The multiplication map G× G → G defines an action of G. Let M be finitelygenerated OS-submodule of OG which generates OG as an OS-algebra. By ourprevious lemma we know that we can find a coinvariant finitely generated OS-submoduleN ⊃M. Since OG is assumed to be OS-flat it is torsion free. ThusN isalso torsion free and, being finitely generated, projective because S is regular ofdimension at most one.

We now check that the morphism G → AutS(N) constructed above is a closedembedding. Working locally on S we can assume N is free Finally, we need tocheck that when N is an OS-submodule of OG the morphism G → AutS(N) is aclosed embedding. To check this we can work locally on S and therefore assumethatN is free. In this case it is clear that sinceN contans the generators for OG as anOS-algebra the induced ring homomorphism OS[Xij,D−1] → OG is surjective.

1.5. Linear actions and representations

1.5.1. Definition. Fix a base scheme S and Let E be a locally free sheaf on S. Wedenote the S-scheme Spec(Sym E ∗) by V(E ).

We can define a canonical action σ : Aut(E )×S V(E ) → V(E ) of the S-groupAut(E ) on V(E ) as follows: For any T -valued point T t→ S an element g ∈ G(T) bydefinition defines an automorphism of OT -modules g : t∗E → t∗E and thus andinduced automorphism (going in the opposite direction) of graded OT algebrasSym(t∗E ∗) → Sym(t∗E ∗). Applying Spec gives an automorphism V(t∗E )

·g→V(t∗E ) such that if g, g ′ ∈ G(T) then ·gg ′ = ·g ◦ ·g ′.

When F is a free module of rank n we obtain an action of GLn(S) on AnS . Witha choice of basis for F and its dual F∗ the action map is dual to the homomorphism

OS[T1, . . . , Tn] → OS[{Xij},D−1][T1, . . . , Tn], Ti 7→ n∑j=1

XijTj.

Given a morphism of S-group G φ→ GLn(S) we obtain an action of G on AnSinduced by the map of OS algebras

OS[T1, . . . , Tn] → OG[T1, . . . , Tn], Ti

m∑j=1

7→ fijTj

where fij ∈ Γ(G,OG) is the image of Xij under the morphism φ].

1.5.2. Definition. Let G be an S-group. A representation of G is a morphism ofS-group G → GLn(S). The representation is faithful if the morphism is a monomor-phism.

1.5.3. Definition. An action of σ : G × AnS → AnS is linear if there are globalsections {fij}i,j=1,...,n ∈ Γ(G,OG) such that the morphism σ] : OS[T1, . . . , Tn] →OG[T1, . . . , Tn] is given by Ti 7→ ∑

j=1 fijTj.

1.5.4. Exercise. Prove that to give a representation G → GLn(S) is equivalent togiving a linear action of G on ASn. Show that the representation is faithful if andonly if there is a non-empty G-invariant open U ⊂ AnS such that G acts with trivialstabilizers on U.


1.5.5. Example. The action of Ga(S) = Spec OS[t] on AnS given by the morphismOS[t] → OS[t][T1, . . . , Tn], t 7→ t+ Ti is not a linear action. On the level of points apoint map is given by (a, a1, . . . an) 7→ (a1, . . . , ai + a, . . . , an). In other words Gaacts by affine rather than linear transformations.

1.6. Group actions and inertia

1.6.1. Definition. Let G be an S-group acting on a scheme X by the morphismσ : G× X → X. The inertia IGX of the action is the scheme defined by the cartesiandiagram

IGX

��

// X

∆

��G×S X

σ×1X // X×S X

1.6.2. Exercise. Verify that the product µ : G×S G → G induces a natural productstructure on IGX which makes it into an X-group. When X is separated this is aclosed sub-group of X-groupG×X → X obtained by base change fromG → S. [Usethe universal property of the fiber product to show that there is a morphism of groupfunctors, Hom( , IGX)×Hom( ,X) → Hom( , IGX), etc. which makes Hom( , IGX) asubgroup functor of Hom( , G× X).]

On the level of points IG(X) = {(g, x)|gx = x} ⊂ G × X with the productIGX×X IGX → IGX given by ((g, x), (g ′, x)) 7→ (gg ′, x).

1.6.3. Example. We define the conjugation action c : G×G → G as the composition

G×G µ×i→ G×G µ→ G. (Hopefully this makes conjugation a left action, as that’s allwe’ve defined.) This allows us to define a G-action IGX = (G×S X)X×XX which onthe level of sets the action is simply h · (g, x) = (hgh−1, hx).

This action will be important because the quotient stack (which will be definedlater) [IGX/G] is the inertia stack of the quotient stack [X/G].

1.6.4. Definition. With the notation as above, if x ∈ X(T) is a T -valued pointthen we denote the T -group scheme obtained by base change from IGX as AutT (x),so IGX = AutX(1X). When T = Speck is the spectrum of field we will just writeAut(x).

Likewise, given x, y ∈ X(T) then we define IsomT (x, y) (or Isom(x, y) if T =Speck) by the cartesian diagram

IsomT (x, y)

��

// T

(x,y)

��G×S X

(σ×1X) // X×S X

With this notation AutT (x) = IsomT (x, x).

1.6.5. Observation. If a group G acts on sets X, Y and φ : Y → X is a G-equivariantmap then, since φ(gy) = gφ(y), we observe that if x = φ(y) StabyG is a subgroupof StabxG. Hence, for example, if S acts freely on the set X it acts freely on Y for any


G-equivariant map Y → X. The next proposition verifies that the expected resultholds for group schemes.

1.6.6. Proposition. Let Y φ→ X be a G-equivariant morphism. Then there is amonomorphism of Y-groups IG(Y) → IG(X)×X Y which is a closed embedding ifφ is separated.

PROOF. Since the morphism φ is G-equivariant, the diagram below is commu-tative.

G× YσY×1Y //

1G×φ��

Y × Y

φ×φ��

G× XσX×1X// X× X

Hence the morphism (σY × 1X) factors as

G× Y θ→ (G× X)X×X(Y × Y) → (Y × Y).

For any S-scheme T → S the first morphism can be described on the set of T -valuedpoints by the rule (g, y) 7→ ((g,φ(y)), (gy, y)).

The universal property of fiber products yields an isomorphismG×(Y×XY) →(G× X)X×X(Y × Y) which is on T -valued points is given by the formula

(g, y1, y2) 7→ ((g,φ(y1)), (gy1, y2)) .

Under this identification of fiber products the morphism θ corresponds 1G × ∆φ.Hence θ is a monomorphism which is a closed emebedding if if φ is separated.

Now the diagram

IGY

��

// IGX×X Y //

��

Y

∆

��G× Y

1G×θ// G× (Y ×X Y) ////

��

Y × Y

��G× X // X× X

has cartesian squares. Hence the morphism IGY → IGX×X Y is a monomorphismwhich is a closed embedding when φ is separated. �

1.7. Actions with finite stabilizers

We will be primarily interested actions where the fibers of IGX → X are finitebecause we can often construct quotients by these actions. However there are subtledifferences between various kinds of actions.

1.7.1. Definition. (i) G acts with finite stabilizers if for every point Speck x→ X themorphism Ix := Speck×X IGX → Speck is finite.

(ii) G acts with finite stabilizer if the morphism IGX → X is finite.

(iii) G acts properly if the morphism G× X σ×1x→ X× X is proper.(iv) G acts with trivial stabilizer if the morphism IGX → X is an isomorphism.(v) G acts with freely if the morphism G× X → X× X is a closed embedding.


1.7.2. Remark. We are essentially only interested in affine (or quasi-affine) groupschemes, so the morphism G× X → X× X is automatically (quasi)-affine. Henceif it is proper then it is finite. Thus, for (quasi-)affine group schemes any properaction has finite stabilizer and hence also finite stabilizers.

1.7.3. Remark. Proposition 1.7.4 below implies that IGX → X is an isomorphism ifan only if the stabilizer of every point is trivial (Condition (ii)). Hence the notionthat a group acts with trivial stabilizers is equivalent to the notion that the groupacts with trivial stabilizer which is why it was not included in Definition 1.7.1.

1.7.4. Proposition. The following conditions equivalent.(i) G acts with trivial stabilizer.(ii) For every point Speck x→ X the morphism Aut(x) is the trivial k-group.(iii) The morphism G× X → X× X is a monomorphism.

PROOF OF PROPOSITION 1.7.4. (i) =⇒ (ii) is trivial.(ii) =⇒ (iii) To prove that the morphism G×S X → X×S X is an immersion

we will show that its geometric fibers are either empty or consist of a single reducedpoint. This will imply that it is an immersion by [?][Proposition 17.2.6].

Let Speck(x,y)→ X×S X be a geometric point of X×S X, so the fiber is Isom(x, y).

If y = x then the morphism Speck → X ×S X factors through the diagonalIsom(x, y) = Aut(x) which is a single reduced point by hypothesis. Now k-valuedpoint of the the fiber is pair (g, x) where g is a k-valued point of G and y = g · x.Since k is algebraically closed we see that the fiber is empty unless y = gx, sinceany scheme defined over an algebraically closed field must have a rational point byNullstelensatz.

Hence we may assume y = gx. Note that the points x, y, g are all S-morphismsthen Aut(x, y) also defined by the cartesian diagram.

Aut(x, x)

��

// Speck

(x,y)

��Gk ×k Xk

(σk×1Xk)

// Xk ×k Xk

On the other hand, the element g ∈ G(k) defines an automorphism Xk·→ Xk

which maps x ∈ X(k) y = gx ∈ X(k). This induces an automorphism of k-schemesIsom(x) = Aut(x, x) → Aut(x, y). By hypothesis Isom(x) = Speck so it followsthat Aut(x, y) also consists of a single reduced point.

(iii) =⇒ (i) Since G×S X → X×S X is a monomorphism its geometric fibersare either trivial or empty (loc. cit.). Since IGX has an identity section it follows thatAut(x) = IGX×X Speck cannot be empty and must therefore be the trivial groupscheme. �

1.7.5. Example. If X is separated then the action of any finite group scheme G isproper. Note that if X is not separated then no group can act properly.

1.7.6. Observation. Let f : Y → X be a G-equivariant morphism. Proposition 1.6.6implies that if G acts on X with finite stabilizers (resp. trivial stabilizer) then it alsoacts on Y with finite stabilizers (resp. trivial stabilizer). If in addition f is separated


then if G acts with finite stabilizer on X (resp. properly or freely) then it acts withfinite stabilizer (resp. properly or freely) on Y.

1.7.7. Example. Given a vector (a0, . . . an) ∈ (Z+)n+1 there is an action ofGm on X = An+1 r {0} by the morphism σ : Gm × X → X, (λ, (z0, . . . , zn, ) 7→(λa0z0, λ

a1z1, . . . , λanzn). We will show that the action is proper. In this case the

action is free if and only if all the weights are equal to 1.The proof that the action is proper is similar to the proof that projective

space is separated1. The product X × X is covered by affines Ui × Uj whereUi is the principal open set D(xi) ⊂ A2. The inverse (σ × 1x)−1(Ui × Uj) is theaffine open set Gm × Uij where Uij = D(xixj). Hence the morphism is affine.The dual map (σ × 1X)] is the homomorphism k[x0, . . . , xn, x

−1i , y0, . . . , yn] →

k[t, t−1, z0, . . . , zn, ziz−1j ] given by xk 7→ zk, yl 7→ t−alzl. A direct calculation

shows that a power of every generator of the ring k[t, t−1, z0, . . . , zn, ziz−1j ] is in theimage of this homorphism so k[x0, . . . , xn, x−1

i , y0, . . . , yn] surjects onto a subringR ⊂ k[t, t−1, z0, . . . , zn, (zizj)−1] such that the extension R ⊂ k[t, t−1, z0, . . . , zn, (zizj)−1]is module finite. Hence the morphism is finite.

1.7.8. Example. The action of Gm on X = A2 r {0} has trivial stabilizer but it is notfree. To see that note the closure of the image of the morphism Gm × X

σ×1→ X× Xgiven by (λ, (z0, z1)) 7→ (z0, z1, λz0, λ

−1z1) is subvariety of X × X defined by theequation x0x1 = x2x3 which contains the point (1, 0, 0, 1) which is not in the imageof σ× 1.

Similarly, any action on X with weights (a, b) such that a > 0 and b < 0 hasfinite stabilizer but is not proper.

1.7.9. Example. To produce an action which has finite stabilizers but not finitestabilizer requires some work. The example we give comes from geometric invari-ant theory. Let G = PSL2(C) act on the projective space X = P(H0(P1,OP1(4)) ofquartic forms in two variables.

We consider three G-invariant open subsets of X: U1 which parametrizes formswith of 4 distinct roots, U2 which parametrizes forms with at least 3 roots, andU3 which parametrizes forms where each root has multiplicity at most 2. In [?,GIT] Mumford shows that U1 is the set of stable points for the action of G on Xwhich implies that G acts properly on U1. The open set U3 ⊃ U1 is the set ofsemi-stable points. The stabilizer of the square of a quadratic form is positivedimensional because there is a positive dimensional subgroup of PSL2 fixing anytwo points in P1, so G does not act with finite stabilizers on U3. NeverthelessU3 is an interesting open set because U3 = P(Sym4 V)ss is the set of semi-stablepoints, so a good projective quotient of U3 by G exists. The open set U2 is the locusof semi-stable points where G acts with finite stabilizers. One might expect thatG-action on U2 is reasonably well behaved, but in fact we cshow that G does notact with finite stabilizer on U2.

1Note that if all the weights are equal to 1 then the morphism Gm× (An+1r {0}) → (An+1r {0})is obtained from the diagonal map Pn → Pn × Pn by base change along the quotient morphismAn+1 r {0} → Pn


To see this consider the 1-parameter family in U2 corresponding to the formsxy(x − y)(x − λy) with λ ∈ C. If λ /∈ {0, 1} then the stabilizer of the point corre-sponding to xy(x− y)(x− λy) contains the 4-element subgroup of PSL2 consistingof consisting of

{1,

(0 1

λ 0

),

(1 −1/λ1 −1

),

(1 −1λ −1

)}.

However, the stabilizer of the point corresponding to the form xy(x − y)2 is the

two element subgroup generated by(0 1

1 0

).

If the G-acted with finite stabilizer on U2 then the restriction of the finitemorphism IGU2 → U2 to a morphism I → A1λ, would be finite, where A1λ is theλ-line and I is the pullback of IGU2 to an A1λ-group. Let I1 be the union of theconnected components of I which surject onto A1λ. Since the the target is a smoothcurve the finite map I1 → A1λ is flat. Since the geometric fibers are reduced (beinggroup-schemes over a field of characteristic 0) the map I1 → A1λ is unramified andhence etale. Thus the number of points in the fibers of I1 → A1λ is constant. This isa contradiction since the generic fiber has 4 points, but the fiber of I1 over λ has nomore than 2 points

1.8. Free actions and torsors

1.8.1. Proposition. Let G be a flat2 S-group acting on a scheme P. Let p : P → X bea G-invariant faithfully flat morphism of S-schemes. The following conditions areequivalent:

(i) The diagram

G× Pp2

��

σ // P

p

��P

p // Xis cartesian.

(ii) There exists a faitfhully flat morphism U → X such that U ×X P is G-equivariantly isomorphism to G×U where G acts on U by the morphism G×G×Uµ×1U→ G×Uwhere µ is the composition in G. (Ie h(g, u) = (hg, u).)

PROOF. (i) =⇒ (ii) is trivial.(ii) =⇒ (i). Since P p→ X isG-invariant, p◦p2 = p◦σ as morphismsG×P → X.

Hence by the universal property of the fiber product there is a morphism of X-schemes G× P → P ×X P. By hypothesis this morphism becomes an isomorphismafter base change by the faithfully flat morphism U → X. Hence by descent theorythe map must be an isomorphism (Give a reference). �

1.8.2. Definition. Let G be a flat S-group. A G-invariant S-morphism Pp→ X is a

G-torsor if it satisfies the equivalent conditions of Proposition 1.8.1. A G-torsor istrivial if it is G-equivariantly isomorphic to the torsor G× X p2→ X.

2Note that since an S-group has the identity section e : S → G, iany flat S-group of finite type isautomatically faithfully flat since it is flat and surjective. Check hypothesis here!!


1.8.3. Remark. In the literature ([, SGA3], etc) a torsor P → X is defined as ascheme with the action of an X-group; ie the base is taken to be X rather than S.This definition is equivalent to ours since a torsor P → X for the S-group G is alsoa torsor for the X-group G × X. The reason for our point of view is that we will(attempt to) construct torsors as quotients of free group actions on S-schemes P, sothe base scheme X is not directly defined.

1.8.4. Proposition. Let P p→ X and P ′ p′→ X be G-torsors. If φ : P ′ → P is a

G-equivariant morphism of X-schemes then then φ is an isomorphism.

PROOF. By descent, we may check that p is an isomorphism after faithfullyflat base change. Hence we may assume P ′ and P are both trivial. Given a G-equivariant X-morphism φ : G× X → G× X let f : X → G be the morphism definedon points by p1(φ(e, x)). Since φ is G-equivariant, φ(g, x) = (gf(x), x). Then themorphism ψ : G × X → G timesX given by (g, x) 7→ (gf(x)−1, x) is the inverse toφ. �

1.8.5. Remark. As a consequence of Proposition 1.8.4 we obtain the well knownfact that any commutative diagram of torsors P ′ //

��

P

��X ′ // X

is in fact cartesian. We

can also use Proposition 1.8.4 to obtain a simple characterization of trivial torsors.

1.8.6. Proposition. A G-torsor P p→ X is trivial if and only there is a section ofs : X → P of the morphism p.

PROOF. The section defines a morphism of torsors over X, G×X → P given by(g, x) 7→ g · s(x). �

Proposition 1.8.1 implies that any torsor is locally trivial in the flat topology.

1.8.7. Definition. A G-torsor is P → X locally trivial in the etale topology if there isan etale surjective morphism U → X such that the torsor U ×X P → U is trivial.A G-torsor is locally trivial if there is a Zariski open cover {Ui} of X such that therestriction of P → X to each Ui is trivial.

1.8.8. Proposition. If G → S is smooth then every G-torsor is locally trivial in etaletopology.

PROOF. Since G → S is smooth the projection G × X → X is also smooth. By[?, EGA4] any smooth morphism has an etale local section. Hence there is an etalesurjective morphism U → X such that after base change to U the morphism P → X

has a section. �

1.8.9. Exercise. If G is finite and G = Spec⊕g∈G OS is finite then any G-torsorP → Xwhich is locally trivial is automatically trivial.


1.9. Torsors and vector bundles

1.9.1. Construction of a vector bundle from a principalG-bundle and a represen-tation of G.. Let G → S be a flat S-group and let P → X be a G-torsor. Given arepresentation G → GLn(S), descent theory for affine morphisms (resp. locally freesheaves) produces a vector bundle V → X (resp. locally free sheaf) whose transitionfunctions are determined by the transition functions for P → X. We now describethis construction.

Let U p→ X be a flat covering which trivializes P and choose an isomorphismPU := U×XP → G×U. LetU ′ = U×XU andU" = U ′×UU ′ = U×XU×XU. Denotethe projections U ′ → U by p1, p2 and the projections U ′′ → U ′ by p12, p13, p23respectively. By base change, each of these morphisms is faithfully flat (sinceU → X

is a flat covering) and we obtain the following diagram of covers and morphisms

U"

p12 //p13 //p23 //

U ′p1 //p2

// U

Since p1 ◦ p = p2 ◦ p there is a canonical isomorphism of G-torsors betweenthe fiber products PU ×p1

U ′ and PU ×p2U ′ where the notation ×pi

refers tothe fiber product taken with the respect to the map pi : U ′ → U. Since we haveidentified PU = G × U, this canonical isomorphism gives an isomorphism of G-torsors θ : G×U ′ → G×U ′. Since the morphism θ is G-invariant and commuteswith the projection to U, it is uniquely determined by a morphism φ : U ′ → G suchthat on points θ(g, u ′) = gθ(1, u ′) = (gφ(u ′), u ′). Since in general U ′ need not beconnected, we call this morphism the transition functions for the torsor P → X.Note that φ necessarly maps the image of ∆(U) ⊂ U ′ to the identity.

Since p1 ◦p12 = p1 ◦p13 and p2 ◦p12 = p1 ◦p23, etc. as morphismsU ′′ → U theisomorphism θ : G×U ′ → G×U ′ satisfies the cocycle condition p∗23θ◦p∗12θ = p∗13θ

as isomorphisms G × U ′′ → G × U ′′. Equivalently p13 ◦ φ = (p23 ◦ φ)(p12 ◦ φ)as morphisms U ′′ → G, where (p23 ◦ φ)(p12 ◦ φ) is the product of the morphismsp23 ◦ φ and p12 ◦ φ; ie the composition

U ′(p23◦φ,p12◦φ)→ G×G µ→ G.

1.9.2. Remark. When U =⋃iUi is a Zariski open covering then an isomorphism

PU → G×U is specified by giving an isomorphism for P|Ui→ G×Ui for each i.

If we set Uij := Ui ∩ Uj then the transition functions are φij : Uij → G such thatUii → G is constant with image the identity. These morphism satisfy the cocyclecondition φijφjk = φik. where φijφjk is the product of the morphisms φij andφjk.

Now if G → GLn(S) is a representation then there is a corresponding linearaction G × AnS

ρ→ AnS (or equivalently a coaction F → OG ⊗OSF where F is a

free OS module of rank n.) This action, together with the transition functions forP → X will produce descent data. The theory of descent implies that faithfulyflat morphisms (of finite presentation) are of effective for affine morphisms (or for


locally free sheaves) so the descent data will produce a rank n-vector bundle V → X

(or equivalently3 a locally free sheaf of rank n on X.)The construction is quite simple. We define a linear isomorphism θ ′ : An×U ′ →

An × U ′ by the formula (v, u ′) 7→ (φ(u ′) · v, u ′) on T -valued points v ∈ An(T),u ′ ∈ U ′(T). We leave it to the reader to express this morphism in terms of themorphism θ and action morphism ρ. The axioms for a G-action imply that φsatisfies the cocycle condition if θ, or equivalently φ does.

1.9.3. Construction of principal GLn(S)-bundle from rankn-vector bundle. Givena vector bundle V = V(E ) for some locally free sheaf E of rank n on an S-scheme Xwe can construct GLn(X)-torsor which is locally trivial in the Zariski topology. Con-sider the functor IsomOX

(E ,OnX ). S ince E is locally trivial in the Zariski topologythe functor is locally represented by the scheme Aut(OnX ) = GLn(X) = X×GLn(S),so IsomOX

(E ,OnX )X is represented by a scheme P = IsomOX(E ,OnX ). If we identify

OnX = OX ⊗ OnS then composition gives an action of GLn(S) = Aut OS(OnS ) on P.Hence P is GLn(S)-torsor.

1.9.4. Remark. Since vector bundles are locally trivial in Zariski topology theGLn-torsor P → X constructed above is also locally trivial in the Zariski topology.

1.9.5. Example. The construction of a vector bundle from a torsor and represen-tation is in general not faithful; ie non-isomorphic representations can produceisomorphic vector bundles. For example, consider the µn-torsor C∗ → C∗, given byz 7→ zn. Because µn is a finite group the torsor is not locally trivial in the Zariskitopology. Indeed this torsor is trivialized only after base change along the degree nfinite etale cover C∗ → C∗, z 7→ zn. Now if χ is a character of µn which we viewas a one-dimensional vector space with linear µn-action determined by χ. Theconstruction above produces a line bundle Lχ on C∗ which is necessarily trivialsince Pic(C∗) = 0. Geometrically we can identify Lχ as the quotient of C∗ × A1 bythe free µn action where e2kπi/n(λ, v) = (e2kπi/nλ, χ(e2πki/n)v) with the quotientmap given by (λ, v) 7→ λn, χ(λ)v) where we view χ as a character of C∗ whichextends4 the morphism χµn → C∗. The isomorphism morphism C∗ × A1 givenby (λ, v) 7→ (λ, χ(λ), v) is µn-equivariant when the action on the source given byg(λ, v) = (gλ, v) and the action on the target is given by g(λ, v) = (gλ, χ(g)v). Thisisomorphism induces an isomorphism of quotients C∗ × A1 → Lχ.

1.9.6. Example. Consider the one-parameter family of non-degenerate quadraticforms on C2 given by qλ((v1, v2), (w1, w2)) = v1w1 + λv2w2 where λ ∈ C∗. Theqλ define a non-degenerate quadratic form q on the trivial bundle V = C∗ × C2; amorphism q : V ×C∗ V → C which is bilinear and non-degenerate o each fiber.

Given λ ∈ C∗ and µ ∈ C∗ with µ2 = λ, the linear automorphism (v1, v2) 7→(v1, µv2) identifies qλ with the standard hyperbolic quadratic form q((v1, v2), (w1, w2)) =v1w1 + v2w2. However, because there is no canonical choice of square root, thepair (V, q) is not isomorphic as a bundle with quadratic form to the trivial bundle

3Although the construction for locally free sheaves is a more basic example of descent, we outlinethe equivalent, but slightly more intuitive geometric construction.

4Viewing µn as the group of n-th roots of unity, note that any character of µn has the form g 7→ gl

with 0 ≤ l ≤ n − 1. This naturally extends to the character of C∗, λ 7→ λl.


with constant quadratic form on each form. To trivialize the pair (V, q) we mustpass to the double cover C∗ → C∗, z 7→ z2.

The bundle V with its quadratic form q determines an O(2,C)-torsor. Preciselyconsider the functor Isom((V, q), (V, q0)) whose sections on a scheme T are theglobal isomorphisms between (VT , qT ) and the standard hyperbolic quadratic form(VT , q0). It is a subfunctor of Aut(V) ' GL2×C∗ and is (etale) locally isomorphic tothe subgroup O(2,C)×C∗ of GL2(C∗). Consequently it is represented by a schemeP which is an O(2,C)-torsor. The bundle V is obtained by the descent constructionapplied to the defining representation O(2,C) ⊂ GL2(C) where O(2,C) is thesubgroup preservering the standard hyperbolic quadratic form on C2.

1.9.7. Vector bundles and GLn-torsors. Although non-trivial torsors can producetrivial vector bundles, the following result states every GLn-torsor is uniquelydetermined by a rank n vector bundle.

1.9.8. Proposition. ?? [, SGA3] If P is a principal bundle GLn(S) and V = V(E )is the vector bundle constructed by descent in ?? from the identity representationof GLn(S) then IsomOX

(E ,OnX ) is isomorphic to P. In particular every principalGLn(S)-bundle is locally trivial in the Zariski topology.

1.9.9. Remark. Following Serre(?) we say that a S-group G is special if everyG-torsor is locally trivial in the Zariski topology. Tus Proposition ?? states thatGLn(S) is special over any base scheme S. Over a field, the groups SLn(k) andSp2n(k) are also special (reference).

1.10. Free actions constructed from representations

A key fact used in the construction of equiviariant Chow groups is the followingfact.

1.10.1. Theorem. Let G be a linear algebraic S-group. For any integer d thereexists a linear action of G of ANS such that G acts freely on an open set U ⊂ ANS andANS rU has codimension more than d in every fiber of ANS → S.

1.10.2. Reduction to GLn(S). Observe that if φ : H → G is a morphism of S-groupsand G acts on a scheme S then the composition

H× X φ×1X→ G× X σ→ X

defines an action σH of H on X. If X = ANS and G acts linearly then the inducedaction of H is necessarily linear. Also if φ is a closed immersion and G acts properly(resp. freely) on X then H acts properly (resp. freely) on X as well. This follows

because the morphism H× X σH×1X→ X× X factors as the composition of the closedimmersion H × X → G × X with the proper (resp. closed immersion) morphism

G× X σ×1X→ X× X.

1.10.3. Proof for GLn(Z). As a consequence of this observation it suffices to proveTheorem 1.10.1 for the group G = GLn(S). By base change it suffices to constructthis representation for GLn := GLn(Spec Z). Let V = An×m parametrize n ×mmatrices. There is a linear action of GLn on V corresponding to left multiplicationby an n× nmatrix. Let U be the G-invariant open set paramatrizing matrices with


a least one non-vanishing minor. Thus if k is a field then the U(k) is the set of n× kmatrices of maximal rank. The complement V(k) ⊂ U(k) parametrizes matrices ofrank n− 1 and has codimensionm− n− 1 in each fiber over Spec Z.

The open set U is covered by open sets UI as I runs through all n elementsubsets of {1, . . . ,m}. Each UI parametrizes n ×m matrices whose n × n minorobtained by selecting the n-columns whose index is in I is non-vanishing. The openset UI is G-equivariantly isomorphic GLn×An×(m−n) where g ∈ GLn acts on apair (h,A) by g · (h,A) = (gh, gA). This action is readily seen to be free and themorphism UI → An×(m−n), (h,A) 7→ hA is a trivial torsor. If we denote the imageAn×(m−n) by AI, the AI glue to form the Grassmannian Gr(n,m) (see Nitsure’sarticle) and we have global torsor U → Gr(n,m) which is trivialized over each AI.Since the Grassmannian is separated (since it is a projective variety) it follows thatthe action of GLn on U is free.

By taking m sufficiently large we can ensure that V r U has arbitrarily highcodimension in each fiber over Spec Z.

1.10.4. Remark. The GLn-torsor U → Gr(n,m) is the GLn torsor associated to theuniversal rank n bundle on Gr(n,m).

CHAPTER 2

Equivalence Relations and Algebraic Spaces

2.1. Introduction and Motiviation

We now come to a fundamental problem in equivariant algebraic geometry.Given a free action of an S-group G on a scheme X construct a quotient objectY = X/G such that X → Y is a G-torsor. Remarkabely there are very few generalresults which ensure the existence of a quotient scheme Y = X/G. For example, if Gis a smooth algebraic group andH ⊂ G is a closed subgroup then classical results inthe theory of algebraic groups (due to Cartietr??) ensure the existence of a schemeG/H and a morphism G → G/Hwith the following properties:(i) G/H has an ample line bundle. (ii) G → G/H is an H-torsor (iii) The multiplica-tion map G×G → G descends to a morphism G×G/H → G/H which defines anaction of G on G/H.

The proof of this result [?, Borel]s geometric. Given a subgroup H ⊂ G aquasi-projective G-variety X is constructed containing a point xH whose orbit isclosed and whose stabilizers group is H. The G-orbit GxH with its reduced inducedscheme structre is the quotient sscheme G/H.

Unfortunately, this argument does not extend to groups defined over higherdimensional bases and there are no general results about the existence of quotientschemes G/H evene when G and H are smooth.

To construct quotients by free actions we need to expand the category ofschemes to the category of algebraic spaces. This category is essentially the minimalextension of the category of schemes which admits a large class of quotients byequivalence relations.

2.2. Equivalence relations in algebraic geometry

2.2.1. Equivalence relations in the category of sets. — If X is a set then anequivalence relation on X is a subset R ⊂ X× X such that(i) R ⊃ ∆(X) (ii) If (x1, x2) ∈ R if and only if (x2, x1) ∈ R. (iii) If (x1, x2) and (x2, x3)are both in R then so is (x1, x3). Given an equivalence relation Y ⊂ X × X thequotient Q = X/R is the set of equivalence classes of R in X. There is a naturalquotient epimorphism q : X → Q taking an element to its equivalence class. Thequotient Q and the map X → Q are characterized by the folllowing universalproperty: Let p1 : R → X and p2 : R → X be the compositions of the inclusionR → X× X with the two projections X× X → X. The map q is the coequalizer of themaps p1, p2. This means that q ◦ p1 = q ◦ p2 and given any map of sets g : X → Z

such that g ◦ p1 = g ◦ p2 there is a unique map h : Q → Z such that g = h ◦ q. Inaddition there is a bijection of sets R → X×Q X.

21


2.2.2. Example. — Given the action of a group G on a set X we can define anequivalence relation on X by setttng R = {(x, gx)} ⊂ X× X. The equivalence classesare the G orbits and the quotient is usually denoted X/G.

2.2.3. Equivalence relations in an arbitrary category. — The notion of equivalencerelation can be defined in any (small) category C with fiber products by requiringthe Hom sets to give equivalence relations of sets. Precisely a pair of morphismsR ⇒p1

p2 X defines an equivalence relation if for every object T of C the map of sets

Y(T)(p1(T),p2(T))→ X(T) × X(T) is injective and the image defines an equivalence

relation of sets. (Here Y(T) = HomC(T, Y) and X(T) = HomC(T, X).) A epimorphismq : X → Q is a categorical quotient of the equivalence relation p : R → X×X if q is thecoequalizer of the morphisms p1, p2, meaning that q ◦ p1 = q ◦ p2 and given anymorphism q ′ : X → Z such that q ′ ◦ p1 = q ′ ◦ p2 there is a map αQ → Z such thatα ◦ q = q ′ The universal property and the fact that q is an epimorphism ensuresthat Q is unique up to unique isomorphism in C.

If the category has fiber products and X → Q is a categorical quotient, we saythat the equivalence relation is effective if, in addition, the morphism R → X×Q X(induced by the property of being a categorical quotient) is an isomorphism.

In particular if S is a fixed base scheme and C is the category of S-schemes thenan equivalence relation Y → X× X in C is called an S-equivalence relation.

2.2.4. Actions with trivial stabilizers and equivalence relations. — If an S-groupacts with trivial stabilizers then the morphism G× X → X× X is a monomorphismand defines an equivalence relation in the category of schemes.

2.2.5. Example. — If P → X is a G-torsor then equivalence relation G× P → P × Pis effective with categorical quotient X.

2.2.6. Warning. — If R → X×X is an effective S-equivalence relation with quotientQ it is not always the case that there is a bijection of sets Q(T) → X(T)/R(T) for anS-scheme T . Indeed the map of sets X(T) → Q(T) need not always be surjective asthe next example shows.

2.2.7. Example of an effective equivalence relation in the categroy of schemes.— [?, Exercise p134] Let S = Spec Z and let X ⊂ An+1

Z be the complement of thezero section. The action of Gm on An+1 restricts to a free action on X and sothe morphism Gm × X

σ×1X→ X × X is an equivalence relation in the category ofschemes. This equivalence relation is effective and the quotient is PnZ . Observethat the morphism q : X → PnZ is a non-trivial Gm-torsor which means that q hasno section.Hence the map HomZ(Pn, X) → Hom(Pn,Pn) is not surjective as theidentity morphism Pn → Pn doesn’t have a lift to a morphism Pn → X. Hence it isnot the case that Pn(Pn) is the set quotient (Gm × X)(Pn)/X(Pn).

2.2.8. Example of a non-effective equivalence relation with a categorical quo-tient. —

2.2.9. Definition. — A schematic equivalence relation R ⇒ X is flat (resp. smooth,etale) if the projections p1 : R → X and p2 : R → X are flat (resp. smooth, etale).Likewise we say that the equivalence relation is finite (resp. affine) if the morphisms,p1, p2 are finite (resp. affine).


2.2.10. Example. — If G is an affine algebraic group acting with trivial stabilizerthen G× X → X× X is a flat affine equivalence relation which is finite if and only ifG is finite.

2.2.11. Proposition. — Any finite flat equivalence relation R ⇒ X with X (andhence R) affine is effective and the quotient morphism X → Q is finite and flat.

PROOF. Let X = SpecA and R = SpecB. Then we have two inclusionsp

]1, p

]2 : A → B. LetA0 be the difference kernel of p]

1 and p]2; ieA0 = {a ∈ A|p

]1(a) =

p]2(a)} and letQ = SpecA0. The inclusion A0 ⊂ A induces a morphism q : X → Q

such that q ◦ p1 = q ◦ p2.Moreover if X → Z is any morphism such that To be continued.... �

Bibliography

[Al] P. Aluffi, Algebra, Chapter 0, Grad. Stud. in Math. 104, Amer. Math. Soc., Providence, RI,2009.

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