algebraic k-theory and local chern charactersesben.bistruphalvorsen.dk/papers/master.pdf · chapter...

Department of Mathematics

University of Copenhagen

Submitted: January 20, 2004

Adviser: Hans-Bjørn Foxby

Master thesis for the cand. scient. degree in mathematics

Esben Bistrup Halvorsen

Algebraic K-theory andlocal Chern charactersapplied to Serre’s conjectures on

intersection multiplicity

Abstract

If M and N are finitely generated modules over a commutative,Noetherian, local ring R, such that M has finite projective dimen-sion and M ⊗R N has finite length, then Serre’s intersection mul-tiplicity is defined as χR(M, N) =

∑`∈Z(−1)` length TorR

` (M,N).It has been conjectured that (0) dimR M + dimR N ≤ dimR;that (1) χR(M, N) = 0 when this not an equality; and that (2)χR(M,N) > 0 when it is. All three conjectures are proved inthe case that dimR N ≤ 1 by replacing M by a simpler module.This simplification is obtained by factoring χR(−, N) through aGrothendieck group and exploiting one among many surprisingisomorphisms between these. Conjecture (1) is proved in the casewhere R is a complete intersection and both modules have finiteprojective dimension, using the properties of local Chern charac-ters. The Grothendieck groups and local Chern characters turnout to be connected.

Contents

Preface v

Preliminaries ix

1 Introduction 1

2 Groups of complexes: K0 52.1 Categories of complexes . . . . . . . . . . . . . . . . . . . . . . . 52.2 Grothendieck groups . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Grothendieck group isomorphisms 173.1 Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Koszul complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 The main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Consequences of the main theorem . . . . . . . . . . . . . . . . . 46

4 Groups of matrices: K1 554.1 The first algebraic K-group . . . . . . . . . . . . . . . . . . . . . 554.2 The localization sequence . . . . . . . . . . . . . . . . . . . . . . . 63

5 Local Chern characters 755.1 Chow groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2 Local Chern characters . . . . . . . . . . . . . . . . . . . . . . . . 77

6 The intersection conjectures 876.1 Serre’s intersection multiplicity . . . . . . . . . . . . . . . . . . . 876.2 Applying algebraic K-theory . . . . . . . . . . . . . . . . . . . . . 886.3 Applying local Chern characters . . . . . . . . . . . . . . . . . . . 916.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Bibliography 97

iii

Preface

This text constitutes my master thesis for the cand. scient. degree in mathematicsat the University of Copenhagen. It is the result of many months of work underthe adept supervision of Hans-Bjørn Foxby.

Already during my first year as a mathematics student, I realized that mymain interest was in algebra, but it was not until I was an exchange student atthe University of California at Berkeley in the academic year 2001/2002 that Iwas able to narrow it down: it had to be commutative algebra, preferably with aflavor of category theory to it. Homological algebra satisfied these requirementsperfectly, so before I returned to Copenhagen I asked Hans-Bjørn Foxby to be myadviser and he accepted. He then chose the subject of intersection multiplicities,and I accepted, eventually putting emphasis on the algebraic K-theory, as itsatisfied my craving for category theory to an even greater extent.

The reader is assumed to be knowledgeable about rings, modules and com-plexes at a level corresponding to graduate courses in homological algebra. Nev-ertheless, to assist the reader, the preliminaries outline basic definitions, notationand terms together with a variety of fundamental results that are presented with-out proof. The remainder of the thesis is structured as follows.

Chapter 1 discusses the motivation behind the work in this thesis: Serre’sconjectures on intersection multiplicity. Chapter 2 introduces the zeroth alge-braic K-groups together with the more general concept of Grothendieck groupsof complexes, and Chapter 3 establishes a multitude of surprising isomorphismsbetween various Grothendieck groups. Chapter 4 presents the first algebraicK-groups and connects them to the Grothendieck groups via the localizationsequence; this leads to the introduction of an invariant known as the MacRaeideal. Chapter 5 changes the subject slightly by digressing into a superficial dis-cussion of the theory of local Chern characters. Finally, Chapter 6 compiles theresults established in previous chapters to prove some of Serre’s conjectures inspecial cases and to discuss the application of algebraic K-theory and local Cherncharacters and the relationship between these in future research.

The core of the thesis is the work done in Chapters 2, 3 and 4. My main sourcesof information for the making of these chapters have been the book [Mag02] byMagurn and the paper [Fox82a] by Foxby, supplemented by the notes [Bas74]by Bass. Most of the theory in these chapters has been gathered from these

v

vi Preface

sources, although the notation has been altered, theorems generalized and lemmasintroduced to comply with the style of the thesis. In many cases I have providedmore detailed proofs than were given in the sources, and at some points I havecome up with proofs myself when none were at hand: this includes Theorem 3.23,which was only proven for d = 1 in [Fox82a]; the first part of Theorem 4.16,which was presented without proof in [Mag02] and proven somewhat differentlyin [Bas74] using a more general version of K1-groups; and Proposition 4.5, whichwas presented without proof in [Mag02] and had a completely different proof in[Bas74].

Chapter 5 should only be seen as supplementing the discussion in Chapters 2,3 and 4. It contains almost no proofs and does not even define the subject ofthe chapter: local Chern characters. The source for this chapter is Roberts’ book[Rob98], supplemented by Foxby’s notes [Fox89].

The box on the following page describes some basic assumptions that remainvalid throughout the entire text. In addition to this, some sections are introducedwith a box describing the additional assumptions made in that section alone. Thereader is advised to pay careful attention to these boxes! The content of a boxremains valid throughout the section in which it is positioned.

I am deeply indebted to my adviser, Hans-Bjørn Foxby, for taking the time toanswer my numerous questions and for provoking me at the right times to come upwith answers myself. I am also grateful to David Jeffrey Breuer for copyeditingthe text. Finally, I would like to thank my wonderful family, especially mymother, Lone Bo Halvorsen, and my father, Christian Henning Bistrup, for theirunlimited love, support and complete faith in me and the projects I undertake.

Esben Bistrup HalvorsenCopenhagen, January 2004

Throughout this thesis, R will denote a nontrivial, unitary, commutative ring.The unit element of R will be denoted by 1 and the zero element by 0. Unlessotherwise stated, all ideals, modules and complexes are assumed to be idealsof R, R-modules and R-complexes, respectively.

Preliminaries

These preliminaries briefly summarize terms, notation and the basic results ofmodules, complexes and various constructions and invariants associated withthese. Readers already familiar with the fundamental concepts of homologicalalgebra will find nothing new here and should be able to quickly skim throughthe following pages.

Basic facts

Throughout this thesis, the symbol ⊆ means “contained in or equal to”, whereasthe symbol ⊂ is used when equality is not an option. The blackboard bold lettersN, N0, Z, Q+ and Q will denote the sets of positive integers, nonnegative integers,integers, positive rational numbers and rational numbers, respectively.

If x = (x1, . . . , xn) is a sequence of elements in a module, the submodulegenerated by these elements will be denoted by 〈x1, . . . , xn〉 or simply 〈x〉. Inparticular, if x = (x1, . . . , xn) is a sequence in R, 〈x〉 denotes the ideal generatedby the elements x1, . . . , xn.

The multiplicative group of units of the ring R is denoted by R∗. A subset Sof R is a multiplicative system if any product of elements from S is contained inS; this includes the “empty product” 1 ∈ R. The complement of a prime idealis a multiplicative system. There is a sort of inverse of this statement: any idealthat is maximal among ideals not intersecting a multiplicative system S is prime(cf. [Eis95, Proposition 2.11]).

When S is a multiplicative system in R and M is a module, the localization ofM at S is denoted by S−1M . If S = R\p for some prime ideal p, then S−1M isreferred to as the localization of M at p and denoted by Mp. Given a multiplicativesystem S in R and a module M , there is an isomorphism S−1M ∼= S−1R⊗R M ,and since S−1R is itself a ring, this means that S−1M has the structure of anR-module as well as of an S−1R-module. The module M is said to be S-torsionif S−1M = 0. We extend these terms to the situation in which there are severalmultiplicative systems: if S = (S1, . . . , Sd) is a family of multiplicative systems,M is said to be S-torsion if M is Sν-torsion for ν = 1, . . . , d.

The set of prime ideals of R is denoted Spec R and referred to as the spectrum

ix

x Preliminaries

of R. If I is an ideal, we define VR(I) to be the set of prime ideals containingI. The intersection of all the primes in VR(I) is denoted by RadR I and referredto as the radical of I; it is the set of elements r ∈ R such that rn ∈ I for somen ∈ N (cf. [Eis95, Corollary 2.12]). A prime ideal p is minimal over I if it isminimal among elements in VR(I) ordered by inclusion. A prime ideal is simplycalled minimal if it is minimal over the zero ideal. When R is Noetherian, thereare only finitely many minimal primes over an ideal I (cf. [Eis95, Exercise 1.2]).

A useful result about prime ideals is the prime avoidance lemma (cf. [Eis95,Lemma 3.3]):

Lemma 0.1 (prime avoidance). If p1, . . . , pn are ideals of which at most twoare not prime, and if I is an ideal contained in ∪ipi, then I is contained in oneof the pi’s.

Suppose that M is a module. A zerodivisor on M is an element r ∈ R suchthat rm = 0 for some nonzero m ∈ M . The set of zerodivisors on M is denotedZdR M . The annihilator of M is the ideal AnnR M = {r ∈ R |rM = 0}, and thesupport of M is the set SuppR M of prime ideals p such that Mp is nontrivial. IfM is finitely generated and S is a multiplicative system, then M is S-torsion ifand only if AnnR M ∩ S 6= ∅; in particular, SuppR M = VR(AnnR M) (cf. [Eis95,Proposition 2.1 and Corollary 2.7]). If m ∈ M , the annihilator of m is theideal AnnR(m) = {r ∈ R | rm = 0}. If the annihilator of an element of Mturns out to be a prime ideal, this ideal is said to be associated to M . The setof associated primes of M is denoted by AssR M . When R is Noetherian andM is finitely generated and nontrivial, AssR M is a nonempty, finite subset ofSuppR M , including all the minimal primes over AnnR M , and the union of theprimes in AssR M equals ZdR M (cf. [Eis95, Theorem 3.1]). If R is Noetherian and0 → L → M → N → 0 is a short exact sequence of modules, then SuppR M =SuppR L ∪ SuppR N (cf. [Rob98, Proposition 1.1.3]). If R is Noetherian andlocal, and M and N are finitely generated modules, then SuppR(M ⊗R N) =SuppR M ∩ SuppR N (cf. [Rob98, Proposition 2.3.2]).

Complexes

A complex X is a family (X`)`∈Z of modules together with a family (∂X` )`∈Z of

homomorphisms ∂X` : X` → X`−1, which are called the differentials of X, such

that ∂X` ∂X

`+1 = 0 for all ` ∈ Z:

X = · · · −→ X`+1

∂X`+1−→ X`

∂X`−→ X`−1 −→ · · · .

The complex X is said to be bounded if X = 0 for all but finitely many `. Morespecifically, X is said to be concentrated in degrees `1, . . . , `t if X` 6= 0 implies

Preliminaries xi

` ∈ {`1, . . . , `t}. From now on, modules will always be thought of as complexesconcentrated in degree 0: that is, complexes X with X` = 0 for all ` 6= 0.

The zero complex is the complex · · · → 0 → 0 → 0 → · · · having the zeromodule in each degree. For n ∈ Z, the complex X shifted n degrees to the left isthe complex ΣnX given by (ΣnX)` = X`−n and ∂ΣnX

` = (−1)n∂X`−n for all ` ∈ Z.

The change of sign of the differentials turns out to be natural in connection withthe mapping cone, which is described below. In the case that n = 1, the operatorΣ1(−) is denoted simply by Σ(−). The direct sum of two complexes X and Y is

the complex X ⊕ Y given by (X ⊕ Y )` = X` ⊕ Y` and ∂X⊕Y` =

(∂X

` 0

0 ∂Y`

)for all

` ∈ Z.

When X and Y are complexes, a morphism φ : X → Y is a family φ = (φ`)`∈Zof homomorphisms φ` : X` → Y`, making the following diagram commutative.

· · · // X`+1

∂X`+1 //

φ`+1

²²

X`

∂X` //

φ`

²²

X`−1//

φ`−1

²²

· · ·

· · · // Y`+1

∂Y`+1 // Y`

∂Y` // Y`−1

// · · ·

The identity morphism on the complex X is the morphism 1X : X → X givenfor each ` ∈ Z by (1X)` = 1X`

, where 1X`: X` → X` is the identity map.

The composition of two morphisms φ : X → Y and ψ : Y → Z is the morphismψφ : X → Z given by (ψφ)` = ψ`φ`.

A morphism φ : X → Y is an isomorphism if a morphism φ−1 : Y → X existssuch that φ−1φ = 1X and φφ−1 = 1Y . In this case, we say that X is isomorphic

to Y , and we write φ : X∼=−→ Y or simply X ∼= Y . A necessary and sufficient

condition for a morphism of complexes to be an isomorphism is that it is anisomorphism of modules in each degree. Note, however, that this does not meanthat complexes with isomorphic modules are isomorphic!

A short exact sequence of complexes is a sequence 0 → X → Y → Z → 0 ofmorphisms such that, for each ` ∈ Z, the sequence 0 → X` → Y` → Z` → 0 ofmodules in degree ` is exact.

A homotopy between two morphisms φ, ψ : X → Y is a family h = (h`)`∈Zof homomorphisms h` : X` → Y`+1, such that φ` − ψ` = ∂Y

`+1h` + h`−1∂X` for all

` ∈ Z. If such a homotopy exists, we say that φ and ψ are homotopic; this definesan equivalence relation of morphisms X → Y . A morphism that is homotopicwith the zero morphism is said to be null-homotopic.

The mapping cone of a morphism φ : X → Y is the complex M(φ) given byM(φ)` = Y` ⊕X`−1 = (Y ⊕ ΣX)` and

∂M(φ)` =

∂Y` φ`−1

0 −∂X`−1

:

Y`

⊕X`−1

−→Y`−1

⊕X`−2

xii Preliminaries

for all ` ∈ Z. An important property of the mapping cone is given in the followingtheorem (cf. [Fox98, (1.24)]).

Theorem 0.2. If φ : X → Y is a morphism of complexes, the (degreewise) in-clusion Y ↪→M(φ) and the (degreewise) projection M(φ) ³ ΣX are both mor-phisms of complexes, and together they form a short exact sequence

0 → Y →M(φ) → ΣX → 0.

The mapping cone construction has other nice properties, including that ofbeing natural and exact in the sense described by the easily established theorembelow.

Theorem 0.3. If there is a commutative diagram

Xγ //

φ

²²

X

φ²²

Yλ // Y

of complexes, then there is an induced morphism M(φ) →M(φ) given in degree` ∈ Z by

λ` 0

0 γ`−1

: M(φ)` =

Y`

⊕X`−1

→Y`

⊕X`−1

= M(φ)`.

If there is a commutative diagram

0 // Xγ //

φ

²²

Xγ //

φ

²²

X

φ²²

// 0

0 // Yλ // Y

λ // Y // 0

with exact rows, then the induced sequence 0 →M(φ) →M(φ) →M(φ) → 0 isexact.

All complexes together with all morphisms form a category denoted by CR.The subcategory of modules is denoted CR

0 for reasons that become obvious later.An important functor in the category CR is the homology functor H: CR → CR,which takes a complex X to the complex H(X) defined by H(X)` = H`(X) =

ker ∂X` / im ∂X

`+1 and ∂H(X)` = 0 for all ` ∈ Z, and a morphism φ : X → Y to the

morphism H(φ) : H(X) → H(Y ) given by

H(φ)`([x]im ∂X`+1

) = H`(φ)([x]im ∂X`+1

) = [φ`(x)]im ∂Y`+1

Preliminaries xiii

for all ` ∈ Z and x ∈ ker ∂X` . H(X) is called the homology complex of X, and

H`(X) is called the homology module in degree `. The complex X is exact ifH(X) = 0.

A short exact sequence 0 → X → Y → Z → 0 of complexes induces an exactsequence of homology modules (cf. [Fox98, (1.21)]):

· · · → H`+1(Z) → H`(X) → H`(Y ) → H`(Z) → · · · .

If φ : X → Y is a morphism of complexes, we say that φ is a homology iso-morphism, and we write φ : X

'−→ Y , if H(φ) : H(X) → H(Y ) is an isomorphism;in other words,

X'−→φ

Ydef⇐⇒ H(X)

∼=−→H(φ)

H(Y ).

The following theorem presents a necessary and sufficient condition for a mor-phism to be a homology isomorphism (cf. [Fox98, Lemma 1.25]).

Theorem 0.4. If φ : X → Y is a morphism of complexes, then φ is a homologyisomorphism if and only if its mapping cone M(φ) is exact.

If R′ is another (nontrivial, unitary and commutative) ring and T : CR0 → CR′

0

is an additive, covariant functor, then T induces a functor CR → CR′ ; the inducedfunctor is also denoted by T . For any complex X ∈ CR, the complex T (X)

is given by T (X)` = T (X`) and ∂T (X)` = T (∂X

` ) for all ` ∈ Z, and for anycomplex morphism φ : X → Y , the induced morphism T (φ) : T (X) → T (Y )is given by T (φ)` = T (φ`) for all ` ∈ Z. If, in addition, T is exact, that is,preserves short exact sequences of modules, then the induced functor commuteswith homology, that is, T (H(X)) ∼= H(T (X)) whenever X is a complex, and Tpreserves homology isomorphisms (cf. [Fox98, (1.51)]). In this case, in particular,T preserves the exactness of complexes.

Similarly, when T : CR0 → CR′

0 is an additive, contravariant functor, there is aninduced functor CR → CR′ also denoted by T . For any complex X, the complex

T (X) is given by T (X)` = T (X−`) and ∂T (X)` = −T (∂X

−`+1), and for any complexmorphism φ : X → Y , the induced morphism T (φ) : T (Y ) → T (X) is givenby T (φ)` = T (φ−`). Once again, if, in addition, T is an exact functor, then theinduced functor commutes with homology and preserves homology isomorphisms.

The tensor product of two complexes X and Y is the complex X⊗R Y , whose`’th module is

(X ⊗R Y )` =∐

n∈ZXn ⊗R Y`−n,

and whose `’th differential ∂X⊗RY` : (X ⊗R Y )` → (X ⊗R Y )`−1 is defined on

generators xn ⊗ y`−n ∈ Xn ⊗R Y`−n ⊆ (X ⊗R Y )` by

∂X⊗RY` (xn ⊗ y`−n) = ∂X

n (xn)⊗ y`−n + (−1)nxn ⊗ ∂Y`−n(y`−n),

xiv Preliminaries

which is an element of (Xn−1⊗R Y`−n)⊕ (Xn⊗R Y`−n−1) ⊆ (X ⊗R Y )`−1. If X isconcentrated in degree 0, then X ⊗R Y is the same as the complex obtained byapplying the additive, covariant functor X⊗R− to the complex Y as described inthe preceding paragraph. Likewise, if Y is concentrated in degree 0, then X⊗R Yis the same as the complex obtained by applying the additive, covariant functor−⊗R Y to the complex X.

An important property of the tensor product of complexes is that if X and Yare complexes and S is a multiplicative system, then S−1(X ⊗R Y ) is isomorphicto S−1X ⊗S−1R S−1Y as S−1R-complexes (cf. [Fox98, (4.24)]).

Any property satisfied by the homology complex H(X) of a complex X issaid to be satisfied homologically by X. If S is a multiplicative system in R, wesay that X is S-torsion if S−1X = 0; in particular, since S−1(−) is a covariant,exact functor, X is homologically S-torsion if and only if S−1X is exact. IfS = (S1, . . . , Sd) is a family of multiplicative systems, we say, as with modules,that X is S-torsion if X is Sν-torsion for ν = 1, . . . , d.

Module invariants

One of the fundamental concepts within homological algebra is that of projectiveresolutions. A resolution of a module M is a complex

X = · · · −→ X2

∂X2−→ X1

∂X1−→ X0 −→ 0,

together with a homomorphism φ0 : X0 → M , such that

· · · −→ X2

∂X2−→ X1

∂X1−→ X0

φ0−→ M −→ 0

is an exact complex; in other words, a resolution of M is a homology isomorphismφ : X

'−→ M . If the modules in X are finitely generated, X is called a finiteresolution of M ; if the modules in X are projective, X is called a projectiveresolution of M ; if the modules in X are flat, X is called a flat resolution of M ;and if the modules in X are free, X is called a free resolution of M . We shalloften combine the word “finite” with the words “projective” and “free”, defininga finite projective resolution to be a resolution that is finite as well as projectiveand a finite free resolution to be a resolution that is finite as well as free.

Any module has a free (and hence projective and flat) resolution, and if R isNoetherian and M is finitely generated, M has a finite free resolution (cf. [Fox98,Theorem 2.6]). If R is local with maximal ideal m and M is finitely generated,we call X a minimal free resolution of M if it is a finite free resolution such thatim ∂X

` ⊆ mX`−1 for all ` ∈ Z.If M is nontrivial, the projective dimension of M , denoted by pdR M , is the

infimum over all n ∈ N0 for which M has a projective resolution X concentrated

Preliminaries xv

in degrees n, . . . , 0; in particular, pdR M = ∞ if there are no bounded projectiveresolutions of M . Similarly the flat dimension of M , denoted by fdR M , is theinfimum over all n ∈ N0 for which M has a flat resolution concentrated in degreesn, . . . , 0. For technical reasons, we let pdR 0 = fdR 0 = −∞, although we stillconsider the zero module to have finite projective and flat dimensions.

If M and N are modules and X is a projective resolution of M , applicationof the (covariant and right exact) functor −⊗R N to X yields the complex

· · · −→ X2 ⊗R N∂X2 ⊗RN−→ X1 ⊗R N

∂X1 ⊗N−→ X0 ⊗R N −→ 0

concentrated in nonnegative degrees. The homology module H`(X ⊗R N) of thisis called the `’th Tor-module of M and N and is denoted by TorR

` (M,N). Hadwe instead applied the (contravariant and left exact) functor Hom`

R(−, N) to X,we would obtain the complex

0 −→ HomR(X0, N)◦(−∂X

1 )−→ HomR(X1, N)◦(−∂X

2 )−→ HomR(X2, N) −→ · · ·concentrated in nonpositive degrees. The homology module H−`(HomR(X, N))of this is called the `’th Ext-module and is denoted by Ext`

R(M,N).There is an alternative definition of TorR

` (M,N) using a projective resolutionof N and an alternative definition of Ext`

R(M, N) using an injective resolution ofN , giving two possible interpretations of TorR

` (M,N) and Ext`R(M, N), but no

ambiguities since the two interpretations are isomorphic. For our purposes, onedefinition suffices, so we will not give further mention to this in the sequel.

The zeroth Tor- and Ext-modules reflect the functors from which they werederived: if M and N are modules, TorR

0 (M, N) ∼= M ⊗R N and Ext0R(M, N) ∼=

HomR(M, N) (cf. [HS97, page 139 and 161]). Tor also maintains the propertythat the tensor product is commutative up to isomorphism, in the sense thatTorR

` (M,N) ∼= TorR` (N, M) for all ` ∈ Z. We also have that TorR

` (M,N)p∼=

TorRp

` (Mp, Np) for any prime ideal p (cf. [Eis95, page 161-162]); in particular,

SuppR(TorR` (M, N)) ⊆ SuppR M ∩ SuppR N.

TorR` (−, N), TorR

` (M,−) and ExtR(M,−) are covariant functors, whereasExt`

R(−, N) is a contravariant functor. These functors take short exact sequencesof modules to long exact sequences of modules: if A, B, C, M and N are modulesand the sequence 0 → A → B → C → 0 is exact, then the following sequencesare exact and referred to as the long exact Tor- and Ext-sequences in the firstand second variable (cf. [HS97, (7.1), (7.4), (11.1) and (11.4)]).

· · · → TorR`+1(M, C) → TorR

` (M, A) → TorR` (M, B) → TorR

` (M, C) → · · · ,

· · · → TorR`+1(C, N) → TorR

` (A, N) → TorR` (B,N) → TorR

` (C,N) → · · · ,

· · · → Ext`−1R (M,C) → Ext`

R(M, A) → Ext`R(M, B) → Ext`

R(M,C) → · · · ,

· · · → Ext`−1R (A,N) → Ext`

R(C, N) → Ext`R(B, N) → Ext`

R(A,N) → · · · .

xvi Preliminaries

Projective dimension is naturally interlinked with Ext as described by thefollowing theorem (cf. [Fox99, 9.1(1) and 9.3(1)]).

Theorem 0.5. For a module M and an nonnegative integer n, the followingconditions are equivalent.

(i) pdR M ≤ n.

(ii) ExtmR (M,−) = 0 for all m > n.

(iii) Extn+1R (M,−) = 0.

(iv) If there is an exact sequence 0 → Xn → Xn−1 → · · ·X0 → M → 0 in whichX0, . . . , Xn−1 are projective, then Xn must be projective.

In particular, pdR M = sup{n ∈ N0 |ExtnR(M,−) 6= 0}.

Likewise, flat dimension is naturally interlinked with Tor as described by thefollowing theorem (cf. [Fox99, 11.4(4) and 11.5(1)]).

Theorem 0.6. For a module M and a nonnegative integer n, the following con-ditions are equivalent.

(i) fdR M ≤ n.

(ii) TorRm(M,−) = 0 for all m > n.

(iii) TorRn+1(M,−) = 0.

(iv) TorRn+1(M,R/I) = 0 for all finitely generated ideals I in R.

(v) If there is an exact sequence 0 → Xn → Xn−1 → · · ·X0 → M → 0 in whichX0, . . . , Xn−1 are flat, then Xn must be flat.

In particular, fdR M = sup{n ∈ N0 |TorRn (M,−) 6= 0}.

In case R is Noetherian, flat and projective are the same for a finitely gener-ated module M , and hence in this case fdR M = pdR M .

The following theorem allows us in some situations to determine the projectivedimension of one of the modules in a short exact sequence from the projectivedimension of the other two. (cf. [Fox99, HA071197-1(1)]):

Theorem 0.7. Suppose 0 → L → M → N → 0 is a short exact sequence ofmodules. Then if two of the modules have finite projective dimension, so does thethird. In any case, pdR L = pdR N − 1 whenever pdR M < pdR N .

Preliminaries xvii

The module M is said to be simple if M 6= 0 and M has no nontrivial propersubmodules: that is, if 0 is the only proper submodule of M . A filtration of Mis a descending chain of submodules

M = M0 ⊇ M1 ⊇ · · · ⊇ Mn = 0

of finite length n. If M is finitely generated, one can always find a filtrationsuch that the factors Mi−1/Mi are isomorphic to R/p for prime ideals p; theset of prime ideals thus obtained is contained in SuppR M and contains AssR M(cf. [Rob98, Theorem 1.1.4]).

The length of a module M , denoted by lengthR M , is the supremum of thelengths of all filtrations M = M0 ⊃ M1 ⊃ · · · ⊃ Mn = 0 in which the inclusionsare strict. One can show that, when the length of M is finite, the numberlengthR M is the length of any filtration of M in which the quotient modulesMi−1/Mi, i = 1, . . . , n, are simple (cf. [Eis95, Theorem 2.13]); in this case, thefiltration is called a composition series for M . If 0 → L → M → N → 0 is ashort exact sequence of modules, then lengthR M = lengthR L+lengthR N . (Herewe follow the usual conventions that n+∞ = ∞+n = ∞ for all n ∈ N0∪{∞}.)In addition, a module of finite length must be finitely generated. Note that ifk is a field and V is a finitely generated k-vector space, lengthk V = lengthR Vis nothing but the usual dimension of V as a vector space over k. The notation“dimk V ”, however, is reserved for an important analog of dimension for modulesover rings, which is described below.

Suppose that R is Noetherian and local with maximal ideal m and quotientfield k = R/m, and that M is a finitely generated module. The n’th Betti numberof M is then the nonnegative integer βR

n (M) = lengthR TorRn (M, k): that is, the

dimension of TorRn (M, k) as a vector space over k. We know from Theorem 0.6

that fdR M = pdR M = sup{n ∈ N0 | βRn (M) 6= 0}, and it follows that, if M

has finite projective dimension, then βRn (M) 6= 0 for only finitely many n. In

this case, the Euler characteristic of M is defined to be the alternating sumχR(M) =

∑n∈Z(−1)nβR

n (M). One can show that the n’th Betti number is therank of the n’th module in a minimal free resolution of M and that the Eulercharacteristic equals the alternating sum

∑`∈Z(−1)` rankR X` for any bounded

finite free resolution X of M (cf. [Hal03, Proposition 4.2 and Theorem 5.2]).The Euler characteristic has many nice properties. One is that it is always

nonnegative, and another is that, if 0 → L → M → N → 0 is a short ex-act sequence of finitely generated modules of finite projective dimension, thenχR(M) = χR(L) + χR(N) (cf. [Hal03, Proposition 5.6 and Theorem 5.5]). The-orem 0.8 below describes another interesting property of the Euler characteristic(cf. [Hal03, Theorem 5.7]).

Theorem 0.8. If R is Noetherian and local, and M is a finitely generated modulewith pdR M < ∞, then the following conditions are equivalent.

(i) χR(M) > 0.

xviii Preliminaries

(ii) SuppR M = Spec R.

(iii) AnnR M = 0.

(iv) AnnR M ⊆ Zd R.

The definition of Euler characteristics can be generalized to complexes. IfR is Noetherian and local and X is a complex such that there is a homologyisomorphism X

'←− F ∈ CR¤(f,P) (in which case we say that X has finite projective

dimension and that F is a bounded finite free resolution of X), then the Eulercharacteristic of X is the sum χR(X) =

∑`∈Z(−1)` lengthR H`(k⊗R F ), which is

equal to the sum∑

`∈Z(−1)` rankR F` (cf. [Hal03, Theorem 5.2]). From this, itfollows that, if p is a prime ideal, then χR(X) = χRp(Xp).

The height of a prime ideal p, denoted by heightR p, is the supremum ofnonnegative integers n for which there is a strictly descending chain

p = p0 ⊃ p1 ⊃ · · · ⊃ pn

of prime ideals. For an arbitrary ideal I, one defines the height of I by

heightR I = inf{heightR p |p ∈ VR(I)}.

The (Krull) dimension of M , denoted by dimR M , is the supremum of thelengths of all strictly descending chains p0 ⊃ p1 · · · ⊃ pn of prime ideals inSuppR M . If M is finitely generated, SuppR M = VR(AnnR M), so in this casewe must have dimR M = dimR R/ AnnR M . Considering R as a module overitself, we see that the Krull dimension of R is

dim R = sup{heightR p |p ∈ Spec R},

and it follows that dim Rp = heightR p for all p ∈ Spec R. In particular, when Ris local with maximal ideal m, dim R = heightR m.

A fundamental theorem about heights is Krull’s principal ideal theorem pre-sented below (cf. [BH93, Theorem A.1]).

Theorem 0.9 (Krull). If R is Noetherian and p is a minimal prime over anideal I generated by n elements, then heightR p ≤ n.

Krull’s principal ideal theorem shows that the height of any ideal is boundedby the number of generators for that ideal. In particular, when R is Noetherianand local, the Krull dimension of R (and hence of any R-module) is finite.

Krull dimension and length are integrated in the following theorem (cf. [Eis95,Theorem 2.17 and Lemma 2.8b]).

Theorem 0.10. If R is Noetherian and M is a finitely generated module, thenthe following are equivalent.

Preliminaries xix

(i) dimR M = 0.

(ii) SuppR M is nonempty and contains only maximal ideals.

(iii) 0 < lengthR M < ∞.

The following theorem relates dimension to height (cf. [BH93, page 413]).

Theorem 0.11. If I is a proper ideal, then heightR I + dimR R/I ≤ dim R.

Suppose R is Noetherian and local with maximal ideal m, and let M be anontrivial and finitely generated module. Then dimR M is the minimum of alld ∈ N0 for which there exist elements x1, . . . , xd ∈ m such that M/〈x1, . . . , xd〉Mhas finite length, and in this case, the sequence (x1, . . . , xd) is called a systemof parameters for M (cf. [Rob98, Theorem 2.3.7]). Systems of parameters arecharacterized by the following theorem (cf. [BH93, Proposition A.4]).

Theorem 0.12. If R is Noetherian and local with maximal ideal m, M is anontrivial and finitely generated module, and x1, . . . , xn are elements in m, then

dimR(M/〈x1, . . . , xn〉M) ≥ dimR M − n

with equality holding if and only if x1, . . . , xn is part of a system of parametersfor M .

If R is Noetherian and local with maximal ideal m, R is said to be regular ifit has a system of parameters generating m; such a system of parameters is calleda regular system of parameters. A Noetherian local ring is regular if and onlyif every finitely generated module has finite projective dimension, and a regularlocal ring is always an integral domain (cf. [BH93, Theorem 2.2.7 and Proposi-tion 2.2.3]). If R equals the quotient Q/I of a regular local ring Q with an idealI of Q generated by heightQ I elements, then we say that R is a complete inter-section.1 This definition clearly shows that any regular local ring is a completeintersection.

If M is a module, an M-regular sequence, or simply an M-sequence, is asequence x = (x1, . . . , xn) of elements of R such that 〈x〉M 6= M and such thatxi is a non-zerodivisor on M/〈x1, . . . , xi−1〉M for i = 1, . . . , n. In the case i = 1,the last condition simply says that x1 must be a non-zerodivisor on M . Thesequence x = (x1, . . . , xn) is called a weak M-sequence if it only satisfies the lattercondition: that xi is a non-zerodivisor on M/〈x1, . . . , xi−1〉M for i = 1, . . . , n. AnR-sequence is simply called a regular sequence.

1This is a more restricted definition than that traditionally given. Normally, one defines acomplete intersection to be a ring whose completion is the quotient of a regular local ring by aregular sequence. This definition, however, suffices for the present purposes, and using it avoidshaving to introduce the concept of completion of rings.

xx Preliminaries

If R is Noetherian, M is finitely generated and x = (x1 . . . , xn) is an M -sequence contained in a prime ideal p ∈ SuppR M , then x/1 = (x1/1, . . . , xn/1)is an Mp-sequence contained in pp (cf. [BH93, Corollary 1.1.3(i)]). If, in addition,R is local, the next theorem shows that the elements of a regular sequence canbe permuted and raised to powers (cf. [BH93, Proposition 1.1.6]).

Theorem 0.13. If R is Noetherian and local, M is a finitely generated module,and x = (x1, . . . , xn) is an M-sequence, then every permutation of x is an M-sequence, and so is (xN1

1 , . . . , xNnn ) for any N1, . . . , Nn ∈ N.

A maximal M -sequence in an ideal I of R is an M -sequence x = (x1, . . . , xn)contained in I, such that (x1, . . . , xn, xn+1) is not an M -sequence for any xn+1 ∈ I.Under the conditions of the next theorem, all maximal M -sequences in I havethe same length (cf. [BH93, Theorem 1.2.5]).

Theorem 0.14 (Rees). If R is Noetherian, M is a finitely generated moduleand I is an ideal such that IM 6= M , then all maximal M-sequences in I havethe same length n given by n = min{i ∈ N0 |Exti

R(R/I,M) 6= 0}.Whenever the conditions of the theorem hold, one defines the grade of I on M ,

denoted by gradeR(I, M), to be the common length of all maximal M -sequences inI. In case IM = M , we complement the definition by setting gradeR(I, M) = ∞;this is consistent with Theorem 0.14, since

gradeR(I, M) = ∞ ⇐⇒ IM = M ⇐⇒ ExtiR(R/I, M) = 0 for all i ∈ N0

(cf. [BH93, page 10]). Thus, we have

gradeR(I,M) = inf{i ∈ N0 |ExtiR(R/I,M) 6= 0}.

With this in mind, one defines the grade of M by

gradeR M = inf{i ∈ N0 |ExtiR(M, R) 6= 0}.

From Theorem 0.5 we immediately derive that gradeR M ≤ pdR M wheneverR is Noetherian and M is nontrivial. For a nonnegative integer p, we say thatM is p-perfect if gradeR M = p = pdR M or M = 0.

If R is Noetherian and local with maximal ideal m and M is a finitely gen-erated module, we refer to gradeR(m,M) as the depth of M and denote it bydepthR M . Depth and dimension come together in the following proposition(cf. [BH93, Proposition 1.2.12]).

Proposition 0.15. If R is Noetherian and local, and M is a nontrivial, finitelygenerated module, then every M-sequence is part of a system of parameters forM . In particular, depthR M ≤ dimR M .

Preliminaries xxi

If R is Noetherian and local, and M is a nontrivial, finitely generated module,we say that M is Cohen–Macaulay if depthR M = dimR M ; in particular, R isCohen–Macaulay if depth R = dim R. Any complete intersection ring is Cohen–Macaulay (cf. [BH93, Proposition 3.1.20]).

Depth is also interlinked with projective dimension in the famous Auslander–Buchsbaum formula (cf. [BH93, Theorem 1.3.3]):

Theorem 0.16 (Auslander–Buchsbaum). If R is Noetherian and local, andM is a nontrivial, finitely generated module with pdR M < ∞, then

pdR M + depthR M = depth R.

Another interesting theorem related to depth is the following lemma by Pe-skine and Szpiro, which is commonly known as the acyclicity lemma (cf. [Eis95,Lemma 20.11]).

Lemma 0.17 (acyclicity). Suppose that R is Noetherian and local, and thatX = 0 → Xn → · · · → X1 → X0 → 0 is a complex of finitely generated modulessuch that depthR X` ≥ ` for ` = 0, . . . , n. Then, if H`(X) 6= 0 for some ` > 0,then for the largest such ` we have depthR H`(X) ≥ 1.

Although grade is not additive on short exact sequences, a lower bound forthe grade of one of the modules in a short exact sequence can always be estimatedif the grades of the two others are known (cf. [BH93, Proposition 1.2.9]):

Proposition 0.18. If R is Noetherian, I is an ideal, and 0 → A → B → C → 0is a short exact sequence of finitely generated modules, then

(i) gradeR(I, A) ≥ min{gradeR(I, B), gradeR(I, C) + 1}.(ii) gradeR(I, B) ≥ min{gradeR(I, A), gradeR(I, C)}.(iii) gradeR(I, C) ≥ min{gradeR(I, A)− 1, gradeR(I, B)}.

The proposition below compiles other nice properties of grade (cf. [BH93,Proposition 1.2.10]).

Proposition 0.19. If R is Noetherian, I and J are ideals and M is a finitelygenerated module, then

(i) gradeR(I, M) = inf{depthRpMp |p ∈ VR(I)}.

(ii) gradeR(I, M) = gradeR(Rad I, M).

(iii) gradeR(I ∩ J,M) = min{gradeR(I, M), gradeR(J,M)}.(iv) If x = (x1, . . . , xn) is an M-sequence in I, then

gradeR(I/〈x〉,M/〈x〉M) = gradeR(I, M/〈x〉M) = gradeR(I, M)− n.

xxii Preliminaries

(v) If N is a finitely generated module with SuppR N = VR(I), then

gradeR(I, M) = inf{i |ExtiR(N,M) 6= 0}.

If R is Noetherian and M is finitely generated, it follows immediately frompart (v) of Proposition 0.19 that gradeR M = gradeR(AnnR M,R). If, in ad-dition, R is local and M is nontrivial, part (i) together with Proposition 0.15imply that gradeR M ≤ height AnnR M , and it then follows from Theorem 0.11that gradeR M + dimR M ≤ dim R. The grade conjecture states that if, in ad-dition, pdR M < ∞, this is actually an equality. Theorem 0.8 verifies the gradeconjecture in the case that gradeR M = 0, since we apparently have

gradeR M = 0 ⇐⇒ χR(M) 6= 0 ⇐⇒ dimR M = dim R.

We shall later verify the grade conjecture in the case that gradeR M = 1.

1 Chapter 1

Introduction

One of the fundamental results within algebraic geometry is that of Bezout’stheorem, stating that if two plane algebraic curves of degree m and n, respectively,in complex projective plane have no common components, then the number ofintersection points of the two curves when counted with multiplicity is mn. Herethe “multiplicity” of an intersection point of two curves is obtained by calculatingthe dimension of a certain complex vector space associated with the polynomialsdefining the curves. Substantial work has been put into defining intersectionmultiplicities in a more general and strictly algebraic setting so that Bezout’stheorem is satisfied. Serre introduced such a generalization in [Ser65] more than40 years ago.

Let R be a regular local ring, and let M and N be finitely generated R-modules such that M⊗RN has finite length. Under these hypotheses, the moduleTorR

` (M,N) has finite length for all ` and is zero for all but finitely many `, andone can therefore define the intersection multiplicity of M and N to be the number

χR(M, N) =∞∑

`=0

(−1)` lengthR TorR` (M,N).

Having presented this new definition, Serre proved that it satisfied many of theproperties required by intersection multiplicities, including Bezout’s theorem.Serre also conjectured that the following conditions hold.

(0) dimR M + dimR N ≤ dim R,

(1) χR(M, N) ≥ 0,

(2) χR(M, N) 6= 0 if and only if dimR M + dimR N = dim R.

These have later been known as the intersection conjectures. They are partof a collection of conjectures known as the homological conjectures, originallyintroduced by Hochster, among which we find the grade conjecture, which is alsodiscussed briefly in this thesis.

Serre proved that condition (0) holds in general and that all the conjectureshold in many cases (namely for equicharacteristic rings and unramified rings of

1

2 Introduction

mixed characteristics—two concepts that are not treated in this thesis). Theconjectures and generalizations of them have been among the central problemsin commutative algebra since they were published.

The second and third condition can be replaced by

(1′) χR(M, N) = 0 if dimR M + dimR N < dim R,

(2′) χR(M, N) > 0 if dimR M + dimR N = dim R.

The claim that condition (1) of the original statement holds is known as thenonnegativity conjecture. The claim that condition (1′) above holds is known asthe vanishing conjecture, and the claim that condition (2′) holds is known as thepositivity conjecture.

Great progress has been achieved since Serre proposed his conjectures. Thevanishing conjecture was proven in about 1985 by Roberts in [Rob85] and, inde-pendently, by Gillet and Soule, and Gabber proved the nonnegativity conjecturein about 1996. So of the original intersection conjectures, only positivity re-mains open—in fact, it only remains to be shown that χR(M,N) 6= 0 wheneverdimR M + dimR N = dim R (and only in the case where R is ramified and ofmixed characteristics). However, many other questions remain unanswered as towhether the intersection conjectures hold in more general settings where it makessense to define the intersection multiplicity of modules M and N over a ring R.

One such typical generalization is to replace the assumption that R is regu-lar by the weaker assumption that R is Noetherian and local and that M andN have finite projective dimensions. Under these assumptions, the intersectionconjectures are still open. One need only assume that one of the modules hasfinite projective dimension to define the intersection multiplicity, but as shownby a counterexample discovered in 1985 by Dutta, Hochster and McLaughlin,vanishing and positivity fail in this generality. However, if one of the moduleshas finite projective dimension and the other has (Krull) dimension 0 or 1, theintersection conjectures hold; Foxby showed this in [Fox82b] in about 1981.

The purpose of this thesis is to present the algebraic K-theory behind Foxby’sproof of the intersection conjectures; to outline the methods used by Roberts inhis proof of the vanishing conjecture; and to discuss how these two apparentlyunrelated approaches are interlinked. The intersection multiplicity is naturallyinvestigated in a K-theoretical setting, since χR(M,−) and χR(−, N) are additiveon short exact sequences of modules for which they are defined and hence factorthrough the Grothendieck groups of categories of such modules. In his proof,Foxby exploited an isomorphism between certain Grothendieck groups to showthat a calculation of χR(M, N) can be reduced to the case where one replaces Mby R/GR(M), where GR(M) is a principal ideal originally introduced by MacRaeand referred to in his honor as the MacRae ideal. Since some of the properties ofthis ideal are known, the intersection conjectures are then seen to hold.

Introduction 3

Roberts took another approach in his proof of the vanishing conjecture, usinglocal Chern characters. Although the mere definition of local Chern charactersis quite unaccessible and lies beyond the scope of this thesis, they satisfy a num-ber of very nice functorial conditions, allowing one to present the intersectionmultiplicity of two modules as an element in a Q-vector space obtained by anapplication of the local Chern characters associated to projective resolutions ofthe two modules. When the ring is a complete intersection (which is the case ifit is a regular local ring), simplification is possible and the vanishing conjectureis seen to hold.

The relationship between the two approaches to the intersection conjectures isapparent from the role played by the Euler characteristic and the MacRae ideal.Not only do these two invariants appear in the simplifications of modules in theGrothendieck groups Foxby used in his proof, they also uniquely determine thezeroth and first local Chern characters of projective resolutions of modules. Thisfact gives hope that, perhaps in future research, the discovery of a new invariant,related to the second local Chern character, will allow the intersection conjecturesto be proven in one more case.

2 Chapter 2

Groups of complexes: K0

Vector spaces over a field are classified according to their dimension. This clas-sification is sufficiently coarse-grained that each vector space is represented bysomething as simple as a cardinal number and sufficiently fine-grained that eachvector space is uniquely determined up to isomorphism by its dimension.

The “weaker” structure of a ring compared with that of a field makes con-structing a similarly powerful classification of modules over a ring impossible.A multitude of coarse-grained classifications of modules do exist, however, usinginvariants such as length, depth, grade, Krull dimension, projective dimensionand Euler characteristic. At the other end of the scale we find the ultimate fine-grained classification of modules, simply taking each module to its isomorphismclass. Somewhere before that we encounter the Grothendieck groups.

If c(V ) denotes the isomorphism class of a finitely generated vector space Vover a field k, assigning to c(V ) the vector space dimension of V yields a one-to-one correspondence between the collection I of isomorphism classes of finitelygenerated k-vector spaces and the set N0 of nonnegative integers. The additionon N0 is thus imposed on I, making it an Abelian monoid with identity elementc(0) and addition given by c(V )+c(W ) = c(V ⊕W ). By including all differencesc(V )− c(W ), I is completed to an Abelian group K0(k) isomorphic to Z; this isthe Grothendieck group of k.

In the following sections, this construction will be generalized to modules andcomplexes over rings, thereby yielding the Grothendieck groups.

2.1 Categories of complexes

This thesis covers multiple categories of complexes. The notation used has the ad-vantage that it is very flexible and thus can be applied to describe a vast numberof categories of complexes. Furthermore, the symbols involved are very descrip-tive, so that one can almost guess what category a certain symbol representswithout having to look it up in the definition.

Definition 2.1. We define the following full subcategories of the category CR ofcomplexes by specifying their objects:

5

6 Groups of complexes: K0

CR¤ : complexes that are bounded;

CR[b,a]: complexes that are concentrated in degrees b, b − 1, . . . , a; here

b, a are integers with b ≥ a;CR

]∞,a]: complexes that are concentrated in degrees b, b−1, . . . , a for someb; here b, a are integers with b ≥ a;

CRc : complexes that are concentrated in degrees c, c− 1, . . . , 0; here c

is a nonnegative integer;CR(f): complexes of finitely generated modules;CR(l): complexes of modules of finite length;CR(P): complexes of projective modules;CR(F): complexes of free modules;CR(pd): complexes of modules of finite projective dimension;

CR(S-tor): complexes of S-torsion modules; here S = (S1, . . . , Sd) is a familyof multiplicative systems;

CR(gr ≥ g): complexes of modules of grade greater than or equal to g; here gis a nonnegative integer; and

CR(p-perf): complexes X of p-perfect modules; here p is a nonnegative inte-ger.

(Such terms as “concentrated”, “S-torsion”, “grade” and “p-perfect” are de-fined in the preliminaries.)

This list could be continued indefinitely by introducing new symbols to de-scribe the “shape” of the complexes and the properties of the modules in thecomplexes. Here we have only included the symbols that are used in the sequel.

We will freely use any combination of the symbols above by setting

CR? (#1, . . . )

def= CR

? ∩ CR(#1) ∩ . . . ,

so that, for example, CR¤(f,P) denotes the category of bounded complexes of

finitely generated projective modules. (Note that we do not consider the casewhere we have several of the subscripts “?”, since any combination of subscriptscan be expressed by a single one of them.)

The homology functor H entails a new series of subcategories of CR. Theseare defined by

X ∈ CR? (#1, . . . |#′

1, . . . )def⇐⇒ X ∈ CR

? (#1, . . . ) and H(X) ∈ CR(#′1, . . . ),

so that, for instance, CR¤(f,P|S-tor) denotes the category of bounded, homologi-

cally S-torsion complexes of finitely generated projective modules.Before introducing the Grothendieck groups, some technical problems con-

cerning isomorphism classes of objects in a category need to be cleared up. Theproblem is that an isomorphism class need not be a set, nor does the collec-tion of isomorphism classes (even if each isomorphism class, in fact, is a set).Both of these set-theoretical problems can be overcome if the category is chosensufficiently “small” that its objects can be represented by a set.

2.2 Grothendieck groups 7

Definition 2.2. Let C be a category and suppose O is a set of objects of C. Wesay that C is represented by O if O contains an object from each isomorphismclass of C. A category C is said to be modest if it is represented by a set. In thiscase, the restricted isomorphism class of an object A of C is the nonempty setc(A) = {B ∈ O|A ∼= B} consisting of elements of O that are in the isomorphismclass of A. The set of restricted isomorphism classes of C will be denoted by I(C).

Henceforth, we shall suppress the word “restricted” and simply refer to c(A)as the isomorphism class of A and to I(C) as the set of isomorphism classes of C .

The categories considered here are all modest, since they are all full subcat-egories of the category of bounded complexes of finitely generated modules. Westate this as a proposition, but leave the (easy) proof as an exercise for thoseinterested in set theory.

Proposition 2.3. The category CR¤(f) of bounded complexes of finitely generated

modules is modest, and so is any full subcategory of CR¤(f).

Note the importance of the requirement in the above proposition that a sub-category C of CR

¤(f) must be full to be modest. Constructing a subcategory ofCR

¤(f) that is not modest is easy: the subcategory consisting of all the objects ofCR

¤(f) and no morphisms but the identities is not modest, since no two distinctobjects are isomorphic and the collection of objects do not form a set.

2.2 Grothendieck groups

Now that we have avoided getting ourselves into any kind of set-theoretical prob-lems, we are finally ready to present the Grothendieck groups.

Definition 2.4. Suppose C is a full subcategory of CR¤(f). The Grothendieck group

K0(C) of C is the group presented by generators [X], one for each isomorphismclass c(X), and relations [X] = 0 whenever X is exact and [Y ] = [X] + [Z]whenever there is an exact sequence 0 → X → Y → Z → 0 in C ; in other words,K0(C) = FZ(I(C))/E(C), where FZ(I(C)) is the free Abelian group based on theset I(C) of isomorphism classes of C , and E(C) is the subgroup generated byelements c(X) for which X is exact and elements c(Y )− c(X)− c(Z) for whichthere is a short exact sequence 0 → X → Y → Z → 0 in C .

Two categories are important enough that their Grothendieck groups havespecial names and notation in the literature. The Grothendieck group of R is

given by G0(R)def= K0(CR

0 (f)), and the zeroth algebraic K-group of R is given by

K0(R)def= K0(CR

0 (f,P)). This thesis, however, investigates many other Grothen-dieck groups, and for these we shall follow the notation from Definition 2.1.

Definition 2.5. If CR? (#1, . . . |#′

1, . . . ) is one of the categories from Definition 2.1,we denote its Grothendieck group by GR

? (#1, . . . |#′1, . . . ).


If C is a full subcategory of CR¤(f) and C does not contain the zero complex,

then there are no exact sequences in K0(C), so in this case we have K0(C) =FZ(I(C))/E(C), where E(C) is the subgroup generated by isomorphism classesc(X) for which X is exact; in particular, if C is a subcategory of the category CR

0 (f)of finitely generated modules, then K0(C) is the free Abelian group FZ(I(C)). Onthe other hand, if the zero complex is an object of C , the short exact sequence0 → 0 → 0 → 0 → 0 of complexes shows that [0] = 0 in K0(C). All the categoriesof complexes treated in this thesis contain the zero complex.

The fact that the existence of a short exact sequence 0 → X → Y → Z → 0 inC forces the equation [Y ] = [X] + [Z] to hold in K0(C) means, in particular, that[X ⊕ Z] = [X] + [Z] whenever 0 → X → X ⊕ Z → Z → 0 is an exact sequencein C . Consequently, if C is a full subcategory of CR

¤(f) that is closed under directsum (which is the case for all the categories considered here), then any finite sumof “nonnegative” elements in K0(C) collapses to a single term, and it follows thatany element of K0(C) can be written as [X]− [X ′] for complexes X,X ′ ∈ C . Thefollowing proposition enables us to determine when such an element is trivial.

Proposition 2.6. Let C be a full subcategory of CR¤(f) containing the zero com-

plex, and suppose that C is closed under direct sum. Furthermore, let X and X ′

be complexes in C . Then [X] = [X ′] in K0(C) if and only if C contains exactcomplexes Y and Y ′ and short exact sequences 0 → U → V → W → 0 and0 → U → V ′ → W → 0 such that X ⊕ Y ′ ⊕ V ′ ∼= X ′ ⊕ Y ⊕ V .

Proof: “if” is clear, since in this case we have [V ′] = [U ] + [W ] = [V ] and

[X] + [V ′] = [X] + [Y ′] + [V ′]

= [X ⊕ Y ′ ⊕ V ′]

= [X ′ ⊕ Y ⊕ V ]

= [X ′] + [Y ] + [V ]

= [X ′] + [V ].

For the other direction, suppose that X and X ′ are complexes in C such that[X] = [X ′] in K0(C). Then the element c(X) − c(X ′) ∈ FZ(I(C)) is in thesubgroup E(C) described in Definition 2.4, and hence we can write

c(X)− c(X ′) =∑

i

c(Yi)−∑

j

c(Y ′j )

+∑

k

(c(Vk)− c(Uk)− c(Wk))−∑

`

(c(V ′` )− c(U ′

`)− c(W ′`)),

where Yi and Y ′j are exact for all i and j, and there are short exact sequences

0 → Uk → Vk → Wk → 0 and 0 → U ′` → V ′

` → W ′` → 0 for all k and `,


respectively. Moving negative terms to the other side of the equation, we obtain

c(X) +∑

j

c(Y ′j ) +

∑

k

(c(Uk) + c(Wk)) +∑

`

c(V ′` )

= c(X ′) +∑

i

c(Yi) +∑

k

c(Vk) +∑

`

(c(U ′`) + c(W ′

`)).

This is an equation within the free Abelian group FZ(I(C)). The terms on eachside of the equation are all elements of the generating set, so the fact that theequation holds implies that there is termwise equality. In particular, taking thedirect sum of the terms on each side turns the above equation into an isomorphism

X ⊕ (∐

j

Y ′j )⊕ (

∐

k

(Uk ⊕Wk))⊕ (∐

`

V ′` )

∼= X ′ ⊕ (∐

i

Yi)⊕ (∐

k

Vk)⊕ (∐

`

(U ′` ⊕W ′

`)).

The direct sums Y ′ =∐

j Y ′j and Y =

∐i Yi are exact since their summands are.

Furthermore, the direct sum of short exact sequences of complexes is again ashort exact sequence, so letting

U =∐

k

Uk, V =∐

k

Vk, W =∐

k

Wk,

U ′ =∐

`

U ′`, V ′ =

∐

`

V ′` , W ′ =

∐

`

W ′`,

we have short exact sequences

0 → U ⊕ U ′ → V ⊕ U ′ ⊕ W ′ → W ⊕ W ′ → 0

and0 → U ′ ⊕ U → V ′ ⊕ U ⊕ W → W ′ ⊕ W → 0.

Since C is closed under direct sum, we can now let U = U ⊕ U ′, W = W ⊕ W ′,V = U ⊕ W ⊕ V ′ and V ′ = V ⊕ U ′ ⊕ W ′, and the proposition is proved.

In the traditional zeroth algebraic K-group K0(R), things are much nicer.

Corollary 2.7. If M and N are modules in CR0 (f,P), then [M ] = [N ] in K0(R)

if and only if M ⊕Rn ∼= N ⊕Rn for some n ∈ N0.

Proof: “if” is clear. For the other direction, note that, since short exact se-quences of projective modules split, Proposition 2.6 says that [M ] = [N ] impliesthe existence of a finitely generated projective module L such that M⊕L ∼= N⊕L.Adding a finitely generated projective module Q such that L⊕Q ∼= Rn for somen ∈ N0 (this exists; see, for example, [Mag02, Proposition 2.21]), we therefore getM ⊕Rn ∼= N ⊕Rn as desired.

A basic result that is used repeatedly is



plex, and suppose that X is a complex in C such that the shifted complex ΣX andthe mapping cone M(1X) are in C . Then [ΣX] = −[X] in K0(C).

Proof: Because of the assumptions, the exact sequence 0 → X → M(1X) →ΣX → 0 from Theorem 0.2 is in C , implying that [M(1X)] = [X] + [ΣX] inK0(C). Since the identity map 1X is a homology isomorphism, its mapping coneM(1X) is exact according to Theorem 0.4, and hence [M(1X)] = 0 and weconclude that [ΣX] = −[X].

When Proposition 2.8 is applied to the complex Σ−1X, the complex X shiftedone degree to the right, one obtains that [Σ−1X] = −[X]. In general, if ourcategory C is large enough that it contains all the necessary complexes, [ΣnX] =(−1)n[X] in K0(C) for all n ∈ Z.

Proposition 2.8 can be taken one step further.


plex, and suppose that φ : X → Y is a homology isomorphism in C such that theshifted complex ΣX and the mapping cones M(1X) and M(φ) are in C . Then[X] = [Y ] in K0(C).

Proof: According to Theorem 0.4, it follows from φ being a homology isomor-phism that M(φ) is exact, and hence that [M(φ)] = 0. From the exact sequence0 → Y →M(φ) → ΣX → 0 (see Theorem 0.2) and Proposition 2.8 we now get[Y ] = [M(φ)]− [ΣX] = [X].

In this thesis we shall consider several homomorphisms between Grothendieckgroups. Among these are the homomorphisms A, H, R and χ described in The-orems 2.10 through 2.13 below. The first theorem shows how we can switch fromcomplexes to modules at the level of Grothendieck groups in certain circumstancesby taking the alternating sum of the modules in a complex.

Theorem 2.10. Let C be a full subcategory of CR¤(f) containing the zero complex,

and let C0 be a full subcategory of CR0 (f) containing all kernels of its homomorphism

(and therefore also containing the zero module). Suppose that for any X in C , themodules X` are in C0. Then there is a group homomorphism A : K0(C) → K0(C0)given by A([X]) =

∑`∈Z(−1)`[X`].

Proof: The assumptions ensure that, for X in C , a(X) =∑

`∈Z(−1)`[X`] is awell-defined element of K0(C0), so it only remains to verify that the relations inK0(C) are preserved under a, so that a induces the homomorphism A.

So suppose that X is exact, and let Z` = ker ∂X` = im ∂X

`+1. For each ` ∈ Z,we then have an exact sequence 0 → Z` → X` → Z`−1 → 0, which, according tothe assumption, is in C0. It follows that

a(X) =∑

`∈Z(−1)`[X`] =

∑

`∈Z(−1)`([Z`] + [Z`−1]) = 0.


Suppose next that 0 → U → V → W → 0 is an exact sequence in C . Thereis then an exact sequence 0 → U` → V` → W` → 0 of modules in each degree` ∈ Z, proving that

a(V ) =∑

`∈Z(−1)`[V`] =

∑

`∈Z(−1)`([U`] + [W`]) = a(U) + a(W ).

This completes the proof that a preserves the relations in K0(C) and therebyinduces the homomorphism A.

In many other cases we can switch from complexes to modules at the level ofGrothendieck groups by taking the alternating sum of the homology modules ofa complex.

Theorem 2.11. Let C be a full subcategory of CR¤(f) containing the zero complex,

and let C0 be a full subcategory of CR0 (f) containing the homology modules H`(X)

of all X in C (and hence also containing the zero module). Assume either thatall homologies H(X) of complexes X in C are concentrated in degree 0, or that C0

contains all the kernels of its homomorphisms. Then, in either case, there is grouphomomorphism H : K0(C) → K0(C0) given by H([X]) =

∑`∈Z(−1)`[H`(X)].

Proof: The assumptions ensure that, for X in C , h(X) =∑

`∈Z(−1)`[H`(X)] isa well-defined element of K0(C0), so it only remains to verify that the relationsin K0(C) are preserved under h, so that h induces the homomorphism H.

If X is exact, then clearly h(X) = 0. Now let 0 → U → V → W → 0 bean exact sequence of complexes in C , and consider the long exact sequence onhomology:

· · · → H`(U) → H`(V ) → H`(W ) → H`−1(U) → · · · .

If we assume that all homologies are concentrated in degree 0, the above is simplythe exact sequence 0 → H0(U) → H0(V ) → H0(W ) → 0, proving that h(V ) =h(U) + h(W ) in K0(C0). So assume instead that C0 contains all the kernels of itshomomorphisms. For ` ∈ Z, let ZU

` , ZV` and ZW

` denote the kernels of the mapsH`(U) → H`(V ), H`(V ) → H`(W ) and H`(W ) → H`−1(U), respectively. We thenhave exact sequences

0 → ZU` → H`(U) → ZV

` ,

0 → ZV` → H`(V ) → ZW

` and

0 → ZW` → H`(W ) → ZU

`−1,


which, according to the assumption, all are in C0. We now see that

h(V ) =∑

`∈Z(−1)`[H`(V )]

=∑

`∈Z(−1)`([ZV

` ] + [ZW` ])

=∑

`∈Z(−1)`([ZU

` ] + [ZV` ]) +

∑

`∈Z(−1)`([ZW

` ] + [ZU`−1])

= h(U) + h(W ).

This completes the proof that h preserves the relations in K0(C) and therebyinduces the homomorphism H.

Sometimes we can go the other way, switching from modules to complexes,by taking projective resolutions.

Theorem 2.12. Let C0 be a full subcategory of CR0 (f,pd) containing the zero

module, and let C be a full subcategory of CR¤(f) containing the zero complex and

for each exact sequence 0 → L → M → N → 0 in C0 containing projectiveresolutions U , V and W of L, M and N respectively, such that there is an exactsequence 0 → U → V → W → 0 in C . Suppose that C is closed under the shiftoperator Σ(−) and under the formation of mapping cones M(−). Then thereis a group homomorphism R : K0(C0) → K0(C) given by R([M ]) = [X], whereX ∈ C is a projective resolution of M .

Proof: If M is a module in C0, then the assumptions ensure that there is at leastone projective resolution X of M in C . We first verify that [X] = [X ′] wheneverX, X ′ ∈ C are both finite projective resolutions of M . In this case, there is a

homology isomorphism X'−→ X ′ (see, for example, [HS97, Proposition IV.4.3]),

so according to Proposition 2.9 and the assumptions made on C , [X] = [X ′].Thus, it makes sense to define, for any M ∈ C0, an element r(M) in K0(C), givenby r(M) = [X] for any choice X ∈ C of projective resolution of M . To show thatr in fact induces a homomorphismR : K0(C0) → K0(C), it suffices to demonstratethat if 0 → L → M → N → 0 is a short exact sequence in C0, then projectiveresolutions U , V and W of L, M and N , respectively, exist, such that there is ashort exact sequence 0 → U → V → W → 0 in C . But this is exactly what wehave assumed.

Although the requirement that short exact sequences of projective resolutionsof short exact sequences of modules should exist sounds slightly awkward, it is notso rarely satisfied. Indeed, if 0 → L → M → N → 0 is a short exact sequenceof modules and U and W are projective resolutions of L and N respectively,then there is a projective resolution V of M whose modules V` are the directsums U`⊕W`, such that the (degreewise) inclusion U ↪→ V and the (degreewise)


projection V ³ W are both morphisms of complexes (see, for example, [Mag02,(3.51)]). Consequently, the requirement is satisfied by all categories of complexesthat are closed under this construction.

In the case that C0, C ′0 and C are categories such that the homomorphismsR : K0(C0) → K0(C) and A : K0(C) → K0(C ′0) from Theorems 2.12 and 2.10, re-spectively, are defined, the composition χ = A◦R is a homomorphism K0(C0) →K0(C ′0) given by χ([M ]) =

∑`∈Z(−1)`[X`] for a finite projective resolution X of

M . One often skips the intermediate step and defines χ directly as a homomor-phism K0(C0) → K0(C ′0).

Theorem 2.13. Let C0 be a full subcategory of CR0 (f,pd) containing the zero

module, and let C ′0 be a full subcategory of CR0 (f) containing CR

0 (f,P). Supposethat every module in C0 has a finite projective resolution. Then there is a grouphomomorphism χ : K0(C0) → K0(C ′0) given by χ([M ]) =

∑`∈Z(−1)`[X`], where

X is a bounded finite projective resolution of M .

Proof: If M is a module in C0, the assumptions ensure that we can find abounded finite projective resolution X of M and that

∑`∈Z(−1)`[X`] is a well-

defined element of C ′0. Suppose that X and X ′ are both bounded finite projectiveresolutions of M . We then have (see, for example, [Mag02, Corollary 3.47])

(∐

` even

X`)⊕ (∐

` odd

X ′`)∼= (

∐

` odd

X`)⊕ (∐

` even

X ′`),

proving that∑

`∈Z(−1)`[X`] =∑

`∈Z(−1)`[X ′`] in K0(C ′0). Thus, we can asso-

ciate to every element M ∈ C0 the element∑

`∈Z(−1)`[X`] in K0(C ′0) for anychoice of bounded finite projective resolution X of M . The fact that this asso-ciation induces a homomorphism χ follows as in the proof of Theorem 2.12 (and[Mag02] (3.51)), since we are free to construct short exact sequences of projectiveresolutions of short exact sequences of modules.

A homomorphism χ as in Theorem 2.13 is called an Euler characteristic. Thisterm is natural since the usual Euler characteristic χR(−) is defined when R isNoetherian and local, in which case K0(R) = K0(CR

0 (f,P)) is isomorphic to Z(see Example 2.14 below), by an isomorphism taking χ([M ]) =

∑`∈Z(−1)`[X`]

to∑

`∈Z(−1)` rankR X` = χR(M). Thus, the usual Euler characteristic is anapplication of the above theorem to the case where C0 = CR

0 (f,pd) and C ′0 =CR

0 (f,P).If C0 is a full subcategory of CR

0 (f), the Grothendieck group K0(C0) has twofundamental properties: (i) it is an Abelian group and (ii) there is a functionπ : I(C0) → K0(C0) (given by π(c(M)) = [M ]) that is additive on short exactsequences. The definition of Grothendieck groups implies that K0(C0) is univer-sal with respect to these properties, in the sense that if A is any other Abeliangroup equipped with a function r : I(C0) → A that is additive on short exact


sequences, then r factors uniquely through K0(C0): that is, a unique homomor-phism r : K0(C0) → A exists such that r = rπ. Henceforth, when given such afunction r, we shall also denote the induced function by r.

The universal property of Grothendieck groups often enables us to calculatethe Grothendieck group of a category. A few examples of this are listed here.

Example 2.14. The usual free rank is a function I(CR0 (f,F)) → Z, which is

additive on short exact sequences. Since rank R = 1, the induced homomorphismGR

0 (f,F) → Z is surjective. To see that it is in fact also injective, recall that,since CR

0 (f,F) is closed under direct sum, all elements of GR0 (f,F) can be written

in the form [M ] − [N ]. If such an element is in the kernel of rank, then Mand N must have the same free rank: that is, M ∼= Rn ∼= N for some n ∈ N0.Consequently [M ] = [N ], so rank is injective as a map GR

0 (f,F) → Z. We havetherefore shown that GR

0 (f,F) ∼= Z. In particular, if R is local, then any finitelygenerated projective module is free, so CR

0 (f,P) is the same as CR0 (f,F), and it

follows that K0(R) ∼= Z. Note that, regardless of whether R is local or not,the map Z → K0(R) given by 1 7→ [R] is always injective, since [Rm] = [Rn]according to Corollary 2.7 implies that m = n.

Example 2.15. Consider the category CZ0 (l) of finite-length Z-modules; this isexactly the same as the category of finite Abelian groups. The order of finitegroups is a function | − | : I(CZ0 (l)) → Q+, which is multiplicative on short exactsequences. The induced homomorphism | − | : GZ

0 (l) → Q+, which is given by[H] − [K] 7→ |H|/|K|, is surjective since Abelian groups of every positive orderexist. To see that |−| is also injective, we can assume, as in the previous example,that H and K are finitely generated Abelian groups with |H| = |K|. Now, recallthat H has a composition series

H = H0 B H1 B · · ·B Ht−1 B Ht = {0H},

in which each quotient Hi−1/Hi is cyclic of prime order: that is, Hi−1/Hi∼=

Z/piZ for some prime number pi. From the exact sequences 0 → Hi → Hi−1 →Z/piZ → 0 we therefore get [H] =

∑ti=1[Z/piZ] in GZ

0 (l). Similarly, we canreduce [K] into composition factors, so that [K] =

∑sj=1[Z/qjZ] where the qj’s

are prime numbers. Based on the assumption that |H| = |K|, it now followsthat

∏ti=1 pi =

∏sj=1 qj, and the fundamental theorem of arithmetics implies that

t = s and that, after rearrangement of the qj’s, pi = qi for all i. Consequently[H] =

∑ti=1[Z/piZ] = [K] in GZ

0 (l), so | − | is injective. We have therefore provedthat GZ

0 (l) ∼= Q+.

Example 2.16. Suppose that R is local with maximal ideal m and quotient fieldk = R/m, and consider the category CR

0 (l) of modules of finite length. The lengthof modules is then a function lengthR : I(CR

0 (l)) → Z, which is additive on shortexact sequences. The induced homomorphism lengthR : GR

0 (l) → Z is surjective


since lengthR k = 1. To demonstrate that lengthR is injective, we can assume,as in the previous examples, that M and N are modules with lengthR M =lengthR N < ∞. Now, note that in a composition series

M = M0 ⊃ M1 ⊃ · · · ⊃ Mt−1 ⊃ Mt = 0,

where t = lengthR M , each quotient Mi−1/Mi is isomorphic to k. From the exactsequences 0 → Mi → Mi−1 → k → 0, we therefore inductively see that [M ] =∑t

i=1[k] = (lengthR M)[k]. From the assumption that lengthR M = lengthR N ,it now follows that [M ] = (lengthR M)[k] = [N ] in GR

0 (l), so lengthR is indeedinjective. Hence we have proved that GR

0 (l) ∼= Z.

This section concludes by investigating how group homomorphisms betweenGrothendieck groups can be induced from functors between the underlying cat-egories. Suppose R and R′ are (nontrivial, unitary and commutative) ringsand that C and C ′ are full subcategories of CR

¤(f) and CR′¤ (f), respectively. Let

F : C → C ′ be a covariant, exact functor taking exact complexes to exact com-plexes. Then there is an induced group homomorphism K0(F ) : K0(C) → K0(C ′)given by K0(F )([X]) = [F (X)] for X ∈ C ; the relations in K0(C) are preservedbecause of the assumption that F is exact and takes exact complexes to exactcomplexes.

In case C is a subcategory of CR0 (f), the requirement that F must take exact

complexes to exact complexes is superfluous, and the only remaining requirementis that F be exact. When the categories CR

0 (f,P) and CR′0 (f,P) from which we form

the traditional K0-groups K0(R) and K0(R′) are considered, it suffices in some

cases that F is additive.

Theorem 2.17. If R and R′ are (nontrivial, unitary and commutative) rings andF : CR

0 → CR′0 is an additive functor such that F (R) ∈ CR′

0 (f,P), then F restricts toan exact functor CR

0 (f,P) → CR′0 (f,P). Consequently, F induces a homomorphism

K0(F ) : K0(R) → K0(R′).

Proof: If P ∈ CR0 (f,P), then Q ∈ CR

0 (f,P) exists such that P ⊕Q ∼= Rn for somen ∈ N0. Since F is additive,

F (P )⊕ F (Q) ∼= F (P ⊕Q) ∼= F (Rn) ∼= F (R)n,

which, by assumption, is a module in CR′0 (f,P). Consequently, F (P ) is a direct

summand of a finitely generated projective R′-module, and hence it is itself afinitely generated projective R′-module. This proves that F restricts to a functorCR

0 (f,P) → CR′0 (f,P). Since all exact sequences in CR

0 (f,P) split, the additivityof F ensures that this restriction of F is exact. It follows that F induces ahomomorphism K0(F ) : K0(R) → K0(R

′).

An important application of Theorem 2.17 is in the case of a ring homomor-phism ρ : R → R′. We can then consider R′ as an (R′, R)-bimodule, and we get an


induced functor R′⊗R− : CR0 → CR′

0 that satisfies the conditions of Theorem 2.17since it is additive and takes R to R′ ⊗R R ∼= R′ ∈ CR′

0 (f,P).

Definition 2.18. If R and R′ are (nontrivial, unitary and commutative) ringsand ρ : R → R′ is a ring homomorphism, then we denote by K0(ρ) the functorK0(R) → K0(R

′) induced by the functor R′ ⊗R − : CR0 → CR′

0 in the sense ofTheorem 2.17.

3Chapter 3

Grothendieck group isomor-phisms

This chapter establishes a number of surprising isomorphisms between relatedGrothendieck groups. The results obtained are all consequences of the followingtheorem, which is therefore referred to as the main theorem.

Main Theorem. Suppose that d is a nonnegative integer and S = (S1, . . . , Sd)is a d-tuple of multiplicative systems of R. Then the group homomorphism

ι : GRd (f,P|S-tor) → GR

¤(f,P|S-tor)

given by ι([X]) = [X] is an isomorphism.

Note that the two [X]’s are different: one is a member of GRd (f,P|S-tor),

whereas the other is a member of GR¤(f,P|S-tor). Note also that the map ι is

not trivially injective, although it is induced by the inclusion of CRd (f,P|S-tor)

in CR¤(f,P|S-tor). The fact that CR

d (f,P|S-tor) is a subcategory of CR¤(f,P|S-tor),

however, does not help anything but to ensure that ι is well defined.In the case that d = 0, the requirement of being homologically S-torsion is

eliminated, and the main theorem states that K0(R) = GR0 (f,P) ∼= GR

¤(f,P).Establishing the main theorem is a cumbersome task. We will construct an

inverse to ι as follows. Given a complex Y ∈ CR¤(f,P|S-tor), we choose n ∈ Z

so that the shifted complex ΣnY is in CRe (f,P|S-tor) for some e > d. To this

complex we associate an element we(ΣnY ) ∈ GR

e−1(f,P|S-tor). Repeating thisprocess a finite number of times, we end up with an element wd+1 · · ·we(Σ

nY ) inGR

d (f,P|S-tor). This is the image of [Y ] under the inverse of ι.We begin by investigating the concepts of contractions and Koszul complexes.

3.1 Contractions

Throughout this section, d denotes a positive integer and S = (S1, . . . , Sd)denotes a d-tuple of multiplicative systems of R.

Definition 3.1. Let X ∈ CR be a complex. A d-tuple α = (α1, . . . , αd) of familiesαν = (αν

` )`∈Z of homomorphisms αν` : X` → X`+1 is an S-contraction of X with

weight s = (s1, . . . , sd) ∈ S1 × · · · × Sd if

∂X`+1α

ν` + αν

`−1∂X` = sν1X`

17

18 Grothendieck group isomorphisms

for all ` ∈ Z and ν = 1, . . . , d.

The situation is as follows.

· · · // X`+1oo∂X

`+1 //X`

∂X` //

αν`

oo X`−1//

αν`−1

oo · · ·oo .

Note that the existence of an S-contraction of X with weight s = (s1, . . . , sd) isequivalent to the condition that the morphisms sν1X : X → X for ν = 1, . . . , dare null-homotopic.

Proposition 3.2. Each complex X ∈ CR¤(f,P|S-tor) has an S-contraction.

Proof: For each ν the S−1ν R-complex S−1

ν X is exact, bounded and consists offinitely generated projective S−1

ν R-modules, so the identity morphism 1S−1ν X on

S−1ν X is null-homotopic (see, for example, [HS97, Theorem IV.4.1]). Thus, we

can find S−1ν R-homomorphisms bν

` : S−1ν X` → S−1

ν X`+1 such that

∂S−1ν X

`+1 bν` + bν

`−1∂S−1

ν X` = 1S−1

ν X`

for all ` ∈ Z.Because of the natural isomorphism (see, for example, [Eis95, Proposition 2.10])

HomS−1ν R(S−1

ν X`, S−1ν X`+1) ∼= S−1

ν HomR(X`, X`+1)

and because our complex is bounded, the S−1ν R-homomorphism bν

` can be writtenas βν

` /tν for an R-homomorphism βν` : X` → X`+1 and some common denominator

tν ∈ Sν . For any x ∈ X`, we must now have in S−1ν X` that

(∂X`+1β

ν` + βν

`−1∂X` )(x)/tν = x/1.

Consequently, we can find uν,x ∈ Sν depending on x so that in X`,

uν,x(∂X`+1β

ν` + βν

`−1∂X` )(x) = uν,xtνx.

Since X is bounded and consists of finitely generated modules, by multiplying afinite number of uν,x’s, we can obtain an element uν ∈ Sν , independent of x and of`, such that uν(∂

X`+1β

ν` +βν

`−1∂X` )(x) = uνtνx for all ` ∈ Z and all x ∈ X`. Setting

αν` = uνβ

ν` and sν = uνtν , we see that α = (α1, . . . , αd), where αν = (αν

` )`∈Z, isan S-contraction of X with weight s = (s1, . . . , sd).

Definition 3.3. Let X and Y be complexes in CR with S-contractions α and β,respectively, and let φ : X → Y be a morphism of complexes. Then α and β aresaid to be compatible with φ if they have the same weight and φ`+1α

ν` = βν

` φ` forall ` ∈ Z and ν = 1, . . . , d.

3.1 Contractions 19

Theorem 3.4 below provides an example of a situation where an S-contractionon a complex induces an S-contraction on another complex. Although the hy-potheses of the theorem are very specific, the theorem turns out to be applicablein several situations.

Theorem 3.4. Let X be a complex in CRe , where e > 1, and suppose that α is an

S-contraction of X with weight s. Let X be another complex in CRe , and suppose

that the complex X is identical to X except for the modules and differentials indegrees e and e−1. Suppose further that Xe = 0 and that a morphism φ : X → Xexists such that φ` = 1X`

for ` = 0, . . . , e − 2 and such that φe−1 is surjective.Then the S-contraction α on X induces an S-contraction α on X with weight ssuch that α and α are compatible with the morphism φ; for ν = 1, . . . , d, αν isdefined by setting αν

e−2 = φe−1ανe−2 and αν

` = αν` for ` = 0, . . . , e− 3.

Proof: The situation is as follows.

0 // Xe

0

²²

∂Xe // Xe−1

φe−1

²²

∂Xe−1 //

ανe−1

oo Xe−2

1Xe−2

²²

//

ανe−2

oo · · · //oo X1

1X1

²²

∂X1 //

oo X0

1X0

²²

αX0

oo // 0

0 // 00 //

Xe−10

oo∂X

e−1 // Xe−2//

φe−1ανe−2

oo · · · //oo X1

∂X1 //

oo X0//

αν0

oo 0

To demonstrate that α is an S-contraction of X with weight s, we need to verifythat sν1Xe−1

= φe−1ανe−2∂

Xe−1 and that sν1Xe−2 = ∂X

e−1φe−1ανe−2 + αν

e−3∂Xe−2 for

ν = 1, . . . , d. Since φe−1 is surjective, the first of these equations follows from thefollowing calculation.

φe−1ανe−2∂

Xe−1φe−1 = φe−1α

νe−2∂

Xe−1

= φe−1(sν1Xe−1 − ∂Xe αν

e−1)

= sνφe−1.

The second equation is even easier:

∂Xe−1φe−1α

νe−2 + αν

e−3∂Xe−2 = ∂X

e−1ανe−2 + αν

e−3∂Xe−2 = sν1Xe−2 .

By construction of α, α and α are compatible with the morphism φ. Thusthe theorem has been proven.

Given an S-contraction α of X with weight s = (s1, . . . , sd) and a d-tuplet = (t1, . . . , td) ∈ S1 × · · · × Sd, we can construct an S-contraction tα of Xwith weight st = (s1t1, . . . , sdtd) by setting tα = (t1α

1, . . . , tdαd) where tνα

ν =(tνα

ν` )`∈Z. We can also shift α n degrees to the left for some n ∈ Z to form an

S-contraction Σnα of ΣnX with weight s by setting Σnα = (Σnα1, . . . , Σnαd)where Σnαν = ((−1)nαν

`−n)`∈Z.The following theorem shows how to construct a natural S-contraction of

the mapping cone of a morphism between two complexes that both have S-contractions.


Theorem 3.5. Let φ : X → Y be a morphism of complexes and let α and βbe S-contractions of X and Y , respectively, with weights s and t, respectively.Define for ν = 1, . . . , d and ` ∈ Z the homomorphism

(β ∗ α)ν` =

sνβν` βν

` φ`αν`−1

0 −tναν`−1

: M(φ)` =

Y`

⊕X`−1

→Y`+1

⊕X`

= M(φ)`+1.

Then (β ∗ α) = ((β ∗ α)1, . . . , (β ∗ α)d), where (β ∗ α)ν = ((β ∗ α)ν` )`∈Z, is an

S-contraction of the mapping cone M(φ) of φ with weight st = (s1t1, . . . , sdtd),and the S-contractions sβ, (β ∗ α) and Σtα are compatible with the morphismsin the canonical exact sequence

0 → Y →M(φ) → ΣX → 0. (3.1)

Proof: This is just a matter of verification. First we need to show that

∂M(φ)`+1 (β ∗ α)ν

` + (β ∗ α)ν`−1∂

M(φ)` = sνtν1M(φ)`

for ν = 1, . . . , d and ` ∈ Z. This comes down to the following multiplication ofmatrices.

(∂Y

`+1 φ`

0 −∂X`

)(sνβ

ν` βν

` φ`αν`−1

0 −tναν`−1

)+

(sνβ

ν`−1 βν

`−1φ`−1αν`−2

0 −tναν`−2

)(∂Y

` φ`−1

0 −∂X`−1

)

=

(sν(∂

Y`+1β

ν` + βν

`−1∂Y` ) (∂Y

`+1βν` − tν)φ`α

ν`−1 + βν

`−1φ`−1(sν − αν`−2∂

X`−1)

0 tν(∂X` αν

`−1 + αν`−2∂

X`−1)

)

=

(sνtν1Y`

βν`−1(−∂Y

` φ` + φ`−1∂X` )αν

`−1

0 sνtν1X`−1

)

=

(sνtν1Y`

00 sνtν1X`−1

).

Here the second equality follows from α and β being contractions with weightss and t, respectively, and the third equality follows from φ being a morphism ofcomplexes. Consequently, (β ∗ α) is an S-contraction with weight st.

The fact that sβ and (β ∗ α) are compatible with the first morphism in (3.1)follows from the following calculation.

(1Y`+1

0

)sνβ

ν` =

(sνβ

ν` βν

` φ`αν`−1

0 −tναν`−1

)(1Y`

0

).

Similarly, the fact that (β ∗α) and Σtα are compatible with the second morphismin (3.1) follows from

(0 1X`

) (sνβ

ν` βν

` φ`αν`−1

0 −tναν`−1

)= −tνα

ν`−1

(0 1X`−1

).

3.2 Koszul complexes 21

3.2 Koszul complexes

Throughout this section, d denotes a positive integer and S = (S1, . . . , Sd)denotes a d-tuple of multiplicative systems of R. Furthermore, e denotes aninteger with e > d.

The construction of the inverse of ι involves the introduction of a complex∆e(X, s), which would generally be known as the Koszul complex of the se-quence s = (s1, . . . , sd) with coefficients in Xe. More specifically, given a complexX ∈ CR

e and a sequence s = (s1, . . . , sd) ∈ S1 × · · · × Sd, ∆e(X, s) is the com-plex Σe−dK(s,Xe): that is, the Koszul complex of the sequence s = (s1, . . . , sd)with coefficients in Xe and shifted e− d degrees to the left. As the reader is notexpected to be familiar with Koszul complexes, they are briefly introduced anda few basic results that will be needed later are listed. Furthermore, ∆e(X, s) isexplicitly described.

Given an element r ∈ R, the Koszul complex K(r) of r is the complex

0 −→ Rr−→ R −→ 0

concentrated in degrees 1 and 0. Given a sequence x = (x1, . . . , xn) ∈ Rn, theKoszul complex of x is the complex

K(x) = K(x1, . . . , xn) = K(x1)⊗R · · · ⊗R K(xn),

which is concentrated in degrees n, . . . , 0. (The tensor product of complexesis defined in the preliminaries.) Given a sequence x = (x1, . . . , xn) ∈ Rn anda module M , the Koszul complex of x with coefficients in M is the complexK(x,M) = K(x) ⊗R M , which is concentrated in degrees n, . . . , 0. Note thatK(x,R) = K(x).

One of the most important properties of Koszul complexes is that they serveas free resolutions for regular sequences (cf. [BH93, Corollary 1.6.14 and Propo-sition 1.6.5]).

Theorem 3.6. Suppose that x = (x1, . . . , xn) is a sequence of elements from R,and that M is a module.

(i) The homology of K(x,M) is annihilated by the ideal 〈x〉.(ii) If x is an M-sequence, then the homology of K(x,M) is concentrated in

degree 0.

(iii) If x is a regular sequence, then K(x) is a finite free resolution of R/〈x〉.Since gradeR R/〈x〉 ≤ pdR R/〈x〉, it follows from part (iii) of Theorem 3.6

that pdR R/〈x〉 = n whenever x = (x1, . . . , xn) is a regular sequence.


Given a complex X ∈ CRe and a sequence s = (s1, . . . , sd) ∈ S1 × · · · × Sd,

we define, as mentioned above, ∆e(X, s) to be the complex Σe−dK(s,Xe). Forconvenience we will now present a more explicit description of ∆e(X, s).

For any ` ∈ Z, let Υ(`) denote the set of `-element subsets of {1, . . . , d}: thatis, the set of subsets in the form i = {i1, . . . , i`} where 1 ≤ i1 < · · · < i` ≤ d. Inparticular, Υ(0) = {∅}, Υ(d) = {{1, . . . , d}} and Υ(`) = ∅ for all ` /∈ {0, . . . , d}.Thus, in any case, Υ(`) contains

(d`

)elements. An element i ∈ Υ(`) is called a

multi-index . The elements of such a multi-index are always denoted by i1, . . . , i`in increasing order, so that i = {i1, . . . , i`}, where 1 ≤ i1 < · · · < i` ≤ d.

Given a complex X ∈ CRe and a d-tuple s = (s1, . . . , sd) ∈ S1 × · · · × Sd,

∆e(X, s) is the complex whose `’th module is given by

∆e(X, s)` =∐

i∈Υ(e−`)

∆e(X, s)i`, where ∆e(X, s)i

` = Xe,

and whose `’th differential ∂∆e(X,s)` : ∆e(X, s)` → ∆e(X, s)`−1 is given by the fact

that its (j, i)-entry (∂∆e(X,s)` )j,i for i ∈ Υ(e− `) and j ∈ Υ(e− ` + 1) is

(∂∆e(X,s)` )j,i =

{(−1)u+1sju1Xe , if j\i = {ju}

0, if j + i

So ∆e(X, s) is a complex whose `’th module ∆e(X, s)` consists of(

de−`

)copies of

Xe and whose `’th differential as a map from the i’th copy of Xe in ∆e(X, s)` tothe j’th copy of Xe in ∆e(X, s)`+1 is nonzero only when i ⊆ j, in which case it ismultiplication by (−1)u+1sju for the unique ju which is in j and not in i.

The complex ∆e(X, s) comes naturally equipped with an S-contraction.

Theorem 3.7. If X ∈ CRe and s = (s1, . . . , sd) ∈ S1×· · ·×Sd, then ∆e(X, s) has

an S-contraction δe(X, s) with weight s; for each ` ∈ Z and each ν = 1, . . . , d,the homomorphism δe(X, s)ν

` : ∆e(X, s)` → ∆e(X, s)`+1 is given by the fact thatits (j, i)-entry for i ∈ Υ(e− `) and j ∈ Υ(e− `− 1) is

(δe(X, s)ν` )j,i =

{(−1)w+11Xe , if i\j = {iw} = {ν},

0, if i + j.

Proof: This is a matter of verification. For each ν ∈ {1, . . . , d}, ` ∈ Z and

i, i′ ∈ Υ(d− `), the (i′, i)-entry of ∂∆e(X,s)`+1 δe(X, s)ν

` is

sν1Xe , if i = i′ and ν ∈ i,(−1)u+wsi′u1Xe , if i\i′ = {iw} = {ν} and i′\i = {i′u}, and

0, otherwise,

whereas the (i′, i)-entry of δe(X, s)ν`−1∂

∆e(X,s)` is

sν1Xe , if i = i′ and ν /∈ i,(−1)u+w+1si′u1Xe , if i\i′ = {iw} = {ν} and i′\i = {i′u}, and

0, otherwise.

3.3 The main theorem 23

Overall, we see that the (i′, i)-entry of ∂∆e(X,s)`+1 δe(X, s)ν

` + δe(X, s)ν`−1∂

∆e(X,s)` is

sν1Xe if i = i′ and 0 otherwise. This is what we wanted to show.

We will only consider ∆e(X, s) for complexes X that are homologically S-torsion. It is an important fact that ∆e(X, s) is then also homologically S-torsion.

Proposition 3.8. Let X ∈ CRe (f,P|S-tor) and let s = (s1, . . . , sd) ∈ S1×· · ·×Sd.

Then the complex ∆e(X, s) is in CR[e,e−d](f,P|S-tor).

Proof: The definition clearly implies that ∆e(X, s) is concentrated in degreese, . . . , e − d and consists of finitely generated projective modules. Accordingto Theorem 3.6(i), the homology modules of ∆e(X, s) are annihilated by theideal 〈s1, . . . , sd〉. In particular, these homology modules must be Sν-torsion forν = 1, . . . , d.

3.3 The main theorem

Throughout this section, d denotes a positive integer and S = (S1, . . . , Sd)denotes a d-tuple of multiplicative systems of R. Furthermore, X denotesa complex in CR

e (f,P|S-tor) for some integer e > d. Finally, α denotes anS-contraction of X with weight s ∈ S1 × · · · × Sd.

Proving the main theorem requires, within the Grothendieck group GRe (f,P|S-tor),

transforming a complex X ∈ CR¤(f,P|S-tor) into one or more complexes that are

concentrated in fewer degrees than X. We first present the idea behind the proof.

Definition 3.9. Let φe(X, α) denote the family (φe(X, α)`)`∈Z of homomor-phisms φe(X,α)` : X` → ∆e(X, s)` =

∐i∈Υ(e−`) Xe given by the fact that their

i’th entries for i ∈ Υ(e− `) are

φe(X, α)i` = α

ie−`

e−1αie−`−1

e−2 · · ·αi1` .

For ` = e, this means that φe(X, α)e = 1Xe , and for ` /∈ {e, . . . , e− d}, it meansthat φe(X, α)` = 0.

Proposition 3.10. φe(X,α) : X → ∆e(X, s) is a morphism of complexes.

Proof: Let ∆def= ∆e(X, s) and φ

def= φe(X, α). To prove that φ is a morphism,

we need to show that φ`−1∂X` = ∂∆

` φ` for all ` ∈ Z: that is, we need to verify that

the j’th entry, αje−`+1

e−1 · · ·αj1`−1∂

X` , of the left side equals the j’th entry of ∂∆

` φ` foreach j ∈ Υ(e−`+1). Since the (j, i)-entry of ∂∆

` is (−1)u+1sju1Xe whenever i is asubset of j with j\i = {ju}, that is, whenever i = {j1, . . . , ju−1, ju+1, . . . , je−`+1}for some u ∈ {1, . . . , e− ` + 1}, we see that the j’th coordinate of ∂∆

` φ` must be

e−`+1∑u=1

(−1)u+1sjuαje−`+1

e−1 · · ·αju+1

`+u−1αju−1

`+u−2 · · ·αj1` .


So overall, we need to show that

e−`+1∑u=1


e−1 · · ·αju+1

`+u−1αju−1

`+u−2 · · ·αj1` = α

je−`+1

e−1 · · ·αj1`−1∂

X` (3.2)

for all j ∈ Υ(e− ` + 1). We do this by descending induction on `.When ` > e, the equation clearly holds since both sides are trivial, and in the

case that ` = e, (3.2) states that sj11Xe = αj1e−1∂

Xe , which is satisfied since α is an

S-contraction of X with weight s. Suppose now that ` < e is arbitrarily chosenand that (3.2) holds for larger values of `. We then have

αje−`+1

e−1 · · ·αj1`−1∂

X` = α

je−`+1

e−1 · · ·αj2` (sj11X`

− ∂X`+1α

j1` )

= sj1αje−`+1

e−1 · · ·αj2`

− (e−`+1∑u=2

(−1)usjuαje−`+1

e−1 · · ·αju+1

`+u−1αju−1

`+u−2 · · ·αj2`+1)α

j1`

=e−`+1∑u=1


e−1 · · ·αju+1

`+u−1αju−1

`+u−2 · · ·αj1` .

Here the second equality follows from the induction hypothesis. This proves (3.2)by induction, so φ is a morphism of complexes.

Definition 3.11. The mapping cone M(φe(X,α)) of φe(X, α) is denoted byMe(X,α).

Letting ∆def= ∆e(X, s) and φ

def= φe(X,α), we see thatMe(X,α) is the complex

0 // Xe

(φe

−∂Xe

)

//∆e

⊕Xe−1

(∂∆

e φe−1

0 −∂Xe−1

)

//∆e−1

⊕Xe−2

// · · ·

· · · //∆e−d

⊕Xe−d−1

( 0 −∂Xe−d−1 )

// Xe−d−2

−∂Xe−d−2 // · · · // X0

// 0

concentrated in degrees e + 1, . . . , 1.Since X and ∆e(X, s) are equipped with S-contractions, Theorem 3.5 provides

an S-contraction of Me(X, α).

Definition 3.12. The S-contraction δe(X, s) ∗ α of Me(X,α) is denoted byµe(X,α).


Letting ∆def= ∆e(X, s), φ

def= φe(X,α) and δ

def= δe(X, s), we see that µe(X, α)

is given by

µe(X, α)ν` =

sνδν` δν

` φ`αν`−1

0 −sναν`−1

:

∆`

⊕X`−1

→∆`+1

⊕X`

for each ` ∈ Z and ν ∈ {1, . . . , d}. The weight of µe(X, α) is s2 = (s21, . . . , s

2d).

Proposition 3.13. Me(X, α) is an object of CR[e+1,1](f,P|S-tor).

Proof: Me(X,α) is clearly concentrated in degrees e, . . . , 1 and composed offinitely generated projective modules. To see that Me(X, α) is homologicallyS-torsion, recall that the short exact sequence

0 → ∆e(X, s) →Me(X, α) → ΣX → 0 (3.3)

from Theorem 0.2 induces the long exact sequence

· · · → H`(∆e(X, s)) → H`(Me(X,α)) → H`(ΣX) → · · ·on homology. Localization preserves exactness, so by localizing at Sν for ν =1, . . . , d it follows that, since ∆e(X, s) as well as ΣX are homologically S-torsion,Me(X, α) must be homologically S-torsion as well.

The exact sequence in (3.3) can be shifted one degree to the right, therebyyielding the exact sequence

0 → Σ−1∆e(X, s) → Σ−1Me(X, α) → X → 0

in CRe (f,P|S-tor), from which it follows that

[X] = [Σ−1Me(X,α)]− [Σ−1∆e(X, s)]

holds in GRe (f,P|S-tor). We are trying to represent X by smaller complexes,

but Σ−1Me(X,α) is not in any way smaller than X. However, as shown below,Σ−1Me(X, α) can be transformed into something that is smaller.

Definition 3.14. Let ∂Ne−1 denote the homomorphism

∂Ne−1 =

−φe(X, α)e−1

∂Xe−1

: Xe−1 −→

∆e(X, s)e−1

⊕Xe−2

= Me(X,α)e−1,

and let Ne(X,α) denote the complex

0 // Xe−1

∂Ne−1 // Me(X,α)e−1

−∂Me(X,α)e−1 // Me(X, α)e−2

−∂Me(X,α)e−2 // · · · // Me(X, α)1

// 0

concentrated in degrees e− 1, . . . , 0.


(One verifies easily that Ne(X, α) indeed is a complex.)

Proposition 3.15. Ne(X, α) is an object of CRe−1(f,P|S-tor).

Proof: Ne(X, α) is clearly composed of finitely generated projective modules.The fact that Ne(X,α) is homologically S-torsion follows from Proposition 3.13and Theorem 3.16 below, from which it follows that Ne(X, α) is homologicallyisomorphic to Σ−1Me(X,α).

Ne(X,α) is Me(X, α) without the segment 0 → Xe → Xe → 0 and shiftedone degree to the right. The good thing about Ne(X, α) is that it is smaller thanMe(X,α), so we hope that we in some way at the level of Grothendieck groupscan represent Me(X,α) by Ne(X,α). This is achieved in the theorem below.

Theorem 3.16. Let B denote the exact complex 0 → Xe → Xe → 0 concentratedin degrees e and e− 1. There is then an exact sequence

0 → B → Σ−1Me(X,α) → Ne(X,α) → 0.

Proof: Let ∆def= ∆e(X, s) and φ

def= φe(X, α), and recall that ∆e = Xe and

φe = 1Xe . The situation is as follows.

0 // B // Σ−1Me(X,α) // Ne(X,α) // 0

‖ ‖ ‖

degree 0

²²

0

²²

0

²²e 0 // Xe

−1Xe //

1Xe

²²

Xe(−φe

∂Xe

)

²²

// 0 //

²²

0

e−1 0 // Xe

(1Xe

−∂Xe

)

//

²²

∆e

⊕Xe−1

(−∂∆e −φe−1

0 ∂Xe−1

)

²²

( ∂Xe 1Xe−1 )

// Xe−1//

(−φe−1

∂Xe−1

)

²²

0

e−2 0 // 0

²²

//∆e−1

⊕Xe−2

²²

1 //∆e−1

⊕Xe−2

²²

// 0

......

......


It is straightforward to verify that the diagram commutes and that all the rowsare exact.

The morphism Σ−1Me(X, α) → Ne(X, α) from Theorem 3.16 is clearly inthe form described in Theorem 3.4, so we are able to induce an S-contraction onNe(X,α) with weight s2 from the S-contraction Σ−1µe(X,α) on Σ−1Me(X, α).

Definition 3.17. The S-contraction on Ne(X, α) induced in the sense of The-orem 3.4 from Σ−1µe(X, α) through the morphism Σ−1Me(X, α) → Ne(X, α)from Theorem 3.16 is denoted by ηe(X, α).

Letting ∆def= ∆e(X, s), φ

def= φe(X, α) and δ

def= δe(X, s), ηe(X,α) from the

above definition is given by

ηe(X, α)ν` =

−sνδ

ν`+1 −δ`+1φ`+1α

ν`

0 sναν`

:

∆e(X, s)`+1

⊕X`

→∆e(X, s)`+2

⊕X`+1

whenever ` = e− 3, . . . , 0, and, as verified by a small calculation, by

ηe(X,α)νe−2 =

(−sν∂Xe δν

e−1 ανe−2∂

Xe−1α

νe−2

):∆e−1

⊕Xe−2

→ Xe−1

whenever ` = e− 2.

From Theorem 3.16 it follows that

[Σ−1Me(X, α)] = [B] + [Ne(X, α)] = [Ne(X, α)]

in GRe (f,P|S-tor), so together with the previous work, we have now succeeded in

transforming our complex X into something smaller at the level of Grothendieckgroups:

[X] = [Σ−1Me(X,α)]− [Σ−1∆e(X, s)] = [Ne(X, α)]− [Σ−1∆e(X, s)].

Although this is an equation in GRe (f,P|S-tor), the complexes involved in the right

end are both concentrated in fewer than e degrees. This, at least, gives us anidea of how to construct the inverse of ι.

Definition 3.18. By we(X, α) we denote the element

we(X, α) = [Ne(X, α)]− [Σ−1∆e(X, s)]

in GRe−1(f,P|S-tor).


The remainder of this section is devoted to showing that we(X, α) is indepen-dent of the choice of α such that we can simply write we(X); that the mapwe : I(CR

e (f,P|S-tor)) → GRe−1(f,P|S-tor) induces a map ωe : GR

e (f,P|S-tor) →GR

e−1(f,P|S-tor); and that the ωe’s for different e’s can be combined to form aninverse of ι.

We begin with a collection of useful lemmas.

Lemma 3.19. If

0 −→ Yψ−→ Y

ψ−→ Y −→ 0

is an exact sequence in CRe (f,P|S-tor), and if β, β and β are S-contractions of Y ,

Y and Y , respectively, compatible with the morphisms in the above exact sequence(and thereby all having the same weight t), then there are exact sequences

0 → ∆e(Y , t) → ∆e(Y, t) → ∆e(Y , t) → 0, (3.4)

0 →Me(Y , β) →Me(Y, β) →Me(Y , β) → 0 and (3.5)

0 → Ne(Y , β) → Ne(Y, β) → Ne(Y , β) → 0, (3.6)

proving that we(Y, β) = we(Y , β)+we(Y , β) in GRe−1(f,P|S-tor). Furthermore the

S-contractions δe(Y , t), δe(Y, t) and δe(Y , t) are compatible with the morphismsin (3.4); the S-contractions µe(Y , β), µe(Y, β) and µe(Y , β) are compatible withthe morphisms in (3.5); and the S-contractions ηe(Y , β), ηe(Y, β) and ηe(Y , β)are compatible with the morphisms in (3.6).

Proof: According to the assumption, there is an exact sequence of modules

0 −→ Yeψe−→ Ye

ψe−→ Ye −→ 0,

which immediately induces the exact sequence in (3.4), because ψe and ψe clearlycommute with each entry of the differentials in ∆e(Y , t), ∆e(Y, t) and ∆e(Y , t).Since ψe and ψe also commute with each entry of the the S-contractions δe(Y , t),δe(Y, t) and δe(Y , t), these must be compatible with the morphisms in the se-quence. In addition, the compatibility of the S-contractions β, β and β withthe morphisms ψ and ψ means that ψeφe(Y , β)i

` = φe(Y, β)i`ψ` and ψeφe(Y, β)i

` =φe(Y , β)i

`ψ` for each ` ∈ Z and i ∈ Υ(e−`), and hence that there is a commutativediagram with exact rows:

0 // Y //

φe(Y ,β)²²

Y //

φe(Y,β)

²²

Y //

φe(Y ,β)²²

0

0 // ∆e(Y , s) // ∆e(Y, s) // ∆e(Y , s) // 0

(3.7)

From Theorem 0.3 now follows the existence of the exact sequence in (3.5),and straightforward calculation easily verifies that the compatibility of the S-contractions β, β and β with the morphisms ψ and ψ, the compatibility of the


S-contractions δe(Y , t), δe(Y, t) and δe(Y , t) with the morphisms in (3.4) and thecommutativity of diagram (3.7) imply that the S-contractions µe(Y , β), µe(Y, β)and µe(Y , β) are compatible with the morphisms in (3.5).

We now claim that the exact sequence in (3.5) induces the exact sequencein (3.6). To see this, let B, B and B denote the complexes 0 → Ye → Ye → 0,0 → Ye → Ye → 0 and 0 → Ye → Ye → 0 concentrated in degrees e and e−1 fromTheorem 3.16. These three complexes come together in a short exact sequence0 → B → B → B → 0, induced by the short exact sequence 0 → Ye → Ye →Ye → 0. We claim that there is a commutative diagram

0

²²

0

²²

0

²²0 // B

²²

// B

²²

// B

²²

// 0

0 // Σ−1Me(Y , β)

²²

// Σ−1Me(Y, β)

²²

// Σ−1Me(Y , β)

²²

// 0

0 // Ne(Y , β)

²²

// Ne(Y, β)

²²

// Ne(Y , β)

²²

// 0

0 0 0

The columns are exact according to Theorem 3.16 and the top rectangles arereadily verified to be commutative. A little diagram chase now shows that wecan use the morphisms in the middle row to induce the morphisms in the bot-tom row, making the entire diagram commutative by construction. As we haveseen, the two top rows are exact, so the exactness of the bottom row follows fromthe 9-lemma (see, for example, [Eis95, exercise A3.12]) applied in each degree.This establishes the exact sequence in (3.6). Once again, straightforward calcu-lation demonstrates that the S-contractions ηe(Y , β), ηe(Y, β) and ηe(Y , β) arecompatible with the morphisms in (3.6).

From (3.4) and (3.6), we now obtain that

we(Y, β) = [Ne(Y, β)]− [Σ−1∆e(Y, t)]

= [Ne(Y , β)] + [Ne(Y , β)]− [Σ−1∆e(Y , t)]− [Σ−1∆e(Y , t)]

= we(Y , β) + we(Y , β),

and the proof is complete.

Lemma 3.20. If X is exact, then we(X, α) = 0 in GRe−1(f,P|S-tor).


Proof: Let ∂Xe−1 denote the inclusion map im ∂X

e−1 ↪→ Xe−2, and let X denotethe complex

0 −→ im ∂Xe−1

∂Xe−1−→ Xe−2

∂Xe−2−→ Xe−3 −→ · · · −→ X1

∂X1−→ X0 −→ 0

concentrated in degrees e− 1, . . . 0. X is X with the module in degree e removedand the module in degree e− 1 replaced by im ∂X

e−1. Since X is exact, X is exact

and it follows from Theorem 0.5 that im ∂Xe−1 is projective, and hence that X is

a complex in CRe−1(f,P|S-tor).

Letting B denote the exact complex 0 → Xe → Xe → 0 from Theorem 3.16,there is an exact sequence

0 // B // X // X // 0

‖ ‖ ‖

degree 0

²²

0

²²

0

²²e 0 // Xe

1Xe //

1Xe

²²

Xe//

∂Xe

²²

0 //

²²

0

e−1 0 // Xe

∂Xe−1 //

²²

Xe−1

∂Xe−1 //

∂Xe−1

²²

im ∂Xe−1

//

∂Xe−1

²²

0

e−2 0 // 0 //

²²

Xe−2

1Xe−2 //

²²

Xe−2//

²²

0

......

......

and we claim that there is a commutative diagram

0

²²

0

²²

0

²²0 // 0 //

²²

B //

²²

B //

²²

0

0 // Σ−1∆e(X, s) //

²²

Σ−1Me(X, α) //

²²

X //

²²

0

0 // Σ−1∆e(X, s) //

²²

Ne(X, α) //

²²

X //

²²

0

0 0 0


The columns are exact (the middle one according to Theorem 3.16) and the toprectangles are readily verified to be commutative. A little diagram chase showsthat we can use the morphisms in the middle row to induce the morphisms in thebottom row, so that the entire diagram is commutative by construction. Now,the two top rows are exact, so the exactness of the bottom row follows from the9-lemma (see, for example, [Eis95, exercise A3.12]) applied in each degree. Thus,we have constructed an exact sequence

0 → Σ−1∆e(X, s) → Ne(X,α) → X → 0 (3.8)

of complexes in CRe−1(f,P|S-tor). Since X is exact, it follows that

we(X,α) = [Ne(X,α)]− [Σ−1∆e(X, s)] = [X] = 0

in GRe−1(f,P|S-tor) as desired.

In the next lemma and the theorem that follows, we shall work with a numberof similar Koszul complexes. Let us therefore introduce some convenient notation.

Definition 3.21. For r ∈ S1, let ∆(r)def= ∆e(X, (r, s2, . . . , sd)); hence, in partic-

ular, ∆(s1) = ∆e(X, s).

Lemma 3.22. Suppose r, r′ ∈ S1, and define homomorphisms

π(r, r′)` : ∆(rr′)` → ∆(r)` and ξ(r, r′) : ∆(r)` → ∆(rr′)`

for each ` ∈ Z by the fact that their (i′, i)-entries for i, i′ ∈ Υ(e− `) are

(π(r, r′)`)i′,i =

0, if i 6= i′,1Xe , if i = i′ and 1 ∈ i,

r′1Xe , if i = i′ and 1 /∈ i,

and

(ξ(r, r′)`)i′,i =

0, if i 6= i′,r′1Xe , if i = i′ and 1 ∈ i,1Xe , if i = i′ and 1 /∈ i.

Then π(r, r′) = (π(r, r′)`)`∈Z is a morphism of complexes ∆(rr′) → ∆(r) andξ(r, r′) = (ξ(r, r′)`)`∈Z is a morphism of complexes ∆(r) → ∆(rr′).

Proof: Assume that i ∈ Υ(e − `) and j ∈ Υ(e − ` + 1). A direct calculation

then shows that the (j, i)-entries of ∂∆(r)` π(r, r′)` and π(r, r′)`−1∂

∆(rr′)` are both

given by

0, if j + i,(−1)u+1sju1Xe , if j\i = {ju} and 1 ∈ i,

(−1)u+1sjur′1Xe , if j\i = {ju} 6= {1} and 1 /∈ i, and

rr′1Xe , if j\i = {ju} = {1} and 1 /∈ i.


This proves that π(r, r′) is a morphism of complexes.

Similarly, a direct calculation shows that the (j, i)-entries of ∂∆(rr′)` ξ(r, r′)`

and ξ(r, r′)`−1∂∆(r)` are both given by

0, if j + i,(−1)u+1sjur

′1Xe , if j\i = {ju} and 1 ∈ i,(−1)u+1sju1Xe , if j\i = {ju} 6= {1} and 1 /∈ i, and

rr′1Xe , if j\i = {ju} = {1} and 1 /∈ i.

This proves that ξ(r, r′) is a morphism of complexes.

We are now ready to take the first step in proving that we(X, α) is independentof α.

Theorem 3.23. Suppose that t = (t1, . . . , td) ∈ S1 × · · · × Sd and consider theS-contraction tα = (t1α

1, . . . , tdαd) of X with weight st = (s1t1, . . . , sdtd). Then

we(X, tα) = we(X,α) in GRe−1(f,P|S-tor).

Proof: If only we can show the equation in the case where tν = 1 for all butone of the ν’s, then the equation follows since

tα = (t1, . . . , td)α = (t1, 1 . . . , 1) · · · (1, . . . , 1, td)α.

We will therefore assume that t = (t1, 1, . . . , 1); the other cases follow similarly(since we can permute the Sν ’s).

To show the desired equation, it suffices to prove that the following equationshold in GR

e−1(f,P|S-tor).

[Σ−1∆(s1t1)] = [Σ−1∆(s1)] + [Σ−1∆(t1)]. (3.9)

[Ne(X, tα)] = [Ne(X, α)] + [Σ−1∆(t1)]. (3.10)

Since ∆(1) is exact according to Theorem 3.6(i), the first equation follows if wecan show that there is an exact sequence

0 // ∆(s1)

(π(1,s1)ξ(s1,t1)

)

//∆(1)⊕

∆(s1t1)

(−ξ(1,t1) π(t1,s1) )// ∆(t1) // 0.

The two matrices clearly define morphisms of complexes, since π(r, u) and ξ(r, u)are morphisms of complexes for r, u ∈ S1 according to Lemma 3.22. Exactnessat ∆(s1) and ∆(t1) is clear since there is always one identity map involved ineither of π(r, u) and ξ(r, u) for r, u ∈ S1. Furthermore, ξ(1, t1)π(1, s1) as well


as π(t1, s1)ξ(s1, t1) are defined in degree ` by the fact that their (i, i′)-entries fori, i′ ∈ Υ(e− `) are

0, if i 6= i′,t11Xe , if i = i′ and 1 ∈ i, ands11Xe , if i = i′ and 1 /∈ i.

To show the exactness of the sequence above, it therefore only remains to showthat, for each ` ∈ Z, the kernel in degree ` of the second morphism is contained inthe image in degree ` of the first. Since all (i, i′)-entries of the maps involved aretrivial except when i = i′, it suffices to consider an element (x, y) in the i-entry∆(1)i

` ⊕∆(s1t1)i` of the `’th module of ∆(1)⊕∆(s1t1). So suppose that such an

element is in the kernel of the map in degree ` of the second morphism. If 1 ∈ i,this means that t1x = y, and in this case (x, y) is the image of x under the mapin degree ` of the first morphism. If 1 /∈ i, it means that x = s1y, and in thiscase (x, y) is the image of y under the map in degree ` of the first morphism. Inall cases, (x, y) is in the image of the map in degree ` of the first morphism, andhence the sequence is exact and equation (3.9) has been proven.

Moving on to equation (3.10), we first define for each ` ∈ Z a homomorphismγ`−1 : X`−1 → ∆(1)` by letting its i’th entry for i ∈ Υ(e− `) be

γi`−1 =

{0, if 1 ∈ i,

αie−`

e−1 · · ·αi1` α1

`−1, if 1 /∈ i.

Another way of writing this is

γ`−1 =∐

i∈Υ(e−`)1/∈i

αie−`

e−1 · · ·αi1` α1

`−1.

We now claim that there are morphisms

Φ: Me(X,α) −→∆(1)⊕

Me(X, tα)and Ψ:

∆(1)⊕

Me(X, tα)−→ ∆(t1)

given in degree ` by

Φ` =

π(1, s1)` γ`−1

ξ(s1, t1)` 0

0 1X`−1

:∆(s1)`

⊕X`−1

−→

∆(1)`

⊕∆(s1t1)`

⊕X`−1

and

Ψ` =(−ξ(1, t1)` π(t1, s1)` ξ(1, t1)`γ`−1

):

∆(1)`

⊕∆(s1t1)`

⊕X`−1

−→ ∆(t1)`.


Proving that Φ and Ψ indeed are morphisms of complexes means proving that

∂∆(1)`+1 0 0

0 ∂∆(s1t1)`+1 φe(X, tα)`

0 0 −∂X`

Φ`+1 = Φ`

(∂

∆(s1)`+1 φe(X,α)`

0 −∂X`

)

and

∂∆(t1)`+1 Ψ`+1 = Ψ`

∂∆(1)`+1 0 0

0 ∂∆(s1t1)`+1 φe(X, tα)`

0 0 −∂X`

for all ` ∈ Z. Since we already know from Lemma 3.22 that π(r, u) and ξ(r, u)are morphisms for r, u ∈ S1, proving the above equations comes down to showingthat the following hold for all ` ∈ Z.

π(1, s1)`φe(X, α)` = ∂∆(1)`+1 γ` + γ`−1∂

X` . (3.11)

φe(X, tα)` = ξ(s1, t1)`φe(X,α)`. (3.12)

π(t1, s1)`φe(X, tα)` = ξ(1, t1)`γ`−1∂X` + ∂

∆(t1)`+1 ξ(1, t1)`+1γ`. (3.13)

We verify (3.11) by brute force, calculating on the right hand side of the equation:

∂∆(1)`+1 γ` + γ`−1∂

X` = ∂

∆(1)`+1

∐

j∈Υ(e−`−1)1/∈j

αje−`−1

e−1 · · ·αj1`+1α

1`

+∐

i∈Υ(e−`)1/∈i

αie−`

e−1 · · ·αi1` α1

`−1∂X`

=∐

i∈Υ(e−`)1∈i

αie−`

e−1 · · ·αi2`+1α

1`

+∐

i∈Υ(e−`)1/∈i

(e−∑u=1

(−1)u+1siuαie−`

e−1 · · ·αiu+1

`+u αiu−1

`+u−1 · · ·αi1`+1α

1`)

+∐

i∈Υ(e−`)1/∈i

αie−`

e−1 · · ·αi1` α1

`−1∂X`

=∐

i∈Υ(e−`)1∈i

αie−`

e−1 · · ·αi1`

+∐

i∈Υ(e−`)1/∈i

αie−`

e−1 · · ·αi1` (∂X

`+1α1` + α1

`−1∂X` )

=∐

i∈Υ(e−`)1∈i

αie−`

e−1 · · ·αi1` +

∐

i∈Υ(e−`)1/∈i

s1αie−`

e−1 · · ·αi1`

= π(1, s1)`φe(X,α)`.


Here, the third equality follows from (3.2) in Proposition 3.10, and the fourthequality follows from α being an S-contraction with weight s = (s1, . . . , sd). Thisproves the equation in (3.11). The equation in (3.12) is clear, since

ξ(s1, t1)`φe(X, α)` =∐

i∈Υ(e−`)1∈i

t1φe(X,α)i` +

∐

i∈Υ(e−`)1/∈i

φe(X,α)i` = φe(X, tα).

To prove that the equation in (3.13) holds, we apply (3.11) to the right side of(3.13):

ξ(1, t1)`γ`−1∂X` + ∂

∆(t1)`+1 ξ(1, t1)`+1γ`

= ξ(1, t1)`(γ`−1∂X` + ∂

∆(1)`+1 γ`)

= ξ(1, t1)`π(1, s1)`φe(X,α)`.

In contrast, applying (3.12) to the left side of (3.13) yields

π(t1, s1)`φe(X, tα)` = π(t1, s1)`ξ(s1, t1)`φe(X, α)`,

so proving equation (3.13) merely requires showing that

ξ(1, t1)`π(1, s1)` = π(t1, s1)`ξ(s1, t1)`. (3.14)

This, however, follows since, for i, i′ ∈ Υ(e − `), both sides of (3.14) have (i, i′)-entries given by

0, if i 6= i′,s11Xe , if i = i′ and 1 /∈ i, andt11Xe , if i = i′ and 1 ∈ i.

Thus we have verified equation (3.13), and we conclude that Φ and Ψ are mor-phisms of complexes.

We now claim that there is a short exact sequence

0 −→Me(X, α)Φ−→

∆(1)⊕

Me(X, tα)

Ψ−→ ∆(t1) −→ 0. (3.15)

To see that the sequence is exact at Me(X,α), suppose that, for some ` ∈ Z, theelement (x, y) ∈ ∆(s1)` ⊕X`−1 = Me(X,α)` maps to 0 under Φ`: that is,

0 =

π(1, s1)` γ`−1

ξ(s1, t1)` 00 1X`−1

(xy

)=

π(1, s1)`(x) + γ`−1(y)ξ(s1, t1)`(x)

y

.

It immediately follows that y = 0, and we are left with the equations π(1, s1)`(x) =ξ(s1, t1)`(x) = 0 which imply that x = 0. Thus, Φ` is injective and (3.15) is exactat Me(X,α).


To see that the sequence is exact at ∆(t1), suppose that x ∈ ∆(t1)i` for some

` ∈ Z and i ∈ Υ(e− `). Then, if 1 ∈ i,

(−ξ(1, t1)` π(t1, s1)` ξ(1, t1)`γ`−1

)

0x0

= x,

and if 1 /∈ i,

(−ξ(1, t1)` π(t1, s1)` ξ(1, t1)`γ`−1

)−x00

= x.

In either case, x is in the image of Ψ`, and we conclude that Ψ` is surjective andthat (3.15) is exact at ∆(t1).

Equation (3.14) clearly shows that ΨΦ = 0, so to show the exactness of (3.15),it only remains verify that the kernel of Ψ` is contained in the image of Φ` for all` ∈ Z. So suppose that (x, y, z) ∈ ∆(1)`⊕∆(s1t1)`⊕X`−1 = (∆(1)⊕Me(X, tα))`

maps to 0 under Ψ`: that is,

−ξ(1, t1)`(x) + π(t1, s1)`(y) + ξ(1, t1)`γ`−1(z) = 0.

Here x = (xi)i∈Υ(e−`) and y = (yi)i∈Υ(e−`) are Υ(e − `)-tuples, so the aboveequation states that, for i ∈ Υ(e− `),

−t1xi + yi = 0, if 1 ∈ i, and

−xi + s1yi + γi`−1(z) = 0, if 1 /∈ i.

Now let w = (wi)i∈Υ(e−`) ∈ ∆(s1)` be defined by wi = xi whenever 1 ∈ i andwi = yi whenever 1 /∈ i. Then

π(1, s1)` γ`−1

ξ(s1, t1)` 00 1X`−1

(wz

)=

π(1, s1)`(w) + γ`−1(z)ξ(s1, t1)`(w)

z

=

∑

i∈Υ(e−`)1∈i

xi +∑

i∈Υ(e−`)1/∈i

(s1yi + γi`−1(z))

∑

i∈Υ(e−`)1∈i

t1xi +∑

i∈Υ(e−`)1/∈i

yi

z

=

xyz

.


This proves that (x, y, z) is in the image of Φ`. We have now proved that (3.15)is exact.

Denoting by B the exact complex 0 → Xe → Xe → 0, we now claim thatthere is a commutative diagram

0

²²

0

²²

0

²²0 // B //

²²

B //

²²

0 //

²²

0

0 // Σ−1Me(X,α) //

²²

Σ−1∆(1)⊕

Σ−1Me(X, tα)

//

²²

Σ−1∆(t1) //

²²

0

0 // Ne(X, α) //

²²

Σ−1∆(1)⊕

Ne(X, tα)

//

²²

Σ−1∆(t1) //

²²

0

0 0 0

The columns are exact according to Theorem 3.16 and the top rectangles arereadily verified to be commutative. A little diagram chase shows that we canuse the morphisms in the middle row to induce the morphisms in the bottomrow, so that the entire diagram is commutative by construction. Now, the toprow is clearly exact, and we have just seen that the middle row is exact, so theexactness of the bottom row follows from the 9-lemma (see, for example, [Eis95,exercise A3.12]) applied in each degree. Thus we have constructed an exactsequence

0 −→ Ne(X,α) −→Σ−1∆(1)

⊕Ne(X, tα)

−→ Σ−1∆(t1) −→ 0.

in GRe−1(f,P,S-tor), and since ∆(1) is exact according to Theorem 3.6(i), equa-

tion (3.10) follows. This proves the theorem.

We can now take the final step in proving that we(X,α) is independent of thechoice of α. First a lemma.

Lemma 3.24. If Y is an exact complex in CRe+1(f,P|S-tor) and β is an S-

contraction of Y with weight t, then we(Ne+1(Y, β), ηe+1(Y, β)) ∈ GRe−1(f,P|S-tor)

does not depend on the choice of β (but still depends on the weight t).


Proof: Let us consider the complex Y ∈ CRe (f,P|S-tor), constructed from Y

in the way X was constructed from X in Lemma 3.20, and equip Y with theS-contraction β induced from β in the sense of Theorem 3.4:

0 // im ∂Ye

∂Ye // Ye−1

∂Ye−1 //

∂Ye βν

e−1

oo · · · //

βνe−2

oo Y1

∂Y1 //

oo Y0//

βν0

oo 0.

Recall that there is an S-contraction δe+1(Y, t) of ∆e+1(Y, t) with weight tand an S-contraction µe+1(Y, β) of Me+1(Y, β) with weight t2. According toTheorem 3.5, the S-contractions Σ−1tδe+1(Y, t), Σ−1µe+1(Y, β) and tβ are com-patible with the morphisms in the short exact sequence

0 → Σ−1∆e+1(Y, t) → Σ−1Me+1(Y, β) → Y → 0. (3.16)

Now, the S-contraction ηe+1(Y, β) on Ne+1(Y, β) is induced in the sense of The-orem 3.4 by the S-contraction Σ−1µe+1(Y, β) on Σ−1Me+1(Y, β) through themorphism Σ−1Me+1(Y, β) → Ne+1(Y, β); similarly, as described above, the S-contraction β on Y is induced in the sense of Theorem 3.4 by the S-contractionβ on Y through the morphism Y → Y . We claim that this implies that the S-contractions Σ−1tδe+1(Y, t), ηe+1(Y, β) and tβ are compatible with the morphismsin the exact sequence

0 → Σ−1∆e+1(Y, t) → Ne+1(Y, β) → Y → 0

from (3.8) in Lemma 3.20. This is easy: let ∆def= ∆e+1(Y, t), M def

= Me+1(Y, β),

N def= Ne+1(Y, β), δ

def= δe+1(Y, t), µ

def= µe+1(Y, β) and η

def= ηe+1(Y, β). Proving, for

example, that Σ−1tδ and η are compatible with the morphism Σ−1∆ → N meansproving the commutativity of the bottom rectangle of the following diagram forall ` ∈ Z and ν = 1, . . . , d.

M`+1

²²

M`

²²

−µòo

∆`+1

::vvvvvvvvv

1

²²

∆`

<<zzzzzzzz

1

²²

−tδòo

N` N`−1η`−1oo

∆`+1

::vvvvvvvvv∆`

<<zzzzzzzz−tδòo

The top rectangle is commutative since Σ−1tδ and Σ−1µ are compatible with thefirst morphism in (3.16), and the back rectangle is commutative since η is inducedfrom Σ−1µ in the sense of Theorem 3.4. We have constructed the morphism


Σ−1∆ → N by inducing it from Σ−1∆ → Σ−1M via the morphism Σ−1M→N ,so the rectangles on the left and right side must also be commutative. Thus allrectangles except possibly the bottom one are commutative. Since the verticalmaps are all surjective, the bottom rectangle now lifts to the top rectangle, andit follows that the bottom rectangle must be commutative. A similar argumentshows that η and tβ are compatible with the morphism N → Y .

Recalling from Lemma 3.20 that the exactness of Y implies the exactness ofY , we now get, using Lemmas 3.19 and 3.20, that

we(N , η) = we(Σ−1∆, Σ−1tδ) + we(Y , tβ)

= we+1(Σ−1∆, Σ−1tδ),

which does not depend on β (but apparently still depends on t).

Theorem 3.25. The element we(X,α) ∈ GRe−1(f,P|S-tor) does not depend on

the choice of α (nor on the weight s): that is, if β is an S-contraction of X withweight t, then we(X, α) = we(X, β).

Proof: We can assume that the weight s of α equals the weight t of β; for if thisis not the case, we consider instead the S-contractions tα and sβ whose weightsare both st, and we know from Theorem 3.23 that we(X, α) = we(X, tα) andwe(X, sβ) = we(X, β).

Consider the mapping cone M(1X) of the identity morphism 1X : X → Xand the canonical short exact sequence

0 → X →M(1X) → ΣX → 0

from Theorem 0.4. According to Theorem 3.5, the S-contractions sβ, β ∗ α andΣsα all have weight s2 and are compatible with the morphisms in the abovesequence.

Now, the above sequence, which is a sequence in CRe+1(f,P|S-tor), induces by

(3.6) from Lemma 3.19 an exact sequence

0 → Ne+1(X, sβ) → Ne+1(M(1X), β ∗ α) → Ne+1(ΣX, Σsα) → 0

in CRe (f,P|S-tor). According to the same lemma, the S-contractions ηe+1(X, sβ),

ηe+1(M(1X), β ∗α) and ηe+1(ΣX, Σsα), which all have weight s4, are compatiblewith the morphisms in the above sequence.

In the construction of Ne+1(X, sβ) we have considered X as a complex con-centrated in degrees e + 1, . . . , 0. Since Xe+1 is the zero module, ∆e+1(X, s2) isthe zero complex, and therefore Ne+1(X, sβ) = X. Furthermore, it is straightfor-ward to see that ηe+1(X, sβ) is the same as s3β considered as an S-contraction


of X. It now follows from Theorem 3.23 and Lemma 3.19 that

we(X, β) = we(X, s3β)

= we(Ne+1(X, sβ), ηe+1(X, sβ))

= we(Ne+1(M(1X), β ∗ α), ηe+1(M(1X), β ∗ α))

− we(Ne+1(ΣX, Σsα), ηe+1(ΣX, Σsα)).

Since M(1X) is exact, Lemma 3.24 implies that the first term in the abovedifference does not depend on β ∗ α and thereby not on β. The second termdoes not depend on β either, so it follows that the difference depends only on α.Replacing β by α, we therefore find that we(X, α) is equal to the same difference,and hence we(X, α) = we(X, β) as desired.

Definition 3.26. In the light of Theorem 3.25, we shall write we(X) to meanwe(X, α) for any choice of S-contraction α of X.

We have now accomplished the first and hardest task in constructing an in-verse to ι : GR

e−1(f,P|S-tor) → GRe (f,P|S-tor). Our second task is achieved in the

theorem below. First a lemma.

Lemma 3.27. Suppose that

0 −→ Xψ−→ X

ψ−→ X → 0 (3.17)

is an exact sequence in CRe (f,P|S-tor). Then a morphism ρ : Σ−1X → X exists

such that its mapping cone M(ρ) is isomorphic to X.

Proof: The existence of the exact sequence in (3.17) implies the existence of anexact sequence of projective modules in each degree. Such a sequence splits, andhence for each ` ∈ Z we can find homomorphisms κ` : X` → X` and κ : X` → X`

such that ψ`κ` + κ`ψ` = 1X`(from which it easily follows that κ`ψ` = 1X`

andψ`κ` = 1X`

).

Denote the differentials of X, X and X by ∂, ∂ and ∂, respectively, and letfor each ` ∈ Z

ρ` = κ`∂`+1κ`+1 : X`+1 → X`.

We claim that ρ = (ρ`)`∈Z defines a morphism Σ−1X → X. To prove this weneed to show that ∂`ρ` = −ρ`−1∂`+1 for each ` ∈ Z. This means showing that∂`κ`∂`+1κ`+1 = −κ`−1∂`κ`∂`+1, and since ψ`−1 is injective, this is the same asshowing that

ψ`−1∂`κ`∂`+1κ`+1 = −ψ`−1κ`−1∂`κ`∂`+1. (3.18)


The left side of (3.18) is calculated:

ψ`−1∂`κ`∂`+1κ`+1 = ∂`ψ`κ`∂`+1κ`+1 (since ψ is a morphism)

= ∂`(1X`− κ`ψ`)∂`+1κ`+1 (since ψ`κ` + κ`ψ` = 1X`

)

= −∂`κ`ψ`∂`+1κ`+1 (since ∂`∂`+1 = 0)

= −∂`κ`∂`+1ψ`+1κ`+1 (since ψ is a morphism)

= −∂`κ`∂`+1 (since ψ`+1κ`+1 = 1X`+1).

For the right side of (3.18), note that

ψ`−1κ`−1∂`κ`∂`+1 = (1X`−1− κ`−1ψ`−1)∂`κ`∂`+1,

since ψ`−1κ`−1 + κ`−1ψ`−1 = 1X`−1. This again is equal to ∂`κ`∂`+1 because of the

calculation

κ`−1ψ`−1∂`κ`∂`+1 = κ`−1∂`ψ`κ`∂`+1 (since ψ is a morphism)

= κ`−1∂`∂`+1 (since ψ`κ` = 1X`)

= 0 (since ∂`∂`+1 = 0).

This proves that ρ is a morphism of complexes.Now consider the mapping cone M(ρ) of ρ. We see that M(ρ) is the complex

M(ρ) = 0 −→Xe

⊕Xe

(∂e ρe−1

0 ∂e

)

−→Xe−1

⊕Xe−1

−→ · · · −→X0

⊕X0

−→ 0

concentrated in degrees e, . . . , 0, and we claim that there is a morphism λ : X →M(ρ) given in degree ` by

λ` =

(κ`

ψ`

): X` →

X`

⊕X`

= M(ρ)`

for all ` ∈ Z. To see that this in fact defines a morphism of complexes, we needto verify the following equation of matrices.

(∂` ρ`−1

0 ∂`

)(κ`

ψ`

)=

(κ`−1

ψ`−1

)∂`.

Identity at the (2, 1)-entry follows from ψ being a morphism, while identity atthe (1, 1)-entry follows from the following calculation

∂`κ` + ρ`−1ψ` = ∂`κ` + κ`−1∂`κ`ψ` (by definition of ρ)

= ∂`κ` + κ`−1∂`(1X`− ψ`κ`) (since ψ`κ` + κ`ψ` = 1X`

)

= ∂`κ` + κ`−1∂` − κ`−1ψ`−1∂`κ` (since ψ is a morphism)

= κ`−1∂` (since κ`−1ψ`−1 = 1X`−1).


For each ` ∈ Z, λ` must be an isomorphism since it has an inverse map given by

(ψ` κ`

):

X`

⊕X`

→ X`,

and it follows that λ must be an isomorphism of complexes. This proves thelemma.

Theorem 3.28. The map we : CRe (f,P|S-tor) → GR

e−1(f,P|S-tor) induces a grouphomomorphism ωe : GR

e (f,P|S-tor) → GRe−1(f,P|S-tor) defined by ωe([X]) = we(X)

for X ∈ CRe (f,P|S-tor).

Proof: The only thing we need to show is that the relations in GRe (f,P|S-tor)

(as described in Definition 2.4) are preserved under the map we.If X is exact, we already know from Lemma 3.20 that we(X) = 0. Thus, it

only remains to show that if

0 −→ Xψ−→ X

ψ−→ X → 0 (3.19)

is an exact sequence in CRe (f,P|S-tor), then we(X) = we(X) + we(X). According

to Lemma 3.27, the existence of the exact sequence in (3.19) implies the existenceof a morphism ρ : Σ−1X → X with the property that M(ρ) is isomorphic to X.Now choose S-contractions α and α for X and X respectively, and let s ands denote the weights of α and α, respectively. Recall that α ∗ Σ−1α is an S-contraction of M(ρ) with weight ss. We now have

we(X) = we(M(ρ))

= we(M(ρ), α ∗ Σ−1α)

= we(X, sα) + we(X, sα)

= we(X) + we(X),

where the third equality follows from Theorem 3.5 and Lemma 3.19. This provesthe theorem.

We are immediately able to show that our homomorphism ωe in fact is anisomorphism.

Theorem 3.29. The group homomorphism

ιe−1 : GRe−1(f,P|S-tor) → GR

e (f,P|S-tor)

given by ιe−1([X]) = [X] is an isomorphism; in fact, the inverse of ιe−1 is ωe.


Proof: If we shift the canonical exact sequence of the mapping cone Me(X, α)one degree to the right, we get the exact sequence

0 → Σ−1∆e(X, s) → Σ−1Me(X, α) → X → 0 (3.20)

of complexes in CRe (f,P|S-tor). Theorem 3.16 showed that there is an exact se-

quence0 → B → Σ−1Me(X, α) → Ne(X, α) → 0 (3.21)

in CRe (f,P|S-tor), where B is the exact complex 0 → Xe → Xe → 0 concentrated

in degrees e and e − 1. From the exact sequences in (3.20) and (3.21) it nowfollows that the following holds in GR

e (f,P|S-tor).

[X] = [Σ−1Me(X,α)]− [Σ−1∆e(X, s)]

= [Ne(X, α)]− [Σ−1∆e(X, s)]

= ιe−1ωe([X]).

Suppose now that Y is a complex in CRe−1(f,P|S-tor) and that β is an S-

contraction of Y with weight t. Then, considering Y as a complex in CRe (f,P|S-tor),

∆e(Y, t) = 0 and Ne(Y, β) = Y , and therefore

[Y ] = [Ne(Y, β)]− [Σ−1∆e(Y, t)] = ωeιe−1([Y ]).

Thus, ιe−1 and ωe are inverses of each other, and the theorem is proved.

We are now ready to prove the main theorem. To do so, recall that a directedset is a partially ordered set I = (I, 4), satisfying the condition that, for everypair of elements i, j ∈ I, an element k ∈ I exists such that i, j 4 k. A directsystem of Abelian groups is a family (Ai, aij)i4j of Abelian groups Ai and grouphomomorphisms aij : Ai → Aj, indexed by pairs of elements i, j ∈ I with i 4 j,such that aii = 1Ai

for all i ∈ I and ajkaij = aik for all i 4 j 4 k. Such adirect system of Abelian groups has a direct limit, which is an Abelian grouplim−→i∈IAi defined as the quotient of

∐i∈I Ai with the subgroup generated by the

elements m− aij(m) for all m ∈ Ai. The direct limit is uniquely determined (upto isomorphism) by the fact that there are homomorphisms τi : Ai → lim−→i∈IAi

for all i ∈ I with the property that τjaij = τi for all i 4 j and the fact that it isuniversal with respect to this property, in the sense that if A is another Abeliangroup with homomorphisms ai : Ai → A such that ajaij = ai for all i 4 j, aunique homomorphism λ : lim−→i∈IAi → A exists such that ai = λτi for each i ∈ I:

Ai

aij

²²

τi

%%KKKKKKKKKK ai

((lim−→i∈IAi

λ //______ A

Aj

τj

99ssssssssss aj

66


Theorem 3.30 (main theorem, d > 0). The group homomorphism

ι : GRd (f,P|S-tor) → GR

¤(f,P|S-tor)

given by ι([X]) = [X] is an isomorphism.

Proof: The sequence

GRd (f,P|S-tor)

ιd−→ GRd+1(f,P|S-tor)

ιd+1−→ · · ·

of Abelian groups GRf (f,P|S-tor) and homomorphisms ιf for f ≥ d is a direct

system. It is straightforward to see that the Grothendieck group GR]∞,0](f,P|S-tor)

satisfies the universal property required by a direct limit of the above sequence:the homomorphisms τf : GR

f (f,P|S-tor) → GR]∞,0](f,P|S-tor) are the maps [X] 7→

[X], and given a group A and a family (af )d≤f of maps af : GRf (f,P|S-tor) → A

as described previously, the map λ : GR]∞,0](f,P|S-tor) → A is given by λ([X]) =

af ([X]) for f chosen sufficiently large that X ∈ GRf (f,P|S-tor). In contrast, since

all the homomorphisms ιf are isomorphisms according to Theorem 3.29, the directlimit must be isomorphic to each of the groups GR

f (f,P|S-tor) and τf must be anisomorphism for each f ≥ d.

Now, the homomorphism ι∞ : GR]∞,0](f,P|S-tor) → GR

¤(f,P|S-tor) given by

ι∞([X]) = [X] is clearly an isomorphism, the inverse being given by [X] 7→(−1)n[ΣnX] for n chosen sufficiently large that ΣnX ∈ CR

]∞,0](f,P|S-tor). Since

ι is composed of the isomorphisms τd : GRd (f,P|S-tor) → GR

]∞,0](f,P|S-tor) and

ι∞ : GR]∞,0](f,P|S-tor) → GR

¤(f,P|S-tor), it follows that ι is an isomorphism.

Substituting A in the above proof with the group GRd (f,P|S-tor) and the af ’s

with the isomorphisms ωd+1 · · ·ωf (and in particular ad with the identity map)places us in the situation

GRf (f,P|S-tor)

ιf

²²

τf

((QQQQQQQQQQQQ ωd+1···ωf

++GR

]∞,0](f,P|S-tor) λ //______ GRd (f,P|S-tor)

GRf+1(f,P|S-tor)

τf+1

66mmmmmmmmmmmm ωd+1···ωf+1

33

where λ must be an isomorphism given by λ([X]) = ωd+1 · · ·ωf ([X]) for f chosensufficiently large that X ∈ CR

f (f,P|S-tor). The property that the above dia-gram commutes in the case f = d means that λτd is the identity, and hence λis inverse to the homomorphism τd : GR

d (f,P|S-tor) → GR]∞,0](f,P|S-tor). It fol-

lows that the inverse of ι is the map ω : GR¤(f,P|S-tor) → GR

d (f,P|S-tor) given


by ω([X]) = (−1)nωd+1 · · ·ωf ([ΣnX]) for n and f chosen sufficiently large that

ΣnX ∈ CRf (f,P|S-tor).

We still need to prove the main theorem in the case that d = 0 where ι isa homomorphism from GR

0 (f,P) = K0(R) to GR¤(f,P). The only reason that we

have required d to be nonnegative so far is to avoid the consideration of “specialcases”. When d = 0, there simply is no S-contraction; however, the techniqueused previously still works! Given X ∈ CR

e (f,P), the complex ∆e(X) in this caseis concentrated in degree e and identical to the module Xe, and the morphismφe(X) : X → ∆e(X) is given by φe(X)e = 1Xe and φe(X)` = 0 for ` 6= e. Ourmapping cone Me(X) = M(φe(X)) is then the complex

0 −→ Xe

( 1−∂X

e

)

−→Xe

⊕Xe−1

( 0 −∂Xe−1 )−→ Xe−2

−∂Xe−2−→ Xe−3 −→ · · · −→ X0 −→ 0

concentrated in degrees e+1, . . . , 1. Removing the identity Xe → Xe from degreese + 1 and e and shifting the result one degree to the right, we obtain the inducedcomplex

Ne(X) = 0 −→ Xe−1

∂Xe−1−→ Xe−2

∂Xe−2−→ Xe−3 −→ · · · −→ X0 −→ 0,

which is just X without the module in degree e. We now have that

[X] = [Ne(X)]− [Σ−1∆e(X)] = [Ne(X)] + (−1)e[Xe]

in GRe (f,P). Repeating the process on the complex Ne(X), we obtain inductively

that [X] =∑e

`=0(−1)`[X`] in GRe (f,P). The claim is now that the inverse of the

homomorphism ι : GR0 (f,P) → GR

¤(f,P) must be given by taking the alternatingsum of the modules in a complex. We state this in a theorem and provide a proofthat does not require one to look through the last 25 pages to verify that theyindeed make sense in the case d = 0.

Theorem 3.31 (main theorem, d = 0). The group homomorphism

ι : GR0 (f,P) → GR

¤(f,P)

given by ι([M ]) = [M ] is an isomorphism.

Proof: We claim that the inverse of ι is the map A : GR¤(f,P) → GR

0 (f,P) givenby A([Y ]) =

∑`∈Z(−1)`[Y`]. To see that A is well defined, note first that, if

0 → U → V → W → 0 is a short exact sequence in CR¤(f,P), then there is a short

exact sequence of modules in each degree, proving thatA([V ]) = A([U ])+A([W ]).Next, if Y is an exact complex in CR

¤(f,P), then∐

` even Y`∼= ∐

` odd Y` (see, forexample, [Mag02, Corollary 3.47]), proving that A([Y ]) = 0. Thus, A is a well-defined homomorphism.


It is clear that A◦ι is the identity on GR0 (f,P), so it only remains to verify

that ι ◦ A is the identity on GR¤(f,P). Given an arbitrary complex Y in CR

¤(f,P),choose e sufficiently large that Y` = 0 for ` > e. We then have an exact sequence

0

²²

0

²²

0

²²0 // 0 //

²²

Ye1 //

∂Ye

²²

Ye//

²²

0

0 // Ye−11 //

∂Ye−1

²²

Ye−1//

∂Ye−1

²²

0 //

²²

0

0 // Ye−21 //

²²

Ye−2//

²²

0 //

²²

0

......

...

of complexes in CR¤(f,P), proving inductively that

[Y ] =∑

`∈Z[Σ`Y`] =

∑

`∈Z(−1)`[Y`] = ι ◦ A([Y ])

as desired.

3.4 Consequences of the main theorem

Given all the effort to prove the main theorem, we are pleased to present in thissection a collection of results that are established relatively easy as corollaries tothe main theorem.

Definition 3.32. If x is an element of R, we let S(x) denote the multiplicativesystem {xn |n ∈ N0}. If d ∈ N and x = (x1, . . . , xd) is a d-tuple of elements, welet S(x) denote the d-tuple (S(x1), . . . , S(xd)) of multiplicative systems.

Lemma 3.33. Suppose that R is Noetherian and local with maximal ideal m andthat x = (x1, . . . , xd) is a system of parameters (by which d must be equal to thedimension of R). Let M be a finitely generated module. Then lengthR M < ∞ ifand only if M is S(x)-torsion.

Proof: “if”: From the assumption it follows that AnnR M ∩ S(xν) 6= ∅ for ν =1, . . . , d. Thus, we can find N1, . . . , Nd ∈ N0 such that xN1

1 , . . . , xNdd ∈ AnnR M . If

AnnR M ⊆ p, where p is a prime ideal, it therefore follows that x1, . . . , xd ∈ p, andsince dimR R/〈x〉 = 0 this implies that p = m. Consequently SuppR M ⊆ {m},so lengthR M < ∞.

3.4 Consequences of the main theorem 47

“only if”: If M is not S(x)-torsion, then AnnR M ∩ S(xν) = ∅ for someν ∈ {1, . . . , d}. Because of the ascending chain condition on R, AnnR M can beextended to an ideal p maximal with respect to the property of not intersectingS(xν). Such a p is prime (see, for example, [Eis95, Proposition 2.11]), and byconstruction, p ∈ SuppR M . However, p 6= m since xν /∈ p, and it follows thatdimR M ≥ 1 and thereby that lengthR M = ∞.

Corollary 3.34. If R is Noetherian and local with dim R = d, then the group ho-momorphism ι : GR

d (f,P|l) → GR¤(f,P|l) given by ι([X]) = [X] is an isomorphism.

Proof: Letting x = (x1, . . . , xd) be a system of parameters, it follows fromLemma 3.33 that GR

¤(f,P|l) = GR¤(f,P|S(x)-tor) and GR

d (f,P|l) = GRd (f,P|S(x)-tor),

and from Theorem 3.30 we now get the isomorphism

GRd (f,P|l) = GR

d (f,P|S(x)-tor)∼=−→ι

GR¤(f,P|S(x)-tor) = GR

¤(f,P|l).

Corollary 3.34 shows that at the level of Grothendieck groups, the complexesin CR

¤(f,P|l) can be represented by complexes concentrated in degrees dim R, . . . , 0.In some sense, we cannot do better than this: the new intersection theorem(cf. [Rob98, Theorem 13.4.1]) states that, if a complex in CR

¤(f,P|l) is nonexactand concentrated in degrees n, . . . , 0, then n ≥ dim R.

Lemma 3.35. Suppose that R is a local Cohen–Macaulay ring of dimension d >0. Then any module M in CR

0 (l,pd) satisfies the condition that either M = 0 orelse pdR M = d. Furthermore, any complex X in CR

d (f,P|l) satisfies the conditionthat its homology complex H(X) is concentrated in degree 0: that is, H(X) is amodule in CR

0 (l,pd).

Proof: If M is a module in CR0 (l,pd), then M 6= 0 implies depthR M = 0 since

depthR M ≤ dimR M = 0 according to Proposition 0.15 and Theorem 0.10. Wetherefore get pdR M = depth R = dim R = d, where the first equality follows fromthe Auslander–Buchsbaum formula (see Theorem 0.16), and the second equalityfollows from the assumption that R is Cohen–Macaulay.

Now let X be a complex in CRd (f,P|l). The modules in X are free, so for

` = 0, . . . , d we either have X` = 0 and thereby depthR X` = ∞, or X` 6= 0and thereby depthR X` = depth R = dim R = d. If H(X) is not concentrated indegree 0, it therefore follows from the acyclicity lemma (see Lemma 0.17) that1 ≤ depthR H`(X) < ∞ for some ` > 0. However, according to Proposition 0.15and Theorem 0.10, we also have depthR H`(X) ≤ dimR H`(X) = 0, since X hashomologies of finite length. This is a contradiction, so X must be concentratedin degree 0.

Lemma 3.35 shows that we are in a position where we can apply Theorem 2.11to obtain a homomorphism H : GR

d (f,P|l) → GR0 (l,pd) given by H([X]) = [H(X)].


Further, the requirements of Theorem 2.12 are satisfied, so there is a homomor-phism R : GR

0 (l,pd) → GR¤(f,P|l) given by R([M ]) = [X], where X ∈ CR

¤(f,P|l) isa projective resolution of M . But GR

d (f,P|l) is isomorphic to GR¤(f,P|l) according

to Corollary 3.34, and the following diagram commutes.

GRd (f,P|l) ι

∼=//

H²²

GR¤(f,P|l)

GR0 (l,pd)

R

88qqqqqqqqqq

As one could hope for, it turns out that H and R are isomorphisms too.

Corollary 3.36. If R is a local Cohen-Macaulay ring of dimension d > 0, thenthe group homomorphism H : GR

d (f,P|l) → GR0 (l,pd) from Theorem 2.11 is an

isomorphism, and so is the group homomorphism R : GR0 (l,pd) → GR

¤(f,P|l) fromTheorem 2.12. In particular, there are isomorphisms

GRd (f,P|l) ∼= GR

0 (l,pd) ∼= GR¤(f,P|l).

Proof: We have already observed that the homomorphisms involved are welldefined, and that R◦H = ι is an isomorphism. Thus, we already know that His injective and R is surjective. Now, if M is a module in CR

0 (l,pd), according toLemma 3.35, M has a projective resolution X in CR

d (f,P|l), proving that [M ] =H([X]) and thereby that H is surjective. Consequently H is an isomorphism,and it follows that R is an isomorphism as well.

Lemma 3.37. Suppose that R is Noetherian and let x = (x1, . . . , xd) be a regularsequence of length d > 0. Then any complex X in CR

d (f,P|S(x)-tor) satisfies thecondition that its homology complex H(X) is concentrated in degree 0: that is,H(X) is a module in CR

0 (f,pd,S(x)-tor).

Proof: Let X be a nonexact complex in CRd (f,P|S(x)-tor) and let t denote

the largest integer such that Ht(X) 6= 0; this exists since H(X) 6= 0 and X isbounded. We already know that t ≥ 0, so let us assume that t > 0 and try toreach a contradiction.

Let p be an associated prime of Ht(X). Since H(X) is S(x)-torsion, we canfind N1, . . . , Nd ∈ N such that xN1

1 , . . . , xNdd ∈ AnnR Ht(X) ⊆ p and thereby

x1, . . . , xd ∈ p. Consequently, (x1/1, . . . , xd/1) is an Rp-sequence in pp (cf. [BH93,Corollary 1.1.3(i)]), so depth Rp ≥ d > 0.

Now, localization preserves exactness and projectivity of modules, so the pro-jective resolution

0 → Xd → · · · → Xt+1 → im ∂Xt+1 → 0


of im ∂Xt+1 induces a projective resolution

0 → (Xd)p → · · · → (Xt+1)p → (im ∂Xt+1)p → 0

of (im ∂Xt+1)p as an Rp-module, showing that pdRp

(im ∂Xt+1)p ≤ d− (t + 1). From

the Auslander–Buchsbaum formula (Theorem 0.16), it now follows that

depthRp(im ∂X

t+1)p = depth Rp − pdRp(im ∂X

t+1)p ≥ t + 1 ≥ 2

if im ∂Xt+1 is nontrivial, whereas depthRp

(im ∂Xt+1)p = ∞ if im ∂X

t+1 is trivial; in

either case, depthRp(im ∂X

t+1)p ≥ 2.

According to Proposition 0.18(i), depthRp(ker ∂X

t )p ≥ 1, since (ker ∂Xt )p is

a submodule of the nontrivial free Rp-module (Xt)p that has depthRp(Xt)p =

depth Rp ≥ d ≥ 1. From Proposition 0.18(iii) applied to the short exact sequence

0 → (im ∂Xt+1)p → (ker ∂X

t )p → (Ht(X))p → 0,

it now follows that depthRp(Ht(X))p ≥ 1. This is a contradiction, however,

because depthRp(Ht(X))p = 0, since p is associated to Ht(X). Thus, t = 0 as

desired.

Once again, we are in position to apply Theorem 2.11 as well as Theorem 2.12,providing us with homomorphisms H : GR

d (f,P|S(x)-tor) → GR0 (f,pd,S(x)-tor)

and R : GR0 (f,pd,S(x)-tor) → GR

¤(f,P|S(x)-tor) that compose by Theorem 3.30to an isomorphism ι : GR

d (f,P|S(x)-tor) → GR¤(f,P|S(x)-tor) as illustrated by the

following commutative diagram.

GRd (f,P|S(x)-tor)

ι∼=

//

H²²

GR¤(f,P|S(x)-tor)

GR0 (f,pd,S(x)-tor)

R

55kkkkkkkkkkkkkkk

According to Theorem 3.31 this even holds when d = 0. Regardless of whetherd = 0 or not, H and R once again turn out to be isomorphisms.

Corollary 3.38. If R is Noetherian and local, and x = (x1, . . . , xd) is a regularsequence of length d ≥ 0, then the group homomorphism

H : GRd (f,P|S(x)-tor) → GR

0 (f,pd,S(x)-tor)

from Theorem 2.11 is an isomorphism, and so is the group homomorphism

R : GR0 (f,pd,S(x)-tor) → GR

¤(f,P|S(x)-tor)

from Theorem 2.12. In particular, there are isomorphisms

GRd (f,P|S(x)-tor) ∼= GR

0 (f,pd,S(x)-tor) ∼= GR¤(f,P|S(x)-tor).


Proof: We already know that the homomorphisms involved are well defined andthat R◦H = ι is an isomorphism. Thus, it suffices to show that H is surjective.

So let M be a module in CR0 (f,pd,S(x)-tor), and let us show by induction

on p = pdR M that [M ] ∈ imH. If p ≤ d, it is clear that [M ] ∈ imH since,in this case, M has a projective resolution in CR

d (f,P|S(x)-tor), so assume thatp > d. Choose a finitely generated free module F and a surjective homomorphismf : F → M . Next, using the fact that M is S(x)-torsion, choose N1, . . . , Nd ∈ Nso that xN1

1 , . . . , xNdd ∈ AnnR M , and let F = F/〈xN1

1 , . . . , xNdd 〉F . The surjection

f induces a surjection f : F → M . Letting K denote the kernel of f , we thenhave an exact sequence

0 → K → F → M → 0.

Now, F is the direct sum of copies of R/〈xN11 , . . . , xNd

d 〉, so it follows fromTheorems 0.13 and 3.6(iii) that pdR F = d, and Theorem 0.7 therefore givespdR K = pdR F − 1 = d − 1. By construction, F and K are S(x)-torsion, soF and K are modules in CR

0 (f,pd,S(x)-tor), and the induction hypothesis yields[M ] = [F ]− [K] ∈ imH. Consequently H is surjective, and it follows that H andR are isomorphisms.

Note that the case d = 0 simply states that

K0(R) = GR0 (f,P) ∼= GR

0 (f,pd) ∼= GR¤(f,P),

and that we do not need R to be local to prove this. However, if R is local, Ex-ample 2.14 reveals that these groups are all isomorphic to Z through the rank onGR

0 (f,P). The proof of Theorem 3.31 shows that the isomorphism GR0 (f,pd) ∼= Z is

given by taking [M ] ∈ GR0 (f,pd) to the alternating sum of the ranks of the modules

in a bounded, finite, free resolution of M ; in other words, the Euler characteristicχR(−) : GR

0 (f,pd) → Z is an isomorphism. It follows that, in GR0 (f,pd), we have

[M ] = χR(M)[R]. We shall be needing this fact later, so let us state it as acorollary.

Corollary 3.39. If R is Noetherian and local, and M is a module in CR0 (f,pd),

then [M ] = χR(M)[R] in GR0 (f,pd).

The proofs of Lemma 3.37 and Corollary 3.38 in the case d = 1 clearly showthat the multiplicative systems S(x) = S(x1) = {xn

1 | n ∈ N0} can be replacedby any multiplicative system S containing only non-zerodivisors: that is, anymultiplicative system S with S ∩ Zd R = ∅. This is because any element ofsuch a multiplicative system will itself constitute a regular sequence of length 1.Consequently, we can improve Corollary 3.38 slightly in the case d = 1.

Corollary 3.40. Suppose R is Noetherian and S is a multiplicative system withS ∩ Zd R = ∅. Then the group homomorphism

H : GR1 (f,P|S-tor) → GR

0 (f,S-tor)


from Theorem 2.11 is an isomorphism, and so is the group homomorphism

R : GR0 (f,pd,S-tor) → GR

¤(f,P|S-tor)


GR1 (f,P|S-tor) ∼= GR

0 (f,pd,S-tor) ∼= GR¤(f,P|S-tor).

For the next corollary, note that the Grothendieck groups GRd (f,pd|gr ≥ d)

and GR0 (f,pd,gr ≥ d) for d ≥ 1 satisfy the conditions of Theorem 2.11; for if X

is a complex in CRd (f,P|gr ≥ d), we can, using Proposition 0.19(iii), find a regular

sequence x = (x1, . . . , xd) of length d contained in the annihilator of all the homol-ogy modules of X. Then X will be homologically S(x)-torsion, and it follows fromLemma 3.37 that the homology of X is concentrated in degree 0. Consequently,we can consider the homomorphism H : GR

d (f,P|gr ≥ d) → GR0 (f,pd,gr ≥ d) from

Theorem 2.11. Note also that the Grothendieck groups GR0 (f,pd,gr ≥ d) and

GR¤(f,P|gr ≥ d) satisfy the conditions of Theorem 2.12, thereby allowing us to

consider the homomorphism R : GR0 (f,pd,gr ≥ d) → GR

¤(f,P|gr ≥ d). In all thathas just been said, we can replace GR

0 (f,pd,gr ≥ d) by GR0 (f,d-perf), thereby ob-

taining a homology isomorphism H′ : GRd (f,P|gr ≥ d) → GR

0 (f,d-perf) and a ho-momorphism R′ : GR

0 (f,d-perf) → GR¤(f,P|gr ≥ d). There is also the natural ho-

momorphism ι′ : GR0 (f,d-perf) → GR

0 (f,gr ≥ d) induced by the inclusion of theunderlying categories, so the situation is as in the following commutative dia-gram.

GR0 (f,d-perf)

ι′

²²

R′

))RRRRRRRRRRRRRR

GRd (f,P|gr ≥ d)

H′55llllllllllllll

H ))RRRRRRRRRRRRRRGR

¤(f,P|gr ≥ d)

GR0 (f,pd,gr ≥ d)

R

55llllllllllllll

(3.22)

As one could hope, all the homomorphisms in the diagram turn out to be iso-morphisms.

Corollary 3.41. If R is Noetherian and local, and d ≥ 1, then the group homo-morphisms

H : GRd (f,P|gr ≥ d) → GR

0 (f,pd,gr ≥ d)

andH′ : GR

d (f,P|gr ≥ d) → GR0 (f,d-perf)

from Theorem 2.11 are isomorphisms, and so are the group homomorphisms

R : GR0 (f,pd,gr ≥ d) → GR

¤(f,P|gr ≥ d)


andR′ : GR

0 (f,d-perf) → GR¤(f,P|gr ≥ d)


GRd (f,P|gr ≥ d) ∼= GR

0 (f,d-perf) ∼= GR0 (f,pd,gr ≥ d) ∼= GR

¤(f,P|gr ≥ d).

Proof: If d is so large that there are no regular sequences in R of length d,then the involved Grothendieck groups are all trivial and the theorem holds. Wecan therefore assume that regular sequences of length d do exist. Let us definean equivalence relation on the set of such sequences, letting a regular sequencex = (x1, . . . , xd) be equivalent to a regular sequence x′ = (x′1, . . . , x

′d) whenever

RadR〈x1, . . . , xd〉 = RadR〈x′1, . . . , x′d〉.It is clear that this, indeed, is an equivalence relation. Denote the set of equiv-alence classes by E, and let us partially order E by reversed inclusion of radicalideals: that is,

x 4 x′ def⇐⇒ RadR〈x1, . . . , xd〉 ⊇ RadR〈x′1, . . . , x′d〉

for x, x′ ∈ E. (It is of course the equivalence classes of x and x′ that belongto E, but this unimportant technicality will be ignored here.) E = (E, 4) is adirected set, for if x and x′ are regular sequences of length d, then according toProposition 0.19(iii), we can find a regular sequence x′′ of length d contained in〈x〉 ∩ 〈x′〉 and hence satisfying the condition that x, x′ 4 x′′.

Now, the category CR0 (f,pd,S(x)-tor) is uniquely determined by the equiva-

lence class of x in E, since for any finitely generated module M ,

M is S(x)-torsion ⇐⇒ ∀ν ∈ {1, . . . , d}∃Nν ∈ N0 : xNνν ∈ AnnR M

⇐⇒ 〈x1, . . . , xd〉 ⊆ RadR(AnnR M)

⇐⇒ RadR〈x1, . . . , xd〉 ⊆ RadR(AnnR M).

Thus, we can consider the family of Grothendieck groups GR0 (f,pd,S(x)-tor) in-

dexed by the equivalence classes in E. Given x, x′ ∈ E with x 4 x′, there is ahomomorphism

ιx,x′ : GR0 (f,pd,S(x)-tor) → GR

0 (f,pd,S(x′)-tor)

given by ιx,x′([M ]) = [M ]; this is well defined, as seen from the bi-implicationsabove. Consequently (GR

0 (f,pd,S(x)-tor), ιx,x′)x4x′ is a direct system, and it fol-lows that it has a direct limit lim−→x∈EGR

0 (f,pd,S(x)-tor) equipped with homomor-phisms

τx : GR0 (f,pd,S(x)-tor) → lim−→x∈EGR

0 (f,pd,S(x)-tor)

for each x ∈ E.


If x = (x1, . . . , xd) is a regular sequence and M is a finitely generated S(x)-torsion module, we can find N1, . . . , Nd ∈ N such that xN1

1 , . . . , xNdd ∈ AnnR M ,

and it follows from Theorem 0.13 that gradeR M ≥ d. Thus, there are naturalhomomorphisms

ax : GR0 (f,pd,S(x)-tor) → GR

0 (f,pd,gr ≥ d)

given by ax([M ]) = [M ]. Since these commute with the homomorphisms ιx,x′ ,the universal property of the direct limit provides us with a homomorphism

λ : lim−→x∈EGR0 (f,pd,S(x)-tor) → GR

0 (f,pd,gr ≥ d)

such that ax = λτx for all x ∈ E:

GR0 (f,pd,S(x)-tor)

ιx,x′

²²

τx

**TTTTTTTTTTTTTTT ax

++lim−→x∈EGR

0 (f,pd,S(x)-tor) λ //______ GR0 (f,pd,gr ≥ d)

GR0 (f,pd,S(x′)-tor)

τx′

44jjjjjjjjjjjjjjj ax′

33

We claim that λ is an isomorphism. Surjectivity is clear, for if M is a modulein CR

0 (f,pd,gr ≥ d), we can find a regular sequence x = (x1, . . . , xd) of length d inAnnR M , and hence M lies in CR

0 (f,pd,S(x)-tor) and [M ] = ax([M ]) = λτx([M ]) inGR

0 (f,pd,gr ≥ d). To see that λ is injective, note that, since the image of the mapsτx span the direct limit (as can be seen from its concrete construction), it sufficesto prove that the assumption [M ] = [M ′] in GR

0 (f,pd,gr ≥ d) leads to [M ] = [M ′]in some GR

0 (f,pd,S(x)-tor). So suppose that [M ] = [M ′] in GR0 (f,pd,gr ≥ d). It

then follows from Proposition 2.6 that we can find modules U , V , V ′ and W inCR

0 (f,pd,gr ≥ d) such that there are exact sequences

0 → U → V → W → 0 and 0 → U → V ′ → W → 0,

and such that M ⊕ V ∼= M ′ ⊕ V ′. Using Proposition 0.19(iii) repeatedly, we canfind a regular sequence x = (x1, . . . , xd) of length d contained in the annihilatorsof all the modules M , M ′, U , V , V ′ and W , and it follows that [M ] = [M ′] inGR

0 (f,pd,S(x)-tor). Consequently, λ is injective.We have now shown that GR

0 (f,pd,gr ≥ d) is the direct limit of the directsystem (GR

0 (f,pd,S(x)-tor), ιx,x′)x4x′ . By the same methods one can show thatGR

d (f,P|gr ≥ d) and GR¤(f,P|gr ≥ d) are the direct limits of the direct systems

(GRd (f,P|S(x)-tor), ιx,x′)x4x′ and (GR

¤(f,P|S(x)-tor), ιx,x′)x4x′ respectively, wherethe homomorphisms ιx,x′ now are given by ιx,x′([X]) = [X]; getting this farrequires applying Proposition 0.19(iii) repeatedly to obtain a regular sequencecontained in the annihilator of all the homology modules of a complex.


Nevertheless, we already know from Corollary 3.38 that there are isomor-phisms

GRd (f,P|S(x)-tor) ∼= GR

0 (f,pd,S(x)-tor) ∼= GR¤(f,P|S(x)-tor)

for all regular sequences x of length d, the first of these being given by thehomomorphism Hx from Theorem 2.11 and the second being given by the homo-morphism Rx from Theorem 2.12. Hence there must also be isomorphisms

GRd (f,P|gr ≥ d) ∼= GR

0 (f,pd,gr ≥ d) ∼= GR¤(f,P|gr ≥ d)

between the direct limits, the first of these being given by the homomorphism Hfrom Theorem 2.11 and the second being given by the homomorphism R fromTheorem 2.12. In particular, R and H are isomorphisms.

Using the commutativity of diagram (3.22), it now follows that R′H′ is anisomorphism. To show thatR′ as well asH′ are isomorphisms, it therefore sufficesto show thatH′ is surjective. This, however, is clear since any finitely generated d-perfect module by definition has a projective resolution in CR

d (f,P|gr ≥ d). Usingthe commutativity of diagram (3.22) once more, it follows that ι′ also is anisomorphism.

Note that Corollary 3.41 actually holds when d = 0, and that including thiscase is unnecessary, as it is already part of Corollary 3.38.

4 Chapter 4

Groups of matrices: K1

Chapter 3 introduced the zeroth algebraic K-group of R as a special instance ofthe more general concept of Grothendieck groups. Likewise, the first algebraicK-group of R is a special instance of the more general concept of Bass–Whiteheadgroups. However, in this thesis the full generality is not needed, so we discusshere only the traditional first algebraic K-group K1(R).

4.1 The first algebraic K-group

If m,n ∈ N, the set Rm×n of m × n matrices is a free R-module with a basisconsisting of the matrix units εij having (i, j)-entry equal to 1 and all otherentries equal to 0. Considering only square matrices, we obtain the ring Mn(R)of n × n matrices over R, in which the matrix units multiply according to therule

εijεkl =

{εil, if j = k,0, if j 6= k.

The general linear group GLn(R) is the multiplicative group consisting of theinvertible elements of Mn(R): that is, the matrices whose determinants lie in R∗.The neutral element in GLn(R) is the identity matrix In = ε11 + · · ·+ εnn.

One way to determine whether a given n × n matrix A is invertible or notis to apply the row operations in the Gauss–Jordan elimination process and seewhether the identity matrix can be obtained. There are three kinds of row op-erations, and each of these corresponds to left multiplication by an invertiblematrix:

(i) adding r ∈ R times the j’th row to the i’th row for i 6= j, which correspondsto left multiplication by eij(r) = In + rεij;

(ii) interchanging the j’th row with the i’th row, which corresponds to leftmultiplication by pij = In − εii − εjj + εij + εji; and

(iii) multiplying the i’th row by a unit u ∈ R∗, which corresponds to left multi-plication by di(u) = In + (u− 1)εii.

55

56 Groups of matrices: K1

The row operations in (i) are called elementary row transvections, and thesubgroup of GLn(R) that they generate is denoted by En(R) and is referredto as the group of elementary matrices of degree n. Since eij(r)

−1 = eij(−r),En(R) consists of finite products of elementary row transvections. The subgroupof GLn(R) generated by the matrices pij from (ii) is denoted by Pn(R), and itconsists of the permutation matrices

Pσ = εσ(1)1 + · · ·+ εσ(n)n

for σ ∈ Sn, where Sn is the symmetric group of degree n consisting of all permu-tations of the set {1, . . . , n}. The subgroup of GLn(R) generated by the matricesdi(u) from (iii) is denoted by Dn(R), and it consists of the invertible diagonalmatrices

diag(u1, . . . , un) = u1ε11 + · · ·+ unεnn

for u1, . . . , un ∈ R∗.The determinants of the three types of row operations are detR(eij(r)) = 1,

detR(pij) = −1 and detR(di(u)) = u. In particular the matrices of En(R) all havedeterminant 1, so performing elementary row transvections on a matrix does notchange its determinant. A,B ∈ Mn(R) are said to be t-row equivalent if B can beobtained from A by elementary row transvections: that is, if B ∈ En(R)A. This isan equivalence relation. Whenever A is invertible, so is its entire t-row equivalenceclass, and hence t-row equivalence on Mn(R) restricts to an equivalence relationon GLn(R).

For the next theorem, recall that if G is a group and H is a subset of G, HCGmeans that H is a normal subgroup of G. If H and K are subgroups of G, wesay that K normalizes H if xHx−1 ⊆ H for all x ∈ K. In this case, it followsthat HK = KH is a subgroup of G and that H C HK.

Theorem 4.1. For all n ∈ N, Pn(R) normalizes Dn(R) and Dn(R) normalizesEn(R), and hence

MLn(R)def= Dn(R)Pn(R) and GEn(R)

def= Dn(R)En(R)

are subgroups of GLn(R) with Dn(R) C MLn(R) and En(R) C GEn(R).

Proof: The subgroup Pn(R) is generated by the matrices pij. The inverseof pij is pij, so given such a matrix and a matrix diag(u1, . . . , un) ∈ Dn(R),we need to show that pijdk(u)pij is an invertible diagonal matrix. Multipli-cation on the left by pij interchanges rows i and j, and multiplication on theright by pij interchanges columns i and j, and hence taking diag(u1, . . . , un) topij diag(u1, . . . , un)pij interchanges the (i, i)-entry with the (j, j)-entry, and theresulting matrix is an invertible diagonal matrix. Thus, Pn(R) normalizes Dn(R).

The subgroup Dn(R) is generated by the matrices di(u) for u ∈ R∗. So givensuch a matrix and an elementary row transvection ejk(r), it suffices to show that

4.1 The first algebraic K-group 57

di(u)ejk(r)di(u−1) is an elementary row transvection. Multiplication on the left by

di(u) multiplies the i’th row by u, and multiplication on the right by di(u−1) mul-

tiplies the i’th column by u−1; hence taking ejk(r) to di(u)ejk(r)di(u−1) leaves the

diagonal entries of ejk(r) intact while possibly multiplying the only off-diagonalentry by u or u−1. In any case the resulting matrix is one of the elementary rowtransvections ejk(r), ejk(ur) or ejk(u

−1r), so Dn(R) normalizes En(R).

Theorem 4.2. En(R) contains matrices corresponding to the row operations

(ii′) multiplying the i’th row by −1 and interchanging it with the j’th row fori 6= j; and

(iii′) multiplying rows i and j for i 6= j with units u and u−1, respectively.

Furthermore, MLn(R) ⊆ GEn(R), and GEn(R) is exactly the subgroup generatedby the matrices corresponding to row operations in the Gauss–Jordan eliminationprocess.

Proof: Define for i 6= j and u ∈ R∗ matrices p′ij and d′i(u) in En(R) by

p′ij = eij(1)eji(−1)eij(1), and

d′ij(u) = eij(u)eji(−u−1)eij(u)eij(−1)eji(1)eij(−1).

It is straightforward to verify that p′ij and d′ij(u) correspond to the row operations(ii′) and (iii′), respectively. It follows that pij = dj(−1)p′ij ∈ GEn(R), so thatPn(R) ⊆ GEn(R) and thereby MLn(R) ⊆ GEn(R). The matrices correspondingto Gauss–Jordan elimination are products of matrices from En(R), Pn(R) andDn(R), so this shows that any such product is contained in GEn(R).

Theorem 4.2 shows that it is possible to get very far in the Gauss–Jordanelimination process using only elementary row transvections from En(R). Theseare all determinant preserving, so the closest one can hope to get to reducing amatrix A ∈ GLn(R) with row operations from En(R) is to bring it in the formdiag(detR A, 1, . . . , 1). The following proposition describes exactly when this ispossible.

Proposition 4.3. The following are equivalent for n ∈ N.

(i) GEn(R) = GLn(R).

(ii) Each A ∈ GLn(R) can be reduced to In by using the Gauss–Jordan rowoperations (i)-(iii) from above.

(iii) Each A ∈ GLn(R) is t-row equivalent to a matrix d1(u) = diag(u, 1, . . . , 1),where u = detR A.


Proof: “(i) ⇒ (ii)”: The group GEn(R) is generated by Gauss–Jordan rowoperations, so the assumption that GEn(R) = GLn(R) means that any matrixA ∈ GLn(R) can be row operated into In and vice versa.

“(ii) ⇒ (iii)”: Given A ∈ GLn(R), the assumption together with Theorem 4.2allows us to find matrices D ∈ Dn(R) and E ∈ En(R) such that In = DEA.Here D is in the form D = diag(u1, . . . , un) for u1, . . . , un ∈ R∗. Recalling therow operation (iii′) from Theorem 4.2 described by the matrices d′ij(u) ∈ En(R)for u a unit, we see that

D = diag(u1 · · · un, 1, . . . , 1)d′21(u2) · · · d′n1(un).

Letting u = u−11 · · · u−1

n and E ′ = d′21(u2) · · · d′n1(un)E, we now find that

diag(u, 1 . . . , 1) = E ′A.

This means that A is t-row equivalent to diag(u, 1, . . . , 1), and since E ′ is aproduct of matrices from En(R), all of which have determinant 1, we must haveu = detR(diag(u, 1, . . . , 1)) = detR(E ′A) = detR A.

“(iii) ⇒ (i)”: Given A ∈ GLn(R), the assumption provides the existenceof a matrix E ∈ En(R) and a unit u ∈ R∗ such that A = E diag(u, 1, . . . , 1).Since En(R)Dn(R) = Dn(R)En(R) = GEn(R), this shows that A ∈ GEn(R) asdesired.

Definition 4.4. The ring R is said to be generalized Euclidean if it satisfies anyof the equivalent properties from Proposition 4.3.

Proposition 4.5. If R is semilocal, then R is generalized Euclidean.

Proof: Let A = (aij) ∈ GLn(R). We show by induction on n that A canbe reduced to In using Gauss–Jordan row operations. The case n = 1 is clear,so suppose that n > 1 and that the statement holds for all smaller invertiblematrices.

We commence by showing that, using row operations, we can obtain a unit inthe (1, 1)-entry of A. Let M denote the (finite) set of maximal ideals containinga11, and suppose that we have performed row operations so that the cardinalityof M is minimal, in the sense that no sequence of row operations can lead to amatrix whose (1, 1)-entry is contained in fewer maximal ideals than the numberof maximal ideals in M. We want to show that M is empty, so that a11 must be aunit. Suppose therefore that M is nonempty, and let m0 ∈ M. Now, the entries inthe first column of A cannot all be contained in the same maximal ideal, becausethen the determinant of A would also be contained in that maximal ideal andhence not be a unit. Thus we can find i so that ai1 /∈ m0. Using prime avoidance(Lemma 0.1), we can, for each maximal ideal m, find an element xm satisfyingthe condition that xm is contained in m and in no other maximal ideal. We now


perform a row operation on A, adding∏

m/∈M xm times the i’th row to the firstrow. As its (1, 1)-entry, the obtained matrix will have the element

a11 + ai1

∏

m/∈M

xm.

This element cannot be contained in any of the maximal ideals from outsideM, since this would imply that a11 was contained in such a maximal ideal. Fur-thermore, the element cannot be contained in the maximal ideal m0, since thiswould imply that one of the factors of ai1

∏m/∈M xm is contained in m0. Thus, row

operations produced a matrix whose (1, 1) entry is contained in fewer maximalideals than the number of maximal ideals in M. This contradicts our assumption,so M must be empty, and hence a11 is a unit.

With a unit in the (1, 1)-entry, we can now perform row operations to obtaina matrix A′ in the form

A′ =

1 a′12 · · · a′1n

0 a′22 · · · a′2n...

.... . .

...0 a′n2 · · · a′nn

.

The determinant of this matrix is still a unit and equals the determinant of the(n−1)×(n−1) matrix obtained from A by deleting the first row and first column.Thus, according to the induction hypothesis, we can perform row operations onthe bottom n−1 rows of A′, thereby obtaining a matrix that only differs from theunit matrix in the last n−1 entries of the first row. Such a matrix is easily reducedto the unit matrix using row operations, and we have proved the proposition.

Recall that if G is a group and x, y ∈ G, the commutator of x and y is theelement [x, y] = xyx−1y−1. The elements x and y commute if and only if [x, y] =1. The subgroup of G generated by all commutators is called the commutatorsubgroup of G and is denoted by [G,G]. Since [x, y]−1 = [y, x], it consists of allfinite-length products of commutators in G. The quotient Gab = G/[G,G] isknown as the Abelianization of G; it is the largest Abelian quotient of G, in thesense that, for any subgroup H of G, H contains [G,G] if and only if H C G andG/H is Abelian.

The next theorem shows that En(R) in some sense is “completely non-Abelian”.

Theorem 4.6. If n ≥ 3, then [En(R), En(R)] = En(R).

Proof: If i, j ∈ {1, . . . , n}, we can find k ∈ {1, . . . , n} different from i and j. Asimple calculation then shows that, for any r ∈ R, eij(r) = [eik(r), ekj(1)]. Thisproves the theorem.

From Theorem 4.6 it follows that En(R) ⊆ [GLn(R), GLn(R)]: that is, thegroup of elementary matrices forms a subset of the commutator group of GLn(R).


We would like the group of elementary matrices to be exactly the commutatorsubgroup of GLn(R), but as we shall see below, this requires a little more “elbowroom”.

Definition 4.7. The infinite-dimensional general linear group is the multiplica-tive group GL(R) of N× N matrices that are obtained from the matrix

I∞ =

1 0 · · ·0 1 · · ·...

.... . .

by replacing an upper left corner In by an invertible matrix A = (aij) ∈ GLn(R)so that one gets the N× N matrix

A⊕ I∞ =

a11 · · · a1n 0 · · ·...

......

an1 · · · ann 0 · · ·0 · · · 0 1 · · ·...

......

. . .

.

The composition in GL(R) is given as follows. If A ∈ GLm(R) and B ∈ GLn(R),

A⊕ In =

(A 00 In

)and B ⊕ Im =

(B 00 Im

)

are both in GLm+n(R), and we set (A⊕ I∞)(B⊕ I∞) = ((A⊕ In)(B⊕ Im))⊕ I∞.The neutral element of this composition is I∞, and inverse elements are given by(A⊕ I∞)−1 = A−1 ⊕ I∞ for A ∈ GLn(R). By identifying a matrix A in GLn(R)with the matrix A⊕ I∞ in GL(R), GLn(R) can be identified with the subgroup{A⊕ I∞ |A ∈ GLn(R)} of GL(R). With these identifications we find that

R∗ = GL1(R) ⊆ GL2(R) ⊆ · · ·and GL(R) =

⋃∞n=1 GLn(R). The group of elementary matrices is the subgroup

E(R) of GL(R) generated by the elementary transvections

eij(r) = I∞ + rεij

for r ∈ R and i 6= j: that is, the N× N matrices obtained from I∞ by replacingan off-diagonal entry with r. Under the identification of A ∈ GLn(R) withA⊕ I∞ ∈ GL(R), the usual elementary transvections eij(r) ∈ En(R) become theeij(r) ∈ GL(R), and hence

{1} = E1(R) ⊆ E2(R) ⊆ · · ·and E(R) =

⋃∞n=1 En(R).


Lemma 4.8 (Whitehead). If n ≥ 3, then

En(R) ⊆ [GLn(R), GLn(R)] ⊆ E2n(R),

and it follows that [GL(R), GL(R)] = E(R).

Proof: The first inclusion follows from Theorem 4.6. For the second inclusion,let A,B ∈ GLn(R). We want to show that

(ABA−1B−1 0

0 In

)is in E2n(R). Since

(ABA−1B−1 0

0 In

)=

(A 00 A−1

)(B 00 B−1

)((BA)−1 0

0 BA

),

it suffices to show that a matrix in the form(

A 00 A−1

)for A ∈ GLn(R) is in E2n(R).

Now, calculation shows that

(A 00 A−1

)=

(In A0 In

)(In 0

−A−1 In

)(In A0 In

)(In −In

0 In

)(In 0In In

)(In −In

0 In

),

and hence it suffices to show that matrices in the forms(

In A0 In

)and

(In 0A In

)for

A ∈ GLn(R) are in E2n(R). This, however, follows immediately from two morecalculations:

(In A0 In

)= I2n +

∑1≤i,j≤n

aijεi(n+j) =∏

1≤i,j≤n

(I2n + aijεi(n+j)) and

(In 0A In

)= I2n +

∑1≤i,j≤n

aijε(n+i)j) =∏

1≤i,j≤n

(I2n + aijε(n+i)j).

This shows that En(R) ⊆ [GLn(R), GLn(R)] ⊆ E2n(R). Now, if A ∈ GLm(R)and B ∈ GLn(R), then

[A⊕ I∞, B ⊕ I∞] = [A⊕ In, B ⊕ Im]⊕ I∞,

and hence [GL(R), GL(R)] =⋃∞

n=1[GLn(R), GLn(R)]. From this follows that[GL(R), GL(R)] = E(R).

We are finally ready to present

Definition 4.9. The first algebraic K-group of R is the group

K1(R) = GL(R)ab = GL(R)/E(R).

Given A ∈ GLn(R), we denote the coset of A ⊕ I∞ ∈ GL(R) in K1(R) by [A]Ror, when there is no doubt about which ring is being referred to, simply by [A].


For any A ∈ GLn(R) and any m ∈ N, [A] = [A⊕ Im] in K1(R). We also have[A] = [Im ⊕ A]. To see this, note first that [A] = [In ⊕ A], since

In ⊕ A = (A−1 ⊕ A)(A⊕ In)

where A−1⊕A is trivial in K1(R), since it belongs to E2n(R) as seen in the proofof Whitehead’s lemma (Lemma 4.8). It follows that

[A] = [A⊕ I2m] = [I2m+n ⊕ (A⊕ I2m)] = [Im+n ⊕ (Im ⊕ A)] = [Im ⊕ A].

A consequence of this is that, for matrices A ∈ GLm(R) and B ∈ GLn(R),

[AB] = [A][B] = [A⊕ In][Im ⊕B] = [A⊕B].

The usual determinant maps detR : GLn(R) → R∗ induce a determinantdetR : GL(R) → R∗ defined for A ∈ GLn(R) by

detR(A⊕ I∞) = detR A.

This is clearly well defined in the sense that the determinant of A⊕I∞ is indepen-dent of choice of n and A. Since R∗ is commutative, the map detR : GL(R) → R∗

factors through the Abelianization of GL(R), and hence there is an induced de-terminant map K1(R) → R∗, which we also denote detR, given for A ∈ GLn(R)by detR [A] = detR A.

The kernel of the usual determinant detR : GLn(R) → R∗ is denoted bySLn(R); it is the special linear group consisting of n × n matrices over R withdeterminant 1. The kernel of detR : GL(R) → R∗ is denoted by SL(R), and thekernel of detR : K1(R) → R∗ is denoted by SK1(R). Notice that since K1(R) =GL(R)/E(R) where E(R) ⊆ SL(R), we must have SK1(R) = SL(R)/E(R).

The determinant map enables us in certain cases to compute K1(R) quiteeasily. The following proposition is the crucial step.

Proposition 4.10. There is an isomorphism K1(R) ∼= SK1(R)⊕R∗. WheneverSK1(R) is trivial, this isomorphism is given by detR : K1(R) → R∗.

Proof: Let h : R∗ → K1(R) be the homomorphism taking a unit u ∈ R∗ to theelement [u] ∈ K1(R), the coset of the diagonal matrix u ⊕ I∞. It is clear thatdetR(h(u)) = detR([u]) = u, so detR : K1(R) → R∗ is surjective and the followingsequence is split exact.

{1} // SK1(R) // K1(R)detR //

R∗ //h

oo {1}.

It follows that K1(R) ∼= SK1(R)⊕R∗ and that detR : K1(R) → R∗ is an isomor-phism whenever SK1(R) is trivial.

We now realize the importance of Definition 4.4.

4.2 The localization sequence 63

Theorem 4.11. If R is generalized Euclidean, then detR : K1(R) → R∗ is anisomorphism.

Proof: We need to show that SK1(R) = SL(R)/E(R) is trivial, but accordingto Proposition 4.3, any n × n matrix with determinant 1 is t-row equivalent toIn. Thus, SL(R) = E(R).

We have previously seen that a ring homomorphism ρ : R → R′ between(nontrivial, unitary and commutative) rings induces a group homomorphismK0(ρ) : K0(R) → K0(R

′) between the corresponding K0-groups. We shall nowsee that the same property is satisfied when constructing K1-groups. So supposeR and R′ are (nontrivial, unitary and commutative) rings and that ρ : R → R′ isa ring homomorphism. We claim that entry-wise application of ρ defines a grouphomomorphism GL(ρ) : GL(R) → GL(R′). To obtain this, note that applica-tion of ρ commutes with determinant: that is, if A = (aij) is an n × n matrixover R, then A′ = (ρ(aij)) is an n × n matrix over R′ whose determinant isdetR′ A

′ = ρ(detR A). It follows that entry-wise application of ρ takes invertiblematrices to invertible matrices, so entry-wise application of ρ defines a grouphomomorphism GL(ρ) : GL(R) → GL(R′). When Abelianizing domain as wellas co-domain, GL(ρ) collapses to a group homomorphism K1(R) → K1(R

′).

Definition 4.12. If R and R′ are (nontrivial, unitary and commutative) rings,and ρ : R → R′ is a ring homomorphism, then we denote by K1(ρ) the grouphomomorphism K1(R) → K1(R

′) defined by entry-wise application of ρ.

4.2 The localization sequence

Throughout this section, S denotes a (single) multiplicative system such thatS ∩ Zd R = ∅, and R is assumed to be Noetherian.

The first algebraic K-groups are connected to the Grothendieck groups in an exactsequence known as the localization sequence. In this section we shall establishthis fact and see how it allows some very nice reductions in the computation ofsome of the Grothendieck groups of the preceding chapter.

Let ρS denote the ring homomorphism R → S−1R taking an element x ∈ Rto ρS(x) = x/1. Note that since S contains no zerodivisors, ρS must be injective.Note also that the units of S−1R are those x/s such that y ∈ R and t ∈ Sexist with xy = st; in particular, R∗ is mapped into (S−1R)∗ under ρS, andthe denominators as well as the numerators of the fractions in (S−1R)∗ are non-zerodivisors of R.

Recall from Definitions 2.18 and 4.12 that there are induced homomorphismsK0(ρS) : K0(R) → K0(S

−1R) and K1(ρS) : K1(R) → K1(S−1R). We shall con-

nect K1(R) → K1(S−1R) to K0(R) → K0(S

−1) via the group GR0 (f,pd,S-tor) in


an exact sequence

K1(R)K1(ρS)−→ K1(S

−1R)δ−→ GR

0 (f,pd,S-tor)χ−→ K0(R)

K0(ρS)−→ K0(S−1R).

Here χ is the Euler characteristic from Theorem 2.13. The map δ still needs tobe defined and the fact that the above sequence is exact still needs to be shown.

Suppose that we are given an element [A] of K1(S−1R), where A is an invert-

ible matrix in GLn(S−1R). If the entries of A are in R (that is, have 1 as a commondivisor), then the obvious choice for δ([A]) is simply to let δ([A]) = [Rn/ARn].This indeed defines an element of GR

0 (f,pd,S-tor): the module Rn/ARn is finitelygenerated by n elements, has a projective resolution

0 −→ Rn A−→ Rn −→ Rn/ARn −→ 0

(A is injective since detR A is a unit in S−1R and thereby a non-zerodivisor inR), and is S-torsion since A becomes invertible when localized at S.

Now, in case A does not have entries in R, we can always find a commondivisor s ∈ S such that (sA), which is still a member of GLn(S−1R), has entriesin R. Since [A] = [(sA)(s−1In)] = [sA][s]−n in K1(S

−1R), we must then haveδ([A]) = δ([sA])−nδ([s]) = [Rn/(sA)Rn]−n[R/sR]. Thus, the requirement thatδ should map elements [A], where A is a matrix with entries in R, to [Rn/ARn]completely determines δ. It still remains to prove that δ exists.

Theorem 4.13. There is a homomorphism δ : K1(S−1R) → GR

0 (f,pd,S-tor) givenfor A ∈ GLn(S−1R) by δ([A]) = [Rn/(sA)Rn]− n[R/sR], where s ∈ S is chosenso that (sA) has entries in R.

Proof: Let us first try to construct a homomorphism

dn : GLn(S−1R) → GR0 (f,pd,S-tor)

by setting dn(A) = [Rn/(sA)Rn] − n[R/sR], where s ∈ S is chosen so that(sA) has entries in R. We already know that [Rn/(sA)Rn] − n[R/sR] is a well-defined element of GR

0 (f,pd,S-tor), so it only remains to verify that this elementis independent of the choice of s and that dn(AB) = dn(A) + dn(B) wheneverA,B ∈ GLn(S−1R). For the first part, suppose that t ∈ S is another elementsuch that (tA) has entries in R. Since s and t are non-zerodivisors, and (sA) and(tA) are injective over R, we have exact sequences

0 −→ Rn/tRn (sA)−→ Rn/(stA)Rn −→ Rn/(sA)Rn −→ 0

and

0 −→ R/tRs−→ R/stR −→ R/sR −→ 0


in CR0 (f,pd,S-tor), proving that in GR

0 (f,pd,S-tor),

[Rn/(sA)Rn]− n[R/sR] = [Rn/(stA)Rn]− [Rn/tRn]− n([R/stR]− [R/tR])

= [Rn/(stA)Rn]− n[R/stR].

Interchanging s and t, we similarly get

[Rn/(tA)Rn]− n[R/tR] = [Rn/(stA)Rn]− n[R/stR],

proving that dn(A) is independent of the choice of s. Thus, dn is well defined asa function GLn(S−1R) → GR

0 (f,pd,S-tor).Now, if A,B ∈ GLn(S−1R), we choose s ∈ S so that (sA) and (sB) both have

entries in R. The exact sequences

0 −→ Rn/(sB)Rn (sA)−→ Rn/(s2AB)Rn −→ Rn/(sA)Rn −→ 0

and0 −→ R/sR

s−→ R/s2R −→ R/sR −→ 0

in CR0 (f,pd,S-tor) then show that in GR

0 (f,pd,S-tor),

dn(AB) = [Rn/(s2AB)Rn]− n[R/s2R]

= [Rn/(sA)Rn] + [Rn/(sB)Rn]− 2n[R/sR]

= dn(A) + dn(B).

This establishes dn as a homomorphism GLn(S−1R) → GR0 (f,pd,S-tor).

Now, given m ∈ N, projections onto the first n and the last m coordinatesdetermine isomorphisms

Rn+m/s(A⊕ Im)Rn+m ∼= Rn/(sA)Rn ⊕Rm/sRm

andRn+m/sRn+m ∼= Rn/sRn ⊕Rm/sRm,

proving that dn+m(A ⊕ Im) = dn(A). Consequently, the maps dn extend to ahomomorphism d : GL(S−1R) → GR

0 (f,pd,S-tor) given by d(A) = dn(A) for eachA ∈ GLn(S−1R). Since GR

0 (f,pd,S-tor) is Abelian, d induces a homomorphismδ : K1(S

−1R) → GR0 (f,pd,S-tor) given by δ([A]) = dn(A) for A ∈ GLn(S−1R).

Now that all the maps in the localization sequence have been introduced,we are ready to prove that it is exact. The key to this will be Corollary 3.40,which enables us to switch between the groups GR

0 (f,pd,S-tor), GR1 (f,P|S-tor) and

GR¤(f,P|S-tor), using the isomorphismsH,R and ι. Switching from GR

0 (f,pd,S-tor)to GR

1 (f,P|S-tor), the homomorphism δ is replaced by ι−1 ◦ R◦δ, which takes anelement [A] in K1(S

−1R) to the element

[0 −→ Rn (sA)−→ Rn −→ 0]− n[0 −→ Rs−→ R −→ 0]


in GR1 (f,P|S-tor). Meanwhile, the homomorphism χ is replaced by χ ◦ H, which

is equal to the homomorphism A from Theorem 2.10, taking an element [X] inGR

1 (f,P|S-tor) to the element [X0]− [X1] in K0(R). We shall denote these mapsby δ and χ as well, leaving it to the reader to determine which of the groupsGR

0 (f,pd,S-tor) and GR1 (f,P|S-tor) is being referred to.

First, however, we need a few lemmas.

Lemma 4.14. Given a complex Y in CR1 (f,P|S-tor), there exists a complex Y in

CR1 (f,P|S-tor) such that Y ⊕ Y is a complex on the form

0 −→ Rn A−→ Rn −→ 0

for some n ∈ N and some matrix A ∈ GLn(S−1R) with entries in R. In par-ticular, every element of GR

1 (f,P|S-tor) is in the form [X] modulo im δ for someX ∈ CR

1 (f,P|S-tor).

Proof: Let Y be a given complex as above, and recall from Proposition 3.2that Y has an S-contraction: that is, a homomorphism β0 : Y0 → Y1 such that∂Y

1 β0 = s1Y0 and β0∂Y1 = s1Y1 for some fixed s ∈ S. When localized at S, ∂Y

1 and∂Y

1 β0 = s1Y0 become isomorphisms, and hence so does β0, and it follows that thecomplex

Y = 0 −→ Y0β0−→ Y1 −→ 0

concentrated in degrees 1 and 0 must be homologically S-torsion: that is, Y is acomplex in CR

1 (f,P|S-tor).Consider now the complex Y ⊕ Y , which has the module Y0⊕Y1 in both of its

nontrivial degrees. The module Y0⊕Y1 is finitely generated and projective, so wecan find a finitely generated and projective module L such that Y0⊕Y1⊕L ∼= Rn

for some n ∈ N0. Letting Z denote the exact complex 0 → L → L → 0concentrated in degrees 1 and 0, the direct sum Y ⊕ Y ⊕Z is now nothing but acomplex

0 −→ Rn A−→ Rn −→ 0,

which is in GR1 (f,P|S-tor). Since the complex becomes exact when localized at S,

the corresponding matrix A must be in GLn(S−1R). Denoting by Y the directsum Y ⊕ Z, this proves the first part of the lemma.

From previous observations, we know that every element of GR1 (f,P|S-tor) can

be written in the form [X] − [Y ] for X, Y ∈ GR1 (f,P|S-tor). It now follows that

[Y ] + [Y ] = δ(A), and hence [X]− [Y ] modulo im δ equals [X] + [Y ] = [X ⊕ Y ].This proves the last part of the lemma.

Lemma 4.15. If A,B ∈ Mn(R) are injective n× n matrices and φ : Rn/ARn →Rn/BRn is an isomorphism, then invertible matrices C, D ∈ GL2n(R) exist, such


that there is a commutative diagram with exact rows

0 // Rn ⊕Rn

(A 00 In

)

//

C

²²

Rn ⊕Rn( πA 0 )//

D

²²

Rn/ARn //

φ²²

0

0 // Rn ⊕Rn

(B 00 In

)

// Rn ⊕Rn( πB 0 )// Rn/BRn // 0

in which πA and πB denote projection maps.

Proof: The rows of the diagram are clearly exact, so we just need to findthe matrices C and D. We already know that φ lifts to a map α : Rn → Rn,and similarly that φ−1 lifts to a map β : Rn → Rn (see, for example, [HS97,Theorem IV.4.1]). Thus, α, β ∈ Mn(R) are matrices such that πBα = φπA andπAβ = φ−1πB. Now let

D =

(2α− αβα αβ − In

In − βα β

),

which is equal to the product(

In α0 In

) (In 0−β In

)(In α0 In

)(In −In

0 In

)(In 0In In

)(In −In

0 In

)

and therefore is invertible with determinant 1. (Note that D corresponds to thematrix d′12(u) from Theorem 4.2 in which the unit u has been replaced by α andits inverse u−1 by β.) The equations πBα = φπA and πAβ = φ−1πB show that ourchoice of D makes the right square of the diagram above commute. It follows,in particular, that the kernels of

(πA 0

)and

(πB 0

)are isomorphic under D:

that is, an isomorphism ψ : ARn⊕Rn → BRn⊕Rn induced by D exists. But thedomain as well as the co-domain of ψ are isomorphic to R2n by the maps

(A 00 In

)and

(B 00 In

), and hence ψ can be translated into an isomorphism C : R2n → R2n,

making the diagram

Rn ⊕Rn

C²²

(A 00 In

)

// ARn ⊕Rn

ψ²²

Rn ⊕Rn

(B 00 In

)

// BRn ⊕Rn

commutative. Consequently, C ∈ GL2n(R) is an invertible matrix, making theleft square of the previous diagram commutative, and the lemma has been proven.

Theorem 4.16. The sequence

K1(R)K1(ρS)−→ K1(S

−1R)δ−→ GR

0 (f,pd,S-tor)χ−→ K0(R)

K0(ρS)−→ K0(S−1R)

is exact.


Proof: Exactness at K1(S−1R): The composition δ◦K1(ρS) takes [A]R ∈ K1(R)

to [Rn/ARn], where Rn/ARn is the zero module, since A is invertible. Conse-quently, δ ◦K1(ρS) = 0.

Conversely, suppose that [A]S−1R is in the kernel of δ: that is, [Rn/(sA)Rn] =[Rn/sRn] for s ∈ S chosen so that (sA) has entries in R. We want to show that[A]S−1R is in the image of K1(ρS). Let us show even more generally that, if A andB are matrices in GLn(S−1R) with entries in R such that [Rn/ARn] = [Rn/BRn],then [AB−1]S−1R is in the image of K1(ρS).

Suppose first that the equation [Rn/ARn] = [Rn/BRn] is caused by an iso-morphism φ : Rn/ARn → Rn/BRn. According to Lemma 4.15 invertible matricesC, D ∈ GL2n(R) then exist such that there is a commutative diagram

0 // Rn ⊕Rn

(A 00 In

)

//

C

²²

Rn ⊕Rn( πA 0 )//

D

²²

Rn/ARn //

φ²²

0

0 // Rn ⊕Rn

(B 00 In

)

// Rn ⊕Rn( πB 0 )// Rn/BRn // 0

in which πA and πB denote projection maps. It follows that, in K1(S−1R),

[AB−1] = [A⊕ In][B ⊕ In]−1 = [D]−1[B ⊕ In][C][B ⊕ In]−1 = [CD−1]

is in the image of K1(ρS).In the general case, we cannot be sure that the equation [Rn/ARn] = [Rn/BRn]

derives from an isomorphism as above. However, Proposition 2.6 ensures the exis-tence of short exact sequences 0 → L → Mi → N → 0, i = 1, 2, in CR

0 (f,pd,S-tor)such that (Rn/ARn) ⊕ M1

∼= (Rn/BRn) ⊕ M2. By switching temporarily toCR

1 (f,P|S-tor), we can even make sure that the modules L, M1, M2 and N haveprojective dimensions at most 1. Now apply Lemma 4.14 to find modules L′ andN ′ such that L ⊕ L′ and N ⊕ N ′ are co-kernels of square matrices with entriesin R. Adding an identity matrix to either of these matrices, we can even findthat the matrices have the same dimension m ∈ N. By adding the short exactsequences 0 → L′ → L′ → 0 → 0 and 0 → 0 → N ′ → N ′ → 0 to each of the shortexact sequences 0 → L → Mi → N → 0 and redefining Mi to be the moduleMi ⊕ L′ ⊕N ′ for i = 1, 2, we obtain short exact sequences in the form

0 → Rm/ERm → Mi → Rm/FRm → 0,

where E, F ∈ GLm(S−1R) are matrices with entries in R. We still have thatRn/ARn ⊕ M1

∼= Rn/BRn ⊕ M2. From the obvious projective resolutions ofRm/ERm and Rm/FRm, we can construct a projective resolution of Mi whosemodules are the direct sum of the corresponding modules in the resolutions ofRm/ERm and Rm/FRm. It follows that the module Mi is also the co-kernel of amatrix Gi ∈ GL2n(S−1R) with entries in R, so we now have

Rn/ARn ⊕R2m/G1R2m ∼= Rn/BRn ⊕R2m/G2R

2m.


This is the situation from before, so we can immediately conclude that, inK1(S

−1R),[A][G1][B]−1[G2]

−1 = [A⊕G1][B ⊕G2]−1

is in the image of K1(ρS). To derive that [A][B]−1 is in the image of K1(ρS), ittherefore only remains to prove that [G1] = [G2] in K1(S

−1R).By the construction of Gi there is a commutative diagram

0

²²

0

²²

0

²²0 // Rm

E²²

( Im0 )

// R2m( 0 Im ) //

Gi

²²

Rm

F²²

// 0

0 // Rm( Im

0 )//

²²

R2m

²²

( 0 Im ) // Rm //

²²

0

0 // Rm/ERm //

²²

R2m/GiR2m //

²²

Rm/FRm //

²²

0

0 0 0

From the commutativity of the diagram, it follows that Gi must be in the form

Gi =

(E G′

i

0 F

)

for some matrix G′i ∈ Mm(R). The proof of Whitehead’s lemma (Lemma 4.8)

showed that any matrix in the form(

Im ?0 Im

)is trivial in K1(S

−1R), so from thecalculation

Gi =

(E G′

i

0 F

)=

(Im G′

iF−1

0 Im

)(E 00 F

)

it follows that [Gi] = [E ⊕F ] in K1(S−1R). In particular, [G1] = [G2] as desired.

Exactness at GR0 (f,pd,S-tor): Suppose A ∈ GLn(S−1R) such that [A] is an

arbitrary element of K1(S−1R). Then δ([A]) = [Rn/(sA)Rn]−n[R/sR] for s ∈ S

chosen so that (sA) has entries in R, and since we have projective resolutions

0 −→ Rn (sA)−→ Rn −→ Rn/(sA)Rn −→ 0

and0 −→ R

s−→ R −→ R/sR −→ 0,

it follows that χ ◦ δ([A]) = [Rn]− [Rn] + [R]− [R] = 0.To show conversely that the kernel of χ is contained in the image of δ,

we switch from GR0 (f,pd,S-tor) to the isomorphic group GR

1 (f,P|S-tor). UsingLemma 4.14, it suffices to assume that we are given an element [X] ∈ GR

1 (f,P|S-tor)


that is in the kernel of χ: that is, such that [X0] = [X1] in K0(R). It then followsfrom Corollary 2.7 (together with [Mag02, Proposition 2.21]) that we can find afinitely generated projective module L such that X0⊕L ∼= X1⊕L ∼= Rn for somen ∈ N. Letting Z denote the exact complex 0 → L → L → 0 concentrated indegrees 1 and 0, X ⊕ Z is then nothing but a complex 0 → Rn → Rn → 0 thatbecomes exact when localizing at S. The matrix associated with this complexmust therefore be in GLn(S−1R), so [X] = [X ⊕ Z] is in im δ.

Exactness at K0(R): We immediately switch from GR0 (f,pd,S-tor) to the iso-

morphic group GR1 (f,P|S-tor). Suppose X is a complex in CR

1 (f,P|S-tor). ThenS−1X is exact, so S−1X0

∼= S−1X1, and it follows that K0(ρS) ◦ χ([X]) =[S−1X0]− [S−1X1] = 0 in K0(S

−1R).Conversely, suppose that [M ]− [N ] is an element in the kernel of K0(ρS): that

is, such that [S−1M ] = [S−1N ] in K0(S−1R). Using Corollary 2.7, we can then

find n ∈ N0 such that S−1M ⊕ (S−1R)n ∼= S−1N ⊕ (S−1R)n. It follows that thereis a homomorphism f : N ⊕Rn → M ⊕Rn that induces an isomorphism

S−1f : S−1N ⊕ (S−1R)n → S−1M ⊕ (S−1R)n

when localized at S (cf. [Eis95, Proposition 2.10]), and hence

X = 0 −→ N ⊕Rn f−→ M ⊕Rn −→ 0

is a complex in CR1 (f,P|S-tor) with χ([X]) = [M ⊕ Rn] − [N ⊕ Rn] = [M ] − [N ]

as desired.

Corollary 4.17. If R is local, K0(ρS) is injective, and the sequence

K1(R)K1(ρS)−→ K1(S

−1R)δ−→ GR

0 (f,pd,S-tor) −→ 0

is exact.

Proof: As shown in Example 2.14, there is an isomorphism Z → K0(R) givenby 1 7→ [R] and an injection Z→ K0(S

−1R) given by 1 7→ [S−1R]. Since K0(ρS)maps [R] to [S−1R], it follows that K0(ρS) must be injective, and Theorem 4.16implies the exactness of the sequence.

Corollary 4.17 allows a determinant map

detS : GR0 (f,pd,S-tor) → (S−1R)∗/R∗

to be induced from the determinant map detS−1R : K1(S−1R) → (S−1R)∗, such

that the diagram

K1(R)K1(ρS)//

detR

²²

K1(S−1R)

δ //

detS−1R

²²

GR0 (f,pd,S-tor)

detS

²²

// 0

0 // R∗ ρS // (S−1R)∗ // (S−1R)∗/R∗ // 0


is commutative: given [M ] in GR0 (f,pd,S-tor), we choose [A] in K1(S

−1R) sothat δ([A]) = [M ], and we define detS [M ] to be the coset of detS−1R [A] in(S−1R)∗/R∗. This is well defined, for if we choose [A] and [A′] in K1(S

−1R) sothat δ([A]) = δ([A′]) = [M ], then [A−1A′] is in the image of K1(ρS), and hence(detS−1R [A])−1 detS−1R[A′] is in the image of ρS.

Definition 4.18. When R is local, the homomorphism GR0 (f,pd,S-tor) → (S−1R)∗

induced by detS−1R : K1(S−1R) → (S−1R)∗ is denoted by detS. Given a module

M in CR0 (f,pd,S-tor), we denote detS[M ] simply by detS M .

When R is local, the surjection δ shows that GR0 (f,P,S-tor) is generated by

elements [Rn/ARn], where n ∈ N and A is a matrix in GLn(S−1R) with entries inR, or equivalently, an n×n matrix over R that becomes invertible when localizedat S. Given such an element [Rn/ARn], detS [Rn/ARn] is the coset of detR A/1in (S−1R)∗/R∗.

If S−1R is generalized Euclidean, everything looks much nicer:

Corollary 4.19. If R is local and S−1R is generalized Euclidean, then there isa commutative diagram

0 // K1(R)K1(ρS)//

detR

²²

K1(S−1R)

δ //

detS−1R

²²

GR0 (f,pd,S-tor)

detS

²²

// 0

0 // R∗ ρS // (S−1R)∗ // (S−1R)∗/R∗ // 0

in which the rows are exact and the columns are isomorphisms.

Proof: According to Proposition 4.5, both R and S−1R are generalized Eu-clidean, so it follows from Theorem 4.11 that the determinant maps detR anddetS−1R are isomorphisms. This immediately implies that detS is an isomorphismas well and that K1(ρS) is injective.

The fact that detS is an isomorphism in the above setting shows that everyelement in GR

0 (f,pd,S-tor) is in the form [R/u] − [R/s], where u ∈ R and s ∈ Sare elements such that u/s ∈ (S−1R)∗. It turns out that, for any module M inCR

0 (f,pd,S-tor), [M ] = [R/u] for some u ∈ R with u/1 ∈ (S−1R)∗. The remainderof this chapter is dedicated to proving this.

In [Mac65] MacRae introduced an ideal, later known as the MacRae ideal,associated to every finitely generated module of finite projective dimension andgrade greater than or equal to 1. Letting M be such a module, the MacRaeideal is denoted by GR(M). We will not present the actual definition of theMacRae ideal here, nor will we prove the theorem below, which describes some ofthe properties of the MacRae ideal in the case where R is Noetherian and local(cf. [Fox82b, Lemma (0.1)]).


Theorem 4.20. Suppose that R is a Noetherian, local ring and that L, M andN are modules in CR

0 (f,pd,gr ≥ 1). Then

(i) GR(M) is a principal ideal;

(ii) GR(M) is generated by detR A whenever M ∼= Rn/ARn for some injectivematrix A;

(iii) GR(M) 6= R if and only if gradeR M = 1;

(iv) GR(M) = GR(L)GR(N) whenever there is a short exact sequence 0 → L →M → N → 0; and

(v) VR(GR(M)) ⊆ SuppR M .

The theorem below shows that the determinant detS M from Definition 4.18generates the MacRae ideal GR(M). For the proof of this, note that, if Mis a module in CR

0 (f,pd,S-tor), then gradeR M ≥ 1 since S contains only non-zerodivisors, and hence the MacRae ideal is defined for M .

Theorem 4.21. Suppose that R is Noetherian and local, and that M is a modulein CR

0 (f,pd,S-tor). Then detS M is the coset in (S−1R)∗/R∗ of an element u/1such that GR(M) = 〈u〉.Proof: The proof is by induction on p = pdR M . The cases p ≤ 0 are trivial,for in these cases M is projective and hence free, and the only way for M to beS-torsion is then if M = 0, in which case detS M = [1/1]R∗ and GR(M) = R (thelatter by Theorem 4.20(iii)).

In the case that p = 1, M is the homology of a complex

0 −→ Rn A−→ Rm −→ 0

for m,n ∈ N0 and an injective m×n matrix A. Now, the fact that M is S-torsionimplies that AnnR M 6= 0, and from Theorem 0.8 it then follows that the (tradi-tional) Euler characteristic of M is χR(M) = 0. Since the Euler characteristic isadditive on short exact sequences, the exact sequence 0 → Rn → Rm → M → 0shows that m = n. Thus, M is the co-kernel of a matrix A in GLn(S−1R) withentries in R, and it follows that detS M is the coset of detR A/1 in (S−1R)∗/R∗.Conversely, Theorem 4.20(ii) implies that GR(M) = 〈detR A〉.

For the general case, suppose that p > 1 and that the theorem has beenproven for smaller values of p. Choose a finitely generated free module F anda surjective homomorphism f : F → M . Since M is S-torsion, we can chooses ∈ S ∩ AnnR M , and we obtain an induced homomorphism f : F/sF → Mthat is also surjective. Letting K denote the kernel of f , we then have an exactsequence

0 → K → F/sF → M → 0. (4.1)


Since pdR F/sF = 1, Theorem 0.7 yields pdR K = p − 1. By construction,F/sF and K are S-torsion, so F and K are modules in CR

0 (f,pd,S-tor), and theinduction hypothesis implies that detS(F/sF ) = [v/1]R∗ and detS K = [w/1]R∗for elements v, w ∈ R such that GR(F/sF ) = 〈v〉 and GR(K) = 〈w〉.

Let u ∈ R be a generator for GR(M). From the short exact sequence in (4.1)together with Theorem 4.20(iv), we then find that 〈v〉 = 〈wu〉 and that [v/1]R∗ =[w/1]R∗ detS M . From the first equation, it follows that [v/1]R∗ = [wu/1]R∗ , andfrom the second equation it then follows that detS M = [v/1]R∗ [w/1]−1

R∗ = [u/1]R∗as desired.

Theorem 4.21 shows that GR(M) is generated by an element u ∈ R satisfyingthe condition that u/1 is a unit in S−1R. Consequently, there exists an elementu′ ∈ R such that uu′ is in S, and since this element also is in the annihilatorof R/GR(M), it follows that R/GR(M) is a module in CR

0 (f,pd,S-tor). UsingTheorem 4.21 together with Corollary 4.19, we therefore immediately obtain thefollowing corollary.

Corollary 4.22. If R is Noetherian and local, S−1R is generalized Euclidean andM is a module in CR

0 (f,pd,S-tor), then [M ] = [R/GR(M)] in GR0 (f,pd,S-tor).

The grade conjecture states that, if R is Noetherian and local and M is finitelygenerated with finite projective dimension, then gradeR M + dimR M = dim R.We have already noted in the preliminaries that Theorem 0.8 implies that thegrade conjecture holds whenever gradeR M = 0. This chapter concludes by show-ing that the properties of the MacRae ideal can be used to verify the gradeconjecture in the case that gradeR M = 1.

Proposition 4.23. If R is Noetherian and local and M is a finitely generatedmodule with pdR M < ∞ and gradeR M ≥ 1, then

gradeR M = 1 ⇐⇒ GR(M) 6= R ⇐⇒ dimR M = dim R− 1.

Proof: The first bi-implication is stated in Theorem 4.20. In addition, sincegradeR M + dimR M ≤ dim R, the equivalences of Theorem 0.8 ensure thatdimR M = dim R−1 implies gradeR M = 1. For the remaining part of the propo-sition, we assume that gradeR M = 1 and want to show that dimR M = dim R−1.But we cannot have dimR M = dim R according to Theorem 0.8, and sinceSuppR(R/GR(M)) ⊆ SuppR M and GR(M) is a proper principal ideal accordingto (i), (iii) and (v) of Theorem 4.20, it follows from Theorem 0.12 that

dim R− 1 ≤ dimR(R/GR(M)) ≤ dimR M.

Consequently, dimR M = dim R− 1.

5 Chapter 5

Local Chern characters

The previous chapters have associated with a complex X an element [X] of anAbelian group through a map that is additive on short exact sequences of com-plexes. This chapter does something similar, associating with each complex Xa family chR(X) of linear maps between Q-vector spaces in a way that, in somesense, is additive on short exact sequences. The family chR(X) is the local Cherncharacter of X.

The very definition of local Chern characters is by far too intricate to beincluded in this thesis, whereas the many nice properties of local Chern characterscan easily be understood. Consequently, almost none of the theorems here arepresented with proof; for more details, the reader is referred to [Rob98].

5.1 Chow groups

Throughout this section, R is assumed to be Noetherian.

The spectrum of a ring can be endowed with a topology known as the Zariskitopology. The closed subsets of Spec R are the sets VR(I) for all ideals I in R.This does, indeed, define a topology on Spec R, since Spec R = VR(0), ∅ = VR(R),

VR(I1) ∪ · · · ∪ VR(In) = VR(I1 · · · In)

whenever I1, . . . , In are ideals of R, and

⋂j

VR(Ij) = VR(∑

j

Ij)

whenever (Ij)j is a family of ideals of R. (The sum of ideals in the last equationis the set of all finite sums of elements from each of the ideals.)

Given an element p of Spec R, the closure of {p} in the Zariski topology is theset VR(p); this is indeed a closed set containing p, and it must be the smallestsuch set, for if VR(I) is any other closed set containing p, then we must have I ⊆ p

and thereby VR(p) ⊆ VR(I). It follows that the only closed points of Spec R arethe maximal ideals.

75

76 Local Chern characters

If V is a closed subset of Spec R, the dimension of V is the number dimR V

defined by dimR V = dimR R/I for an ideal I such that VR(I) = V: that is,dimR V is the supremum of the lengths of chains of prime ideals in V.

The support of a finitely generated module M is, as mentioned in the pre-liminaries, the closed subset SuppR M = VR(AnnR M) of Spec R. We extend thedefinition of support to complexes, defining the support of a complex X ∈ CR

¤(f)to be the subset

SuppR X = {p ∈ Spec R |Xp is exact}.If X ∈ CR

¤(f,P) and Y ∈ CR¤(f), then SuppR(X ⊗R Y ) = SuppR X ∩ SuppR Y

(cf. [Fox98, Lemma 14.5]). If 0 → X → Y → Z → 0 is a short exact sequence ofcomplexes in CR

¤(f), then SuppR Y ⊆ SuppR X ∪ SuppR Z; this is apparent fromthe long exact sequence of homology modules.

We define the annihilator of X ∈ CR¤(f) to be the product of the ideals

AnnR H`(X), among which all but finitely many are equal to R: that is,

AnnR X = · · · (AnnR H`+1(X))(AnnR H`(X)) · · · .

It follows that SuppR X = VR(AnnR X); in particular, SuppR X is closed in theZariski topology.

Definition 5.1. If V is a closed subset of Spec R and i is an integer, let Vi ={p ∈ V |dimR R/p = i} and let ZR

i (V) denote the free Abelian group whose basisconsists of the symbols [R/p] for p ∈ Vi. In particular, ZR

i (∅) is the trivial group.

The notation “[R/p]” is purely symbolic: even if R/p ∼= R/q for prime idealsp and q, this does not mean that [R/p] is identical with [R/q]. In case R isan integral domain, we shall, however, deviate slightly from this policy of strictsymbolism, writing [R] instead of [R/0].

Let M be a finitely generated module with dimR M ≤ i and let p ∈ Vi forsome closed subset V of Spec R. Recall that the localization Mp is nontrivial ifand only if p contains AnnR M , in which case p is minimal in VR(AnnR M) andMp has finite length as an Rp-module according to Theorem 0.10. As mentionedin the preliminaries, there are only finitely many minimal primes over an idealwhen R is Noetherian, so it follows that the sum

∑p∈Vi

lengthRp(Mp)[R/p] is

finite and therefore determines a well-defined element of ZRi (V).

Definition 5.2. Let V be a given closed subset of Spec R. For a finitely generatedmodule M with dimR M = i, we define an element [M ]V in ZR

i (V) by

[M ]V =∑

p∈Vi

lengthRp(Mp)[R/p].

Furthermore, for a prime q ∈ Vi+1 and an element x ∈ R, we define an elementdivV(q, x) by

divV(q, x) =

{[R/(q + 〈x〉)]V, if x /∈ q,

0, if x ∈ q.

5.2 Local Chern characters 77

The quotient of the group ZRi (V) by the subgroup generated by the elements

divV(q, x) for all q ∈ Vi+1 and x ∈ R is denoted by ARi (V). Two elements of

ZRi (V) are said to be rationally equivalent if they represent the same element in

ARi (V). The Chow group AR(V) of V is the direct sum of the AR

i (V)’s. Therational Chow group is the group (or Q-vector space) AR(V)Q = AR(V) ⊗Z Qwhich is the direct sum of the groups AR

i (V)Q = ARi (V)⊗Z Q. The image of an

element [R/p] ∈ ZRi (V) in AR(V)Q shall also be denoted by [R/p]. Likewise, the

image of [M ]V ∈ ZRi (V) in AR(V)Q shall also be denoted by [M ]V, and the i’th

component of such an element (which is nontrivial only if i = dimR M) shall bedenoted [M ]Vi .

We shall only consider rational Chow groups, so from now on, these aresimply referred to as Chow groups. The i’th component AR

i (V)Q of a Chow groupAR(V)Q is nontrivial only if V contains prime ideals p with dimR R/p = i; henceAR(V)Q is concentrated in degrees 0, . . . , dimR V (or 0, 1, . . . if dimR V = ∞).

Note that, if we consider the module R/p for a prime p contained in Vi forsome closed subset V ⊆ Spec R, then [R/p]V = [R/p] in ZR

i (V) and hence inAR(V)Q. Also note that, by definition, we must have that AR(∅)Q is equal to thetrivial group, and that AR({m})Q for any maximal ideal m is equal to the groupgenerated as a Q-vector space by [R/m]: that is, AR({m})Q ∼= Q.

If V ⊆ W is an inclusion of closed subsets of Spec R, there is a naturalhomomorphism ιV,W

i : ARi (V)Q → AR

i (W)Q given by ιV,Wi ([R/p]) = [R/p] for

all p ∈ Vi. Note here that (similarly to the case of Grothendieck groups) thetwo [R/p]’s are different: one is an element of AR

i (V)Q, whereas the other isan element of AR

i (W)Q. The fact that ιV,Wi is induced by the inclusion map

ZRi (V) → ZR

i (W) does not mean that ιV,Wi is injective—it only ensures that it

is well defined. When the domain ARi (V)Q and co-domain AR

i (W)Q of a naturalmap ιV,W

i are clear from the context, we shall simply denote the natural map byι; indeed, ι can be thought of as a family of maps ιV,W

i indexed by nonnegativeintegers i ∈ N0 and closed subsets V ⊆ W ⊆ Spec R.

5.2 Local Chern characters

Throughout this section, R is assumed to be Noetherian and local with max-imal ideal m and quotient field k = R/m. Furthermore, it is assumed that Ris the homomorphic image of a regular local ring Q: that is, R = Q/I for anideal I of Q.

Suppose X ∈ CR¤(f,P) and let X denote the support of X. Recall that X is closed.

The local Chern character of X is a family chR(X) of homomorphisms

chRi (X)j,V : AR

j (V)Q → ARj−i(V ∩ X)Q


for all i, j ∈ N0 and all closed subsets V of Spec R. Note that, since the groupAR

j (V)Q is nontrivial only if 0 ≤ j ≤ dimR V, the map chRi (X)j,V is nontrivial

only if i ≤ j ≤ dimR V and j − i ≤ dimR V ∩ X. For convenience, we definechR

i (X)j,V to be the zero map ARj (V)Q → AR

j−i(V ∩ X)Q when i is a negativeinteger.

For all i ∈ N0 and all closed subsets V of Spec R, we define chRi (X)V to be the

homomorphism AR(V)Q → AR(V ∩ X)Q given in degree j by chRi (X)j,V. In this

way, chRi (X)V is a homomorphism AR(V)Q → AR(V∩X)Q of degree −i. The i’th

local Chern character of X is the family chRi (X) of homomorphisms chR

i (X)V forall closed subsets V of Spec R. In general, we can think of the i’th local Cherncharacter as an operator of degree −i on Chow groups. For any closed subset V

of Spec R, we define chR(X)V to be the homomorphism AR(V)Q → AR(V ∩X)Qwhose (j, j − i)-entry is chR

i (X)j,V.When given a local Chern character chR(X) it determines a homomorphism

chRi (X)j,V : AR

j (V)Q → ARj−i(V∩X)Q for any choice of i, j ∈ N0 and closed subset

V of Spec R. We shall often compose chRi (X)j,V with a natural homomorphism

ιV∩X,Wj−i , thereby obtaining a map AR

j (V)Q → ARj−i(W)Q for any choice of i, j ∈ N0

and closed subsets V,W of Spec R with V ∩ X ⊆ W. Whenever it is clear fromthe context what the domain and co-domain are (or if the choice of these isunimportant), we shall simply denote such a map by chR(X).

We can add local Chern characters in the following way. Suppose that Xand Y are complexes in CR

¤(f,P) with supports X and Y, respectively. The sumchR(X) + chR(Y ) is then defined to be a family of maps

(chR(X) + chR(Y ))j,Vi : AR

i (V)Q → ARj−i(V ∩ (X ∪Y))Q

for all i, j ∈ N0 and closed subsets V of Spec R, given by

(chR(X) + chR(Y ))j,Vi = chR

i (X) + chRi (Y ),

in which we consider chRi (X) and chR

i (Y ) as maps ARj (V)Q → AR

j−i(V∩(X∪Y))Q.It is clear that the addition is commutative. At first, calling this constructionan addition of local Chern characters may seem odd, since it is not clear thatthe sum of two local Chern characters is itself a local Chern character; this,however, is the case, as follows from Proposition 5.3(iii) below, which shows thatchR(X) + chR(Y ) = chR(X ⊕ Y ).

The local Chern character chR(0) of the zero complex is a neutral elementwith respect to the addition described above. We shall denote this local Cherncharacter by 0 and refer to it as the trivial local Chern character.

A kind of inverse with respect to the addition also exists. Suppose thatX ∈ CR

¤(f,P) has support X. Then − chR(X) is defined to be a family of maps

(− chR(X))j,Vi : AR

j (V)Q → ARj−i(V ∩ X)Q


for all closed subsets V ⊆ Spec R and i, j ∈ N0, given by

(− chR(X))j,Vi = − chR

i (X)j,V.

Again, it is not clear that− chR(X) is itself a local Chern character; this, however,is in fact the case, as follows from Proposition 5.3(iii) and (iv) below, which showthat − chR(X) = chR(ΣX) − chR(M(1X)) = chR(ΣX). It is tempting to thinkthat − chR(X) is an inverse to chR(X) under the addition so that their sum isequal to the trivial local Chern character, but this is not entirely true. Althoughthe sum chR(X) +(− chR(X)) is a family of zero maps AR

j (V)Q → ARj−i(V∩X)Q,

this family is not the same as the family of zero maps ARj (V)Q → AR

j−i(∅)Qdefined by the trivial local Chern character. By abuse of notation, however, weshall allow ourselves to write chR

i (X) = 0 whenever chRi (X) is a family of zero

maps, regardless of whether the support of X is empty.We can multiply local Chern characters in the following way. Suppose that

X and Y are complexes in CR¤(f,P) with supports X and Y, respectively. The

product chR(X) chR(Y ) is then defined to be a family of maps

(chR(X) chR(Y ))j,Vi : AR

j (V)Q → ARj−i(V ∩ X ∩Y)Q

for all i, j ∈ N0 and closed subsets V of Spec R, given by

((chR(X) chR(Y ))j,Vi =

∑m+n=i

chRm(X) chR

n (Y ),

in which we consider chRn (Y ) as a map AR

j (V)Q → ARj−n(V ∩ Y)Q and chR

m(X)as a map AR

j−n(V ∩ Y)Q → ARj−m−n(V ∩ X ∩ Y)Q. Once again, calling this

construction a multiplication may seem odd, since it is not clear that the productof two local Chern characters is itself a local Chern character; this, however,is indeed the case, as follows from Proposition 5.3(iv) below, which states thatchR(X) chR(Y ) = chR(X ⊗R Y ). This said, it is also clear that the local Cherncharacter chR(R) of the complex R concentrated in degree 0 is a neutral elementfor the multiplication of local Chern characters.

We now list some of the properties of local Chern characters (cf. [Rob85] and[Rob98, Theorem 12.1.2 and Corollary 11.4.1]).

Proposition 5.3. If X, Y and Z are complexes in CR¤(f,P) with supports X, Y

and Z, respectively, then the following hold.

(i) chR(X)ι = ι chR(X): that is, if U ⊆ V ⊆ Spec R are closed and i, j ∈ N0,then the diagram

ARj (U)Q

chR(X)//

ι

²²

ARj−i(U ∩ SuppR X)Q

ι

²²AR

j (V)QchR(X)// AR

j−i(V ∩ SuppR X)Q

is commutative.


(ii) chRm(X) chR

n (Y ) = chRn (Y ) chR

m(X): that is, if V ⊆ Spec R is closed andj, m, n ∈ N0, then the diagram

ARj (V)Q

chR(X) //

chR(Y )²²

ARj−m(V ∩ X)Q

chR(Y )²²

ARj−n(V ∩Y)Q

chR(X)// ARj−m−n(V ∩ X ∩Y)Q

is commutative. In particular, chR(X) chR(Y ) = chR(Y ) chR(X).

(iii) chR(Y ) = chR(X) + chR(Z) whenever there is a short exact sequence 0 →X → Y → Z → 0: that is, if V ⊆ Spec R is closed and i, j ∈ N0, then inthe diagram

ARj−i(V ∩ X)Q

ι

))SSSSSSSSSSSSSS

ARj (V)Q

chR(X)55kkkkkkkkkkkkkkk

chR(Y ) //

chR(Z) ))SSSSSSSSSSSSSSSAR

j−i(V ∩Y)Qι // AR

j−i(V ∩ (X ∪ Z))Q

ARj−i(V ∩ Z)Q

ι

55kkkkkkkkkkkkkk

the sum of the upper and lower maps equals the middle map.

(iv) chR(X) = 0 whenever X is exact.

(v) chR(X ⊗R Y ) = chR(X) chR(Y ): that is, if V ⊆ Spec R is closed, then thediagram

AR(V)QchR(Y ) //

chR(X⊗RY ) ))TTTTTTTTTTTTTTTT AR(V ∩Y)Q

chR(X)²²

AR(V ∩ X ∩Y)Q

is commutative. In particular, if i, j ∈ N0, then

chRi (X ⊗R Y )j,V =

∑m+n=i

chRm(X)j−n,V∩Y chR

n (Y )j,V.

Part (i) of Proposition 5.3 states that, when a local Chern character chRi (X)V

is applied to an element η that is a linear combination of [R/p]’s, it does notreally matter in what Chow group we choose to place η. To explain this ingreater detail, let X be a complex in CR

¤(f,P) with support X, and let η be aformal sum in the form

η = n1[R/p1] + · · ·+ nt[R/pt]


for n1, . . . , nt ∈ Z and p1, . . . , pt ∈ Spec R. For any closed subset W of Spec Rwith VR(p1 · · · pt) ∩ X ⊆ W, we can then let

chR(X)(η)def= chR(X)V(η) ∈ AR(W)Q

and

chRi (X)(η)

def= chR

i (X)V(η) ∈ AR(W)Q,

for any choice of closed subset V of Spec R containing {p1, . . . , pt} and satisfyingthe condition that V ∩ X ⊆ W. (On the right side of the equations, we haveconsidered η as an element in AR(V)Q.) This is done completely without referenceto V; the only thing that matters is that we specify W and that W is sufficientlylarge.

Where the local Chern characters are operators on Chow groups associated toevery bounded complex of free and finitely generated modules, there is an elementof a Chow group associated to every bounded complex of finitely generated (notnecessarily free) modules. Given a complex X in CR

¤(f) with support X, the Toddclass of X is an element τR(X) in the Chow group AR(X)Q. The component indegree i of τR(X) is denoted by τR

i (X). By applying the natural homomorphismι, we may choose to consider τR(X) as an element in AR(V)Q for any closedsubset V of Spec R with X ⊆ V.

In order to define the Todd classes, one needs to know that R is the homo-morphic image of a regular local ring; this is why we have made this assumptionthroughout the present section. Todd classes are defined in terms of local Cherncharacters, whose definition we have avoided, so there is no reason to presentthe actual definition of Todd classes here. Instead, we list a few of their manynice properties (cf. [Rob87, page 8] and [Rob98, Corollary 12.4.3 and Proposi-tion 12.4.4]).

Proposition 5.4. If X, Y and Z are complexes in CR¤(f) with supports X, Y and

Z, respectively, then the following hold.

(i) τR(Y ) = τR(X) + τR(Z) in AR(X ∪ Z)Q whenever there is a short exactsequence 0 → X → Y → Z → 0.

(ii) τR(X) = 0 whenever X is exact.

(iii) τR(X ⊗R Y ) = chR(X)(τR(Y )) whenever X ∈ CR¤(f,P).

(iv) τRi (X) =

∑`∈Z(−1)`[H`(X)]Xi whenever the homology modules of X have

dimensions not exceeding i.

(v) τR(R) = [R]Spec R whenever R is a complete intersection.


If X and Y are complexes in CR¤(f) such that there is a homology isomorphism

φ : X'−→ Y , then their Todd classes are equal; this follows by applying (i) and

(ii) of the proposition to the short exact sequences 0 → Y →M(φ) → ΣX → 0and 0 → X →M(1X) → ΣX → 0.

Part (iii) of the proposition is known as the local Riemann–Roch formula.From this follows that, whenever X ∈ CR

¤(f,P),

τR(X) = τR(X ⊗R R) = chR(X)(τR(R)). (5.1)

Combining this with the observation from the previous paragraph we find that,if X ∈ CR

¤(f,P) is a projective resolution of M ∈ CR0 (f), then

τR(M) = chR(X)(τR(R)).

According to part (v) of the proposition, this is particularly interesting when Ris a complete intersection.

From part (iv) of the proposition, it follows that if X ∈ CR¤(f|l),

τR(X) = H([X])[R/m] (5.2)

in AR({m})Q; here H denotes the homomorphism GR¤(f|l) → GR

0 (l) from The-orem 2.11 (which is applicable since GR

0 (l) contains the kernels of all its homomor-phisms) in which we have used the identification of GR

0 (l) with Z as establishedin Example 2.16; thus H([X]) =

∑`∈Z(−1)` lengthR H`(X). From part (iv), it

also follows that, if M ∈ CR0 (f), then

τR(M) = [M ]SuppR M + terms of lower degree. (5.3)

This chapter concludes by describing how the zeroth and first local Cherncharacters are closely related to the Euler characteristic and the MacRae ideal.

Proposition 5.5. If X ∈ CR¤(f,P), then chR

0 (X) is multiplication by χR(X): thatis, if V ⊆ Spec R is closed, j ∈ N0 and X is the support of X, then the mapchR

0 (X)j,V : ARj (V)Q → AR

j (V ∩ X)Q is given by

chR0 (X)j,V[R/p] = χR(X)[R/p]V∩X

for all p ∈ Vj.

Proof: Note first that according to (5.3), [R/p] = τRj (R/p) and τR

i+j(R/p) = 0for all i > 0. Thus, since SuppR R/p = VR(p) ⊆ V, we find in AR

j (V ∩ X)Q that

chR0 (X)j,V[R/p] = chR

0 (X)j,VR(p)[R/p]

= chR0 (X)j,VR(p)(τR

j (R/p))

=∑

i∈N0

chRi (X)i+j,VR(p)(τR

i+j(R/p))


is equal to the j’th component of chR(X)VR(p)(τR(R/p)). By the local Riemann–Roch formula (Proposition 5.4(iii)), this is equal to τR

j (X⊗RR/p) in ARj (V∩X)Q,

and by Proposition 5.4(iv) this again is equal to∑

`∈Z(−1)`[H`(X⊗RR/p)]VR(p)∩Xj .

If p /∈ X, this is equal to 0 in ARj (V ∩ X)Q, and the theorem is proved since

[R/p]V∩X = 0 in this case. If p ∈ X, we continue our calculation:

chR0 (X)j,V[R/p] =

∑

`∈Z(−1)`[H`(X ⊗R R/p)]

VR(p)∩Xj

=∑

`∈Z(−1)` lengthRp

(H`(X ⊗R R/p))p[R/p]

=∑


H`((X ⊗R R/p)p)[R/p]

=∑


H`(Xp ⊗Rp (R/p)p)[R/p]

= χRp(Xp)[R/p]

= χR(X)[R/p].

For the third equality, we have used the fact that localization is an exact functoron modules and therefore induces a functor on complexes that commutes withthe homology functor, as described in the preliminaries. The fourth equality isalso explained in the preliminaries. This proves the proposition.

Note that since X is a projective resolution of itself, χR(X) is just the alter-nating sum of the ranks of the modules in X. Consequently, if p is a prime fromoutside the support of X, then χR(X) = χRp(Xp) = 0. It follows that χR(X)and hence chR

0 (X) is nonzero only if SuppR X = Spec R. Thus, in this case,chR

0 (X) does indeed act as a multiplication, namely multiplication by χR(X) oneach Chow group AR(V)Q.

Corollary 5.6. If X ∈ CR¤(f,P), then chR

0 (X) 6= 0 if and only if χR(X) 6= 0.

Proof: “only if” is clear, and “if” follows since chR0 (X)[R/m] = χR(X)[R/m] is

nonzero in AR({m})Q ∼= Q whenever χR(X) 6= 0.

To describe the first local Chern character, we need to introduce a new op-eration on Chow groups. The operation is commonly known as intersection witha divisor, where the divisor in this case is a principal ideal. The definition isstated below; one should of course verify that it is well defined, that is, thatit preserves rational equivalence, but this will be omitted (see instead [Rob98,Proposition 8.8.1]).

Definition 5.7. For every x ∈ R and every closed subset V ⊆ Spec R, we definea homomorphism 〈x〉∩− : AR(V)Q → AR(V∩VR(x))Q, referred to as intersection


with 〈x〉, by

〈x〉 ∩ [R/p] =

{[R/(p + 〈x〉)]V∩VR(x), if x /∈ p,

0, if x ∈ p,

for every p ∈ V.

If dimR R/p = i, then 〈x〉 ∩ [R/p] is an element in ARi−1(V∩VR(x))Q; in other

words, 〈x〉 ∩ − has degree −1. Note that the element 〈x〉 ∩ [R/p] corresponds tothe element divV(p, x), except that we are evaluating in AR(V ∩ VR(x))Q ratherthan AR(V)Q.

As with local Chern characters, one often composes the intersection map〈x〉 ∩ − : AR(V)Q → AR(V ∩ VR(x))Q with a natural homomorphism ιV∩VR(x),W,thereby obtaining a map AR(V)Q → AR(W)Q for any choice of closed subset W

of Spec R with V ∩ VR(x) ⊆ W. We shall also denote this map by 〈x〉 ∩ −.We can now describe the relation between local Chern characters and the

MacRae ideal—as always without proof (see instead [Rob87]).

Proposition 5.8. If X ∈ CR¤(f,P) is a projective resolution of a module M in

CR0 (f,pd,gr ≥ 1), then chR

1 (X) is intersection with GR(M): that is, if V ⊆ Spec Ris closed, j ∈ N0 and X is the support of X (and hence of M), then the mapchR

1 (X)j,V : ARj (V)Q → AR

j−1(V ∩ X)Q is given by

chR1 (X)j,V[R/p] = GR(M) ∩ [R/p],

for all p ∈ Vj.

Note that, for the proposition to make sense, we are considering GR(M)∩− asa map from AR(V)Q → AR(V∩X)Q; this is allowed according to Theorem 4.20(v),since VR(GR(M)) ⊆ X.

Corollary 5.9. If X ∈ CR¤(f,P) is a projective resolution of a module M in

CR0 (f,pd,gr ≥ 1), then chR

1 (X) 6= 0 if and only if GR(M) 6= R.

Proof: “only if” is clear. For the other direction, let x1 ∈ R be a non-zerodivisorgenerating GR(M). Then, according to Proposition 0.15, x1 is part of a system ofparameters x = (x1, x2, . . . , xd) for R, where d = dim R. From Theorem 0.12, itnow follows that dimR /〈x2, . . . , xd〉 = 1, and hence we can choose a prime p 6= m

that is minimal over 〈x2, . . . , xd〉. Thus, p must satisfy the condition that x1 /∈ p

and dimR R/p = 1; in particular, VR(p)∩ VR(x1) = {m}. Applying the first localChern character of X to [R/p], we now get

chR1 (X)[R/p] = GR(M) ∩ [R/p]

= [R/p + GR(M)]{m}

= lengthR(R/(p + GR(M))[R/m],


which is nonzero in AR({m})Q ∼= Q. Consequently chR1 (X) 6= 0.

We have defined the map 〈x〉∩− for every element x of R. One can generalizethis definition to include all elements of (R\Zd R)−1R by letting 〈r/s〉 ∩ − bedefined by 〈r/s〉 ∩ [R/p] = 〈r〉 ∩ [R/p]− 〈s〉 ∩ [R/p]. This definition depends onr/s only up to multiplication with a unit, so we can define an intersection map forevery element of ((R\Zd R)−1R)/R∗ by letting [r/s]R∗ ∩− be equal to 〈r/s〉∩−.It would be interesting to know whether Proposition 5.8 can be generalized tohold for all complexes in CR

¤(f,P|S-tor) for some multiplicative system S withS ∩ Zd R = ∅, using the determinant detS instead of the MacRae ideal: that is,it would be interesting to know whether chR

1 (X) is the same as detS X ∩ − forall X ∈ CR

¤(f,P|S-tor).The Chern grade of a complex X ∈ CR

¤(f,P) is the number

Ch-gradeR X = inf{i ∈ N0 |chRi (X) 6= 0}.

It is natural to ask whether the Chern grade, similar to what has been proposedfor ordinary grade, satisfies the condition that

Ch-gradeR X = dim R− dimR X,

where dimR X = dimR(SuppR X). Corollaries 5.6 and 5.9 together with The-orems 0.8 and Theorem 4.23 verify that this indeed is true whenever X is a pro-jective resolution of a module and Ch-gradeR X ≤ 1. However, Dutta, Hochsterand McLaughlin’s counterexample for the intersection conjectures in the casewhere only one of the modules has finite projective dimension turns out to serveas a counterexample for this assertion.

Dutta, Hochster and McLaughlin constructed a ring R, a homomorphic imageof a regular local ring, with dim R = 3 and two modules M and N with dimR M =0 and dimR N = 2, such that χR(M, N) = −1. Let X ∈ CR

¤(f,P) be a projectiveresolution of M , so that χR(M,N) = H([X⊗RN ]), whereH is the homomorphismGR

¤(f|l) → GR0 (l) from Theorem 2.11 in which we have identified GR

0 (l) with Z asin Example 2.16. Note that chR

0 (X) = chR1 (X) = 0 according to Corollaries 5.6

and 5.9 together with Theorems 0.8 and Theorem 4.23. We then find that, inAR({m})Q,

−[R/m] = χR(M, N)[R/m]

= H[X ⊗R N ][R/m]

= τR(X ⊗R N) (according to (5.2))

= chR(X)(τR(N)) (using Proposition 5.4(iii))

= chR2 (X)(τR

2 (N)) (using Corollaries 5.6 and 5.9),

and it follows that chR2 (X) 6= 0 and thereby Ch-gradeR X = 2 6= dim R−dimR X.

6 Chapter 6

The intersection conjectures

This chapter returns to the discussion from Chapter 1 and applies the resultsestablished in chapters 2 through 5 to Serre’s intersection conjectures.

6.1 Serre’s intersection multiplicity

In this section, R is assumed to be Noetherian and local with maximal ideal m

and quotient field k = R/m. Furthermore, M and N denote finitely generatedmodules with pdR M < ∞ and dimR(M ⊗R N) = 0.

Let us begin by recalling the definition of the intersection multiplicity and theformulation of the intersection conjectures.

Definition 6.1. The intersection multiplicity of M and N is the number

χR(M, N) =∑

`∈Z(−1)` lengthR TorR

` (M,N).

This is well defined: the fact that pdR M < ∞ ensures that TorR` (M, N)

is nonzero for only finitely many `, and the fact that SuppR M ∩ SuppR N =SuppR(M ⊗R N) = {m} ensures that SuppR(TorR

` (M, N)) ⊆ {m}: that is,TorR

` (M,N) has finite length for all ` ∈ Z.

The intersection conjectures. The following hold.

(0) dimR M + dimR N ≤ dim R.

(1) χR(M, N) ≥ 0. (nonnegativity)

(2) χR(M, N) 6= 0 if and only if dimR M + dimR N = dim R.

Here conditions (1) and (2) can be replaced by

(1′) χR(M, N) = 0 if dimR M + dimR N < dim R. (vanishing)

(2′) χR(M, N) > 0 if dimR M + dimR N = dim R. (positivity)

87

88 The intersection conjectures

As mentioned in the introduction, Serre stated the conjectures in the casewhere R is a regular local ring, and he proved that condition (0) holds in this case.This chapter presents Foxby’s proof in [Fox82b] that the intersection conjectureshold when R is Noetherian and local (but not necessarily regular) and eithergradeR M ≤ 1 or dimR N ≤ 1 and Roberts’ proof in [Rob85] that the vanishingconjecture holds when R is a complete intersection (and hence, in particular,when R is a regular local ring).

First we elaborate a bit on the observations made in the introduction. The firstthing we notice is the trivial fact that the intersection multiplicity is commutative,in the sense that χR(M, N) = χR(N, M); this follows from the commutativity ofTorR(−,−). Defining the intersection multiplicity requires one of the modules tohave finite projective dimension, and because of commutativity, we can alwaysassume that M is such a module.

The second thing we notice is that χR(M,−) is additive on short exact se-quences: that is, if 0 → N ′ → N → N ′′ → 0 is an exact sequence of moduleson which χR(M,−) is defined, then χR(M, N) = χR(M, N ′) + χR(M, N ′′). Thisfollows by an inductive argument on the long exact Tor-sequence

· · · → TorR` (M, N ′) → TorR

` (M, N) → TorR` (M, N ′′) → TorR

`−1(M, N ′) → · · · ,

using methods similar to the ones used in the proof of Theorem 2.11 that thehomomorphism H is well defined. Similarly (or by commutativity), χR(−, N) isadditive on short exact sequences.

Additivity of the intersection multiplicity means that the maps χR(M,−) andχR(−, N) factor through Grothendieck groups of modules on which they can bedefined. This allows the theory obtained in chapters 2, 3 and 4 to be applied.

6.2 Applying algebraic K-theory



The proof of the intersection conjectures in the case dimR N = 0 is not veryhard and can easily be established without the use of theory from the precedingchapters. However, we include a proof here that uses some of this theory to givea flavor of how the use of Grothendieck groups can help compute the intersectionmultiplicity.

Theorem 6.2. The intersection conjectures hold if dimR N = 0.

Proof: The assumption immediately implies that (0) of the intersection con-jecture holds. Since SuppR N = {m}, χR(−, N) is defined on all modules of

6.2 Applying algebraic K-theory 89

finite projective dimension, and hence it factors through the Grothendieck groupGR

0 (f,pd). Thus, according to Corollary 3.39,

χR(M,N) = χR([M ], N) = χR(M)χR([R], N) = χR(M) lengthR N,

since, as seen directly from the definition, χR(R,N) = lengthR N . We haveassumed that N 6= 0 (since dimR(M⊗RN) = 0), so lengthR N is nonzero, and thevanishing and positivity conjectures follow immediately from Theorem 0.8.

We could also prove the conjectures by considering χR(M,−) instead: sinceχR(M,−) is defined on all modules of finite length, it factors through the Grothen-dieck group GR

0 (l). Thus, according to Example 2.16,

χR(M, N) = χR(M, [N ]) = χR(M, [k]) lengthR N = χR(M) lengthR N,

where we have used the fact that the intersection multiplicity χR(M, k) by defi-nition is equal to the Euler characteristic χR(M) of M .

Corollary 6.3. The intersection conjectures hold if gradeR M = 0.

Proof: According to the assumption we must have AnnR M ⊆ Zd R whichby Theorem 0.8 implies SuppR M = Spec R, and from the assumption thatdimR(M ⊗R N) = 0 follows SuppR N = {m}. Thus, the case that gradeR M = 0is covered by the case that dimR N = 0.

Theorem 6.4 (Foxby). The intersection conjectures hold if dimR N = 1.

Proof: We cannot have dimR M = dim R, because this implies SuppR M =Spec R according to Theorem 0.8, contradicting the fact that dimR(M⊗RN) = 0.Thus, we must have dimR M < dim R, so (0) of the intersection conjectures holds.

Now, as described in the preliminaries, N has a filtration

N = N0 ⊇ N1 ⊇ · · · ⊇ Nt = 0

in which the factors Ni−1/Ni are isomorphic to R/qi for prime ideals qi; the set{q1, . . . , qt} of prime ideals thus obtained is contained in SuppR N and containsAssR N . Using additivity of χR(M,−) on each of the exact sequences 0 → Ni →Ni−1 → R/qi → 0, we find that

χR(M,Ni−1) = χR(M,Ni) + χR(M, R/qi),

and it follows inductively that we can write χR(M,N) =∑t

i=1 χR(M, R/qi). Foreach of the qi’s, we have dimR R/qi ≤ dimR N = 1, and using Theorem 6.2 wecan ignore all terms for which dimR R/qi = 0. This shows that we can writeχR(M,N) as a sum of χR(M, R/qi)’s for those qi’s with dimR R/qi = 1. Thissum is nonempty since it includes all terms for which qi is minimal over AnnR N .


To show the vanishing and positivity conjectures, it therefore suffices to considerthe case where N = R/q for some prime ideal q.

We can apply χR(−, R/q) to all finitely generated modules of finite projectivedimension whose annihilator contains something from outside q. In other words,letting S = R\q, χR(−, R/q) can be applied to all modules in CR

0 (f,pd,S-tor),and hence it factors through GR

0 (f,pd,S-tor). Thus, since S−1R = Rq is localand thereby generalized Euclidean according to Proposition 4.5, it follows fromCorollary 4.22 that

χR(M, R/q) = χR([M ], R/q)

= χR([R/GR(M)], R/q)

= χR(R/GR(M), R/q).

This is easily calculated: letting u ∈ R be a generator for GR(M) and tensoringthe obvious projective resolution for R/GR(M) = R/〈u〉 with R/q, we obtain thecomplex

0 −→ R/qu−→ R/q −→ 0.

Now, u/1 is a representative of detS M and is therefore a unit in S−1R = Rq,and hence u does not belong to q. Thus, the homology of the above complex isconcentrated in degree 0 where it is equal to R/(GR(M) + p), and it follows that

χR(R/GR(M), R/q) = lengthR R/(GR(M) + q).

This is nonzero if and only if GR(M) 6= R: that is, according to Proposition 4.23,if and only if dimR M = dim R − 1. This proves the vanishing and positivityconjectures.

Corollary 6.5. The intersection conjectures hold if gradeR M = 1.

Proof: In this case, according to part (i), (iii) and (v) of Theorem 4.20, GR(M)is a proper principal ideal such that SuppR R/GR(M) ⊆ SuppR M , so The-orem 0.12 yields that

dimR N − 1 ≤ dimR N/GR(M)N

= dimR(R/GR(M)⊗R N)

≤ dimR(M ⊗N),

and hence dimR N ∈ {0, 1}. These cases are covered by theorems 6.2 and 6.4.

6.3 Applying local Chern characters 91

6.3 Applying local Chern characters

In this section, R is assumed to be a Noetherian, local ring with maximalideal m and quotient field k = R/m. Furthermore, it is assumed that R is thehomomorphic image of a regular local ring Q: that is, R = Q/I for an ideal Iof Q. Finally, M and N denote finitely generated modules with pdR M < ∞and dimR(M ⊗R N) = 0.

We now prove the vanishing conjecture in the case where R is a complete intersec-tion. Although the proof comes out very short and simple below, one should recallthe vast amount of work lying behind the propositions (and even definitions) ofChapter 5 whose proofs were elegantly avoided.

Theorem 6.6 (Roberts). The vanishing conjecture holds whenever R is a com-plete intersection and N has finite projective dimension.

Proof: Suppose that X,Y ∈ CR¤(f,P) are projective resolutions of M and N ,

respectively, so that χR(M, N) = H([X ⊗R N ]), where H is the homomorphismGR

¤(f|l) → GR0 (l) from Theorem 2.11 in which we have identified GR

0 (l) with Z asin Example 2.16. We then find that, in AR({m})Q,

χR(M, N)[R/m] = H[X ⊗R N ][R/m]



= chR(X) chR(Y )(τR(R)) (according to (5.1))

= chR(X) chR(Y )[R]Spec R (using Proposition 5.4(v))

=∑

i+j=d

chRi (X) chR

j (Y )[R]Spec R,

where d is the dimension of R. Now, as we saw in Chapter 5, chRj (Y )[R]Spec R

is trivial for d − j > dimR N : that is, since i + j = d, it must be trivial fori > dimR N . Similarly, chR

i (X)[R]Spec R is trivial for d − i > dimR M : that is, itis trivial for j > dimR M . It now follows from the commutativity of local Cherncharacters as stated in Proposition 5.3(ii) that the only nontrivial terms in thesum above are the terms corresponding to i ≤ dimR N and j ≤ dimR M . Butsince we are trying to proving the vanishing conjecture, we are assuming thatdimR M + dimR N < d, and hence there are no such terms. This proves thevanishing conjecture.

Since regular implies complete intersection, Theorem 6.6 confirms the van-ishing conjecture as originally formulated by Serre. The assumption that R isa complete intersection is only used to deduce that τR(R) = [R]Spec R. A ringsatisfying this property is called a Roberts ring. Thus, the vanishing conjecture


holds whenever R is a Roberts ring and both M and N have finite projectivedimensions.

Note that the calculation of χR(M, N) in the proof of Theorem 6.6 holds evenwithout the assumption that dimR M + dimR N < d. Letting i = dimR N andj = dimR M , and assuming that condition (0) of the intersection conjecture holds(which it does for regular local rings, as proven by Serre), we generally find that

χR(M,N)[R/m] = chRi (X) chR

j (Y )[R]Spec R.

Proving the positivity conjecture in the case that R is a complete intersectionand N has finite projective dimension is then a matter of showing that the aboveelement is “positive”.

We can also use the theory of local Chern characters to prove the intersectionconjectures in the cases that dimR N = 0, 1. The proofs, however, do not appearto be quite as strong as the ones from the preceding section, since we are assumingthat R is the homomorphic image of a regular local ring. (But, in fact, the proofsare equally strong, since we are allowed to replace the ring by its completion—atechnique that will not be discussed here.)

Let X ∈ CR¤(f,P) be a projective resolution of M . In the case that dimR N = 0,

the essential calculation in the proof of Theorem 6.2 follows from the followingcalculation in AR({m})Q ∼= Q.

χR(M,N)[R/m] = H[X ⊗R N ][R/m]



= chR0 (X)[N ]{m} (according to (5.3))

= χR(M)[N ]{m} (using Proposition 5.5)

= χR(M) lengthR N [R/m].

In the case that dimR N = 1 (by which it follows that dimR M < dim R,so that the MacRae ideal of M is defined), we assume, as in the proof of The-orem 6.4, that N = R/q for a prime q, and the essential calculation in the proofof Theorem 6.4 then follows from the following calculation in AR({m})Q ∼= Q.

χR(M, R/q)[R/m] = H[X ⊗R R/q][R/m]

= τR(X ⊗R R/q) (according to (5.2))

= chR(X)(τR(R/q)) (using Proposition 5.4(iii))

= chR1 (X)(τR

1 (R/q)) (using Corollary 5.6)

= chR1 (X)[R/q] (according to (5.3))

= GR(M) ∩ [R/q] (using Proposition 5.8)

= [R/(q + GR(M))]{m}

= lengthR(R/(q + GR(M)))[R/m].

6.4 Concluding remarks 93

6.4 Concluding remarks



In section 6.2 we proved the intersection conjectures in the case that dimR N = 0by working in the Grothendieck group GR

0 (f,P) in which we could exploit theidentity [M ] = χR(M)[R], whereby we proved that

χR(M, N) = χR(M) lengthR N.

The vanishing and positivity conjectures then followed from the bi-implicationdimR M = dim R ⇐⇒ χR(M) 6= 0. Similarly, we proved the intersectionconjectures in the case dimR N = 1 by working in the Grothendieck groupGR

0 (f,P|S-tor) for a multiplicative system S of R, in which we could exploit theidentity [M ] = [R/GR(M)], whereby we proved, assuming that N = R/q for aprime ideal q, that

χR(M, N) = lengthR R/(GR(M) + q).

The vanishing and positivity conjecture then followed from the bi-implicationdimR M = dim R− 1 ⇐⇒ GR(M) 6= R.

It is tempting to hope that these proofs can be generalized to higher dimen-sions of N , although the counterexample by Dutta, Hochster and McLaughlinshows that we cannot hope for a generalization to all dimensions of N withoutassuming that pdR N < ∞. A natural place to start is the case that dimR N = 2.Since we have already dealt with the cases that gradeR M = 0, 1, we may assumethat gradeR M ≥ 2. It immediately follows that

dimR N + dimR M ≤ dimR N + dim R− gradeR M ≤ dim R,

so (0) of the intersection conjectures holds.In general, one can imagine a situation where the intersection conjectures have

been verified for dimR N < n and gradeR M < n for some n > 1. Consideringthe case dimR N = n, we can then assume that gradeR M ≥ n, and an argumentsimilar to the one above then shows that (0) of the intersection conjectures holds.Moving on to vanishing and positivity along the lines of section 6.2, the firstquestion that needs to be asked is which Grothendieck group χR(−, N) can befactored through. The answer is as follows.

Let N1 denote the (finite) set of minimal primes in SuppR N . Using primeavoidance (Lemma 0.1), we can choose an element

x1 ∈ AnnR M\( Zd R ∪⋃

q∈N1

q),


for otherwise AnnR M would be contained in Zd R, which is definitely not thecase since grade M ≥ n, or in an prime q ∈ N1, contradicting the assumptionsthat dimR N = n and SuppR M ∩ SuppR N = {m}. We now have from Propo-sition 0.19(iv) that gradeR(AnnR M, R/〈x1〉) = gradeR M − 1 ≥ n − 1, and bychoice of x1 we have dimR N/x1N = dimR N − 1 = n− 1.

Letting N2 denote the (finite) set of minimal elements in SuppR N/x1N , wecan next choose, using prime avoidance again, an element

x2 ∈ AnnR M\( ZdR(R/〈x1〉) ∪⋃

q∈N2

q),

for otherwise AnnR M would be contained in ZdR(R/〈x1〉) or in a prime fromN2, both situations contradicting the facts that gradeR(AnnR M, R/〈x1〉) ≥ n−1and dimR N/x1N = n− 1. Continuing this process, we choose at the i’th step anelement

xi ∈ AnnR M\( ZdR(R/〈x1, . . . , xi−1〉) ∪⋃

q∈Ni

q),

where Ni is the (finite) set of minimal elements in SuppR(N/〈x1, . . . , xi−1〉N),and where we know that gradeR(AnnR M, R/〈x1, . . . , xi−1) ≥ n − i + 1 anddimR(N/〈x1, . . . , xi−1〉N) = n− i + 1.

The process terminates at the n’th step, where we have obtained a sequencex = (x1, . . . , xn) of elements in AnnR M such that gradeR(AnnR M, R/〈x〉) ≥ 0and dimR N/〈x〉N = 0. By construction, x is a regular sequence, and apparentlyx constitutes a system of parameters for N . Thus, we can factor χR(−, N)through the Grothendieck group GR

0 (f,P|S(x)-tor): from the way we constructedx, M clearly belongs to this Grothendieck group, and χR(−, N) is defined onany module from it. In attempting to calculate the intersection multiplicityχR(M, N), one can now replace M by any module that is identical to M withinGR

0 (f,pd,S(x)-tor). Following the lines of theorems 6.2 and 6.4, one could hopeto replace M by a module that is sufficiently simple that we can easily calculateits intersection multiplicity with N . According to Corollary 3.38, we know that[M ] can be written as the difference of two elements represented by modules ofprojective dimension n.

This, however, is as far as our theory takes us! To get any further, one couldlook for an invariant to join the Euler characteristic and the MacRae ideal, whichprovided the necessary simplifications in dimensions 0 and 1. The theory of localChern characters might assist in this search.

Proposition 5.5 showed that if X ∈ CR¤(f,P) is a projective resolution of M , the

zeroth local Chern character chR0 (X) is given as multiplication by χR(M), and

Proposition 5.8 showed that the first local Chern character chR1 (X) is given by

intersection with GR(M). Thus, the local Chern characters are definitely relevantin the search for an invariant to join the Euler characteristic and the MacRaeideal; one could imagine that such an invariant is related to the n’th local Cherncharacter chR

n (X).

6.4 Concluding remarks 95

The connection between local Chern characters and the Euler characteristicand MacRae ideal is additionally emphasized in Corollaries 5.6 and 5.9. Togetherwith Theorem 0.8, Corollary 5.6 shows that the following are equivalent for amodule M ∈ CR

0 (f,pd) with projective resolution X ∈ CR¤(f,P).

(i) dimR M = dim R.

(ii) gradeR M = 0.

(iii) χR(M) 6= 0.

(iv) chR0 (X) 6= 0.

Assuming that none of these conditions are satisfied, Corollary 5.9 together withProposition 4.23 show that the following are equivalent.

(i) dimR M = dim R− 1.

(ii) gradeR M = 1.

(iii) GR(M) 6= R.

(iv) chR1 (X) 6= 0.

It is thinkable that one could continue with four more conditions that areequivalent under the assumption that none of the above conditions are satisfied.Not only would this provide a valuable clue for how to attack the intersectionconjectures in higher dimensions, it would also verify the grade conjecture in onemore case. This is left for future studies.

Bibliography

[Bas74] Hyman Bass. Introduction to some methods of algebraic K-theory.American Mathematical Society, Providence, R.I., 1974. ExpositoryLectures from the CBMS Regional Conference held at Colorado StateUniversity, Ft. Collins, Colo., August 24-28, 1973, Conference Board ofthe Mathematical Sciences Regional Conference Series in Mathematics,No. 20.

[Ber01] George M. Bergman. An invitation to general algebra and universalconstructions. 2.2 edition, August 2001. Lecture notes for the graduatecourse Math250a at the University of California at Berkeley.

[BH93] Winfried Bruns and Jurgen Herzog. Cohen-Macaulay rings, volume 39of Cambridge Studies in Advanced Mathematics. Cambridge UniversityPress, Cambridge, 1993.

[Eis95] David Eisenbud. Commutative algebra. With a view toward algebraicgeometry, volume 150 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.

[Fox81] Hans-Bjørn Foxby. Intersection multiplicities of modules. Unpublishedpreprint, May 1981.

[Fox82a] Hans-Bjørn Foxby. K-theory for complexes with finite length homology.Copenhagen University Preprint Series no. 1, 1982.

[Fox82b] Hans-Bjørn Foxby. The Macrae invariant. In Commutative algebra:Durham 1981 (Durham, 1981), volume 72 of London Math. Soc. LectureNote Ser., pages 121–128. Cambridge Univ. Press, Cambridge, 1982.

[Fox89] Hans-Bjørn Foxby. Notes on local Chern characters. Personal notes fora lecture held in Barcelona, 1989.

[Fox98] Hans-Bjørn Foxby. Hyperhomological Algebra & Commutative Rings.Unpublished notes, September 1998.

97

98 Bibliography

[Fox99] Hans-Bjørn Foxby. Homologisk algebra [Homological Algebra]. Lecturenotes for the graduate course Matematik 4ha at the University ofCopenhagen, 1997 (with an addition from 1999).

[Hal03] Esben Bistrup Halvorsen. Betti numbers and Euler characteristics. Canbe downloaded from http://www.math.ku.dk/~ebh/papers.html.Course project related to the graduate course Math253 at the Uni-versity of California at Berkeley, January 2003.

[HS97] Peter J. Hilton and Urs Stammbach. A course in homological algebra,volume 4 of Graduate Texts in Mathematics. Springer-Verlag, NewYork, second edition, 1997.

[Lan93] Serge Lang. Algebra. Addison-Wesley Publishing Company, Reading,MA, third edition, 1993.

[Mac65] Robert E. MacRae. On an application of the Fitting invariants. J.Algebra, 2:153–169, 1965.

[Mag02] Bruce A. Magurn. An algebraic introduction to K-theory, volume 87 ofEncyclopedia of Mathematics and its Applications. Cambridge Univer-sity Press, Cambridge, 2002.

[Rob85] Paul C. Roberts. The vanishing of intersection multiplicities of perfectcomplexes. Bull. Amer. Math. Soc. (N.S.), 13(2):127–130, 1985.

[Rob87] Paul C. Roberts. The MacRae invariant and the first local Chern char-acter. Trans. Amer. Math. Soc., 300(2):583–591, 1987.

[Rob92] Paul C. Roberts. The homological conjectures. In Free resolutions incommutative algebra and algebraic geometry (Sundance, UT, 1990), vol-ume 2 of Res. Notes Math., pages 121–132. Jones and Bartlett, Boston,MA, 1992.

[Rob98] Paul C. Roberts. Multiplicities and Chern classes in local algebra, vol-ume 133 of Cambridge Tracts in Mathematics. Cambridge UniversityPress, Cambridge, 1998.

[Rob03] Paul C. Roberts. Cycles and commutative algebra. Can be down-loaded from http://arxiv.org/abs/math.AC/0310448/. E-print pub-lished on arXiv, 2003.

[Ser65] Jean-Pierre Serre. Algebre locale. Multiplicites [Local algebra. Multiplic-ities], volume 11 of Cours au College de France, 1957–1958, redige parPierre Gabriel. Troisieme edition, 1975. Lecture Notes in Mathematics.Springer-Verlag, Berlin, 1965.

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