Riemann-Roch Algebra (Grundlehren der mathematischen Wissenschaften)

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics

A Selection

207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225 226. 227. 228, 229.

Faith: Algebra: Rings, Modules, and Categories I Faith: Algebra 11, Ring Theory Mal'cev: Algebraic Systems Polya/Szego: Problems and Theorems in Analysis I Igusa: Theta Functions Berberian: Baer*-Rings Athreya/Ney: Branching Processes Benz: Vorlesungen fiber Geometric der Algebren Gaal: Linear Analysis and Representation Theory Nitsche; Vorlesungen uber Minimalfiichen Dold: Lectures on Algebraic Topology Beck: Continuous Flows in the Plane Schmetterer. Introduction to Mathematical Statistics Schoeneberg: Elliptic Modular Functions Popov: Hyperstability of Control Systems Nikol'skii: Approximation of Functions of Several Variables and Imbedding Theorems Andre: Homologie des Algebres Commutativcs Donoghue: Monotone Matrix Functions and Analytic Continuation Lacey: The Isometric Theory of Classical Banach Spaces Ringel: Map Color Theorem Gihman/Skorohod: The Theory of Stochastic Processes 1 Comfort/Negrepontis. The Theory of Ultrafilters Switzer: Algebraic Topology-Homotopy and Homology Shafarevich: Basic Algebraic Geometry van der Waerden: Group Theory and Quantum Mechanics Schaefer: Banach Lattices and Positive Operators Polya/Szego: Problems and Theorems in Analysis II Stenstrbm: Rings of Quotients Gihman/Skorohod: The Theory of Stochastic Process II Duvant/Lions: Inequalities in Mechanics and Physics Kirillov: Elements of the Theory of Representations Mumford: Algebraic Geometry I: Complex Projective. Varieties Lang: Introduction to Modular Forms BerghlLofstrom: Interpolation Spaces. An Introduction Gilbarg/Trudinger. Elliptic Partial Differential Equations of Second Order Schutte: Proof Theory Karoubi: K-Theory, An Introduction Graucrt/Remmert: Theone der Steinschen Raume Segal/Kunze: Integrals and Operators Hasse: Number Theory

230.

Klingenberg: Lectures on Closed Geodesics

231

Lang: Elliptic Curves: Diophantine Analysis Gihman/Skorohod: The Theory of Stochastic Processes III Stroock/Varadhan: Multi-dimensional Diffusion Processes Aigner: Combinatorial Theory

190. 191. 192. 193. 194. 195. 196. 197. 198. 199.

200. 201. 202. 203. 204. 205. 206_

232. 233. 234.

Continued after Index

William Fulton Serge Lang

Riemann Roch Algebra

Springer-Verlag

New York Berlin Heidelberg Tokyo

William Fulton Department of Mathematics Brown University Providence, RI 02912

Serge Lang Department of Mathematics Yale University New Haven, CT 06520

U.S.A.

U.S.A.

AMS Subject Classification: 14C40

Library of Congress Cataloging in Publication Data Fulton, William Riemann-Roch algebra. (Grundlehren der mathematischen Wissenschaften; 277) Bibliography: p. Includes index. 1. Geometry, Algebraic. 2. Riemann-Roch theorems. I. Lang, Serge, 1927II. Title. III. Series. 84-26842 QA564.F85 1985 512'.33

© 1985 by Springer-Verlag New York Inc.

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.

Typeset by Composition House Ltd., Salisbury, England. Printed and bound by R. R. Donnelley & Sons, Harrisonburg, Virginia Printed in the United States of America.

987654321 ISBN 0-387-96086-4 Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-96086-4 Springer-Verlag Berlin Heidelberg New York Tokyo

Contents

Introduction

Vii

CHAPTER I

A-Rings and Chern Classes

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§1. A-Rings with Positive Structure . . . . §2. An Elementary Extension of 2-Rings . . §3. Chern Classes and the Splitting Principle . §4. Chern Character and Todd Classes . . . §5. Involutions . . . . . . . . . . . . §6. Adams Operations . . . . . . . . .

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3 7 11 17

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20 23

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32 37 43

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54 58

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62

CHAPTER II

Riemann-Roch Formalism

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. . . §1. Riemann-Rock Functors . . . . . §2. Grothendieck-Riemann-Roch for Elementary Imbeddings and .

Projections .

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§3. Adams Riemann-Roch for Elementary Imbeddings and Projections §4. An Integral Riemann-Roch Formula . . . . . . . . .

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CHAPTER III

Grothendieck Filtration and Graded K .

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§1. The y-Filtration . . . . . . . . . . . : . . . . . . . §2. Graded K and Chern Classes . . . . . . . . . . . . . . §3. Adams Operations and the Filtration . . . . . . . . . . . §4. An Equivalence Between Adams and Grothendieck Riemann-Roch Theorems . . . . . . . . . . . . . . . . . . . . . CHAPTER IV

Local Complete Intersections

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§1. Vector Bundles and Projective Bundles . . §2. The Koszul Complex and Regular Imbeddings §3. Regular Imbeddings and Morphisms . . . §4. Blowing Up . . . . . . . . . . . . §5. Deformation to the Normal Bundle . . . .

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66

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66 70 77 91 96

Vi

CONTENTS

CHAPTER V

The K-functor in Algebraic Geometry .

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100

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102 104 118 126 134 141 144 149

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§1. The A-Ring K(X) . . . . . . . . . §2. Sheaves on Projective Bundles . . . . . §3. Grothendieck and Topological Filtrations. §4. Resolutions and Regular Imbeddings . . §5. The K-Functor of Regular Morphisms . . §6. Adams Riemann-Roch for Imbeddings . . §7. The Riemann-Roch Theorems . . . . . Appendix. Non-connected Schemes . . . . CHAPTER VI

An Intersection Formula. Variations and Generalizations §1. The Intersection Formula . . . §2. Proof of the Intersection Formula §3. Upper and Lower K . . . . . §4. K of a Blow Up . . . . . . §5. Upper and Lower Filtrations . .

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152 157 164 169 178 184 188 190 192

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§6. The Contravariant Maps fx and f .. §7. Singular Riemann-Roch §8. The Complex Case . . §9. Lefschetz Riemann-Roch

References

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197

Index of Notations

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199

Index

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201

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Introduction

In various contexts of topology, algebraic geometry, and algebra (e.g. group representations), one meets the following situation. One has two contravariant functors K and A from a certain category to the category of rings, and a natural transformation

p:K -.A of contravariant functors. The Chern character being the central example, we call the homomorphisms px: K(X) -, A(X)

characters. Given f : X -> Y, we denote the pull-back homomorphisms by f R: K(Y) _) K(X)

and

fA: A(Y) _ A(X).

As functors to abelian groups, K and A may also be covariant, with push-forward homomorphisms

and

fA:A(X)-,.A(Y).

Usually these maps do not commute with the character, but there is an element T f e A(X) such that the following diagram is commutative: K(X) fC1

K(Y)

T`'px, A(X)

P

JfA

A(Y)

The map in the top line is px multiplied by z f. When such commutativity holds, we say that Riemann-Roch holds for f. This type of formulation was first given by Grothendieck, extending the work of Hirzebruch to such a relative, functorial setting. Since then

Viii

INTRODUCTION

several other theorems of this Riemann-Roch type have appeared. Underlying most of these there is a basic structure having to do only with elementary algebra, independent of the geometry., One purpose of this monograph is to describe this algebra independently of any context, so that it can serve axiomatically as the need arises.

A common feature of these Riemann-Roch theorems is that a given morphism f is factored into poi:

X -IPI+Y, where i is a closed imbedding and p is a bundle projection. One con structs a deformation from i to the zero-section imbedding of X in the normal bundle to X in P, suitably completed at infinity. General proce-

dures, which we axiomatize here, allow one to deduce a general Riemann-Roch theorem from the elementary cases of imbeddings in and projections from bundles; these cases are usually handled by direct calculation.

We illustrate the formalism by giving a complete elementary account of Grothendieck's Riemann-Roch theorem in the context of schemes and local complete intersection morphisms, as first presented in [SGA 6]. Here K(X) is the Grothendieck ring of locally free sheaves on X, and A(X) is an associated graded group of K(X), with rational coefficients. To prepare for this we include self-contained discussions of several important subjects from algebra and algebraic geometry, such as: A-rings, Adams operations, y-filtrations, Chern classes, algebraic K-theory, regular imbeddings and Koszul complexes, sheaves on projective bundles, and local complete intersections. Manin's very useful notes [Man] were also written to give an accessi-

ble account of parts of [SGA 6], for the case of imbeddings of nonsingular varieties. Several developments since then allow us to give both a more elementary and more complete treatment, including a complete

proof of the main theorem, as well as some conjectures left open in [SGA 6]. Most important among these developments are: (a) an understanding of deformation to the normal bundle (cf. [J], [BFM 1], [V], [BFM 2], [FM]); (b) the use of Castelnuovo-Mumford "regular" sheaves on projective bundles (cf. [Q]). Among the resulting improvements we mention : (1)

A proof that the y-filtration on K(X) is finer than the topological filtration (V, §3).

(2)

A Riemann-Roch theorem for the Adams operations /3 without denominators (V, §6).

(3)

An elementary construction of the push-forward fK for a projective local complete intersection morphism f (V, §4).

INTRODUCTION

ix

Of these, (1) and (2) were conjectured in [SGA 6]. Other features included are: (4) An Intersection Formula for K-theory (VI, §1). (5) (6)

A direct proof, using a power-series calculation of R. Howe, for Grothendieck Riemann-Roch for bundle projections (II, §2). An equivalence between forms of Riemann-Roch for the Chern character and Adams operators (III, §4).

Chapter I contains an elementary treatment of 1%-rings and Chern classes; the excellent exposition of Atiyah and Tall [AT] can be referred to for more on 1-rings. We include a proof of a splitting principle for abstract Chern classes; in our application in Chapter V, however, this splitting principle will be evident, so the reader can skip this proof. In Chapter II we develop the abstract Riemann-Roch formalism. The main new feature here is an axiomatic formulation of the deformation to the normal bundle: to prove a Riemann-Roch theorem for a given im-

bedding, it suffices to "deform" it to an "elementary imbedding" for which one knows the theorem. We also axiomatize the dual case of an "elementary projection". Chapter III describes the y-filtration of Grothendieck, and constructs Chern classes in the associated graded ring. Chapter IV is a chapter of "intermediate algebraic geometry", which

could supplement a text such as Hartshorne's [H]. We establish the basic category of algebraic geometry for which we shall prove the Riemann-Roch formula, namely the category of regular morphisms. By this we mean morphisms which can be factored into a local complete intersection imbedding, and a projection from a projective bundle. We include a short proof of Micali's theorem on regular sequences, and basic

facts about regular imbeddings, conormal sheaves, and blowing up. Theorem 4.5 on the residual structure of a proper transform is, we believe, new. The culmination of this chapter is a simple construction of the deformation to the normal bundle. Many of the results of Chapter IV are not needed for the proof of Riemann-Roch proper, but are included for completeness. All these ideas come together in Chapter V, where the A-ring K(X) is

shown to satisfy the abstract properties of the first three chapters. The Grothendieck Riemann-Roch theorem (including the version without denominators), and analogous theorems for the Adams operators, follow quickly.

Chapter VI contains an Intersection Formula in the context of Ktheory which seems to be new in this generality, and which is analogous to the "excess intersection formula" of [FM], see also [F 2], Theorem 6.3. The formula is proved by using the general formalism of basic deformations, together with the geometric construction of the deformation

to the normal bundle. This follows a pattern similar to the proof for

INTRODUCTION

X

Riemann-Roch itself, and provides another striking application of the formalism of Chapter II. In Chapter VI, we also discuss the relation of the Grothendieck group

of locally free sheaves with the Grothendieck group of all coherent sheaves. We give an application to the calculation of an exact sequence for K of a blow up of a regularly imbedded subscheme, relying on the Intersection Formula. Finally, we discuss briefly and incompletely how Riemann-Roch can be extended beyond the case of local complete intersections. In addition, we sketch several other contexts where the formalism developed here can be applied. It would take another book to give a systematic treatment of these topics, including the relations between Ktheory, the Chow group and etale cohomology in a more schemy and sheafy context than [F 2]. We have made our exposition self-contained from [H] for algebraic geometry, [L] for general algebra, and the simpler parts of [Mat] for a little more commutative algebra. Thus we have included proofs of elementary facts whenever necessary to achieve this. At least in first reading, the reader interested only in a fast proof of Riemann-Roch is advised to skim Chapters I, IV, and the first half of Chapter V. More is included in these chapters than is strictly needed for Riemann-Roch, with the hope that this important material will be more accessible than its previous position in SGA and EGA permit. Those interested primarily in the Riemann-Roch theorem should concentrate on Chapters II, III, and V. We have not discussed applications to the theory of group representations. For these, we refer especially to the articles by Atiyah-Tall, Evens, Kahn, Knopfmacher, Thomas, as well as Grothendieck's general discussion as listed in the Bibliography. On the other hand, the applications to group representations are not independent of those to algebraic geometry. Even though the K-groups can be defined in terms of modules, one can analyze them via considerations of topology, classifying spaces, and algebraic geometry, so there is a considerable amount of feedback.

We also do not discuss applications to topology. We refer to the lectures by Atiyah [At] and Bott [Bo] for some K-theory like that of Chapters I and III in a topological context, stopping short of RiemannRoch theorems, however.

We hope that the simpler logical structure of the proofs which emerges in this treatise will make it easier to understand these results,

and to find new situations to which this "Riemann-Roch algebra" applies.

CHAPTER I

A-Rings and Chern Classes

This chapter describes first the basic ring structure of the objects to be encountered later in a more geometric context. The algebra involved is elementary and self-contained. We have axiomatized certain notions which originally arose in the theory of vector bundles. Actually we work with two rings, one of them usually graded. We also develop the formalism of Hirzebruch polynomials, which belongs to the basic theory of symmetric functions. We have preserved original names like Chern classes, Todd character, etc., although the algebra involved here deals only with a pair of rings and some elementary formal manipulation of power series, independently of the geometry from which they came. We now make additional comments concerning the way these notions arise in applications to algebraic geometry and group representations. These are not necessary for a logical understanding of the chapter. However, we may have at least two categories of readers: those who know some Riemann-Roch theory previously and are principally interested in a quick proof of Grothendieck Riemann-Roch; and those who have more limited knowledge in this direction and are thus directly interested in the more elementary material. Our additional comments are addressed to this second category. A fundamental aim of algebraic geometry is to study divisor classes, or equivalently isomorphism classes of line bundles. More generally, one

wishes to study vector bundles, with certain equivalence relations. The Grothendieck relations are those which to each short exact sequence

E"-,0 gives the relation

[E] = [E'] + [E"].

The group of isomorphism classes of vector bundles over a space X modulo these relations is called the Grothendieck group K(X). It has both covariant and contravariant functorial properties, although the covariant ones are much more subtle.

2

d-RINGS AND CHERN CLASSES

The addition is induced by the direct sum, and there is also a multiplication induced by the tensor product, so that K(X) is in fact a ring. The class of the trivial line bundle is the unit element. This ring has various structures. First, it has an augmentation, which to E associates its rank E (E). Then s extends to an augmentation on K(X) (algebra homomorphism into Z). The vector bundles themselves generate a semigroup under addition. In §1, we axiomatize this structure by defining "positive elements" whose properties are modelled on those of vector bundles. The elements of augmentation 1 correspond to line bundles, and are thus called line elements. Second, the ring K(X) has another operation induced by the alternating product. To each integer i >_ 0 we have AE, and therefore its class

[A`E] denoted by 2'(E). A standard elementary formula for the direct sum E = E' (D E" of free modules reads

A"(E) Q (A'E' ® A `E"). i=O

Passing to the classes in the K-group, we get the relation n

A"(X + Y) _ E )'(41, -i(Y) i=o

But this relation amounts to saying that the map x H Y ;t (x) t` = 2L(x)

by definition

is a homomorphism from the additive group of K(X) to the multiplicative group of power series with constant term equal to 1. This gives rise to the notion of 2-ring. A great deal of the formalism of Riemann-Roch algebra can be developed for the general I-rings. The reader should read simultaneously the beginning of Chapter I and the beginning of Chapter V to see the parallelism between the abstract algebra and the geometric construction giving rise to this algebra. In the theory of group representations, one may start with the category of finite-dimensional vector spaces over a field k, and a representation of a (finite) group G on the space. Then again we have direct sums, tensor products of (G, k)-spaces and the analogous definition of 2-ring, formed by the isomorphism classes of such spaces modulo the relations in the Grothendieck group. The positive elements are just the classes of such spaces as distinguished from the group generated by them in the Grothendieck group. In §2 we shall discuss a particular extension of a 2-ring, which gives

an axiomatization for the extension obtained from a projective bundle.

A-RINGS WITH POSITIVE STRUCTURE

[I, §11

3

The corresponding geometric case is discussed in Chapter V, Theorem 2.3 and Corollary 2.4. Since the existence of the extension is proved in a self-contained way by geometric means in Chapter V, the reader interested only in the geometric application can omit the existence proof of Theorem 2.1 in this chapter. The corresponding graded extension will be constructed in §3.

I §1. ?,-Rings with Positive Structure Let K be a commutative ring. For each integer i ? 0 suppose given a mapping

A':K->K such that .l°(x) = 1, A'(x) = x for all x e K, and if we put ;"(x) _ E A'(x)t'

then the map x i- A,(x)

is a homomorphism. This condition is equivalent with the conditions k

(1.1)

2k(x + y)

Y2'(x)2k-`(y)

i=o

for all positive integers k. A ring with such a family of maps A' is called a A-ring.

In addition, we suppose that the A-ring has what we shall call a positive structure. By this we mean : A surjective ring homomorphism s:K -+ Z called the augmentation.

A subset E of the additive group of K called the set of positive elements such that E together with 0 form a semigroup, satisfying the conditions

Z+cE,

EE=E,

K=E-E

so every element of K is the difference of two elements of E; furthermore for e e E we have a(e) > 0, and if a(e) = r then

2'(e) = 0 for i > r

and

2'(e) is a unit in K.

4

A-RINGS AND CHERN CLASSES

CI, §1]

We define L to be the subset of elements u e E such that s(u) = 1. Since 21u = u, it follows that L is a subgroup of the units K*. Elements of L will be called line elements. An extension K' of a A-ring K is a A-ring K' containing K, with Ai and augmentation extending that of K, and with positive elements E' containing E. We shall be concerned with a class SI of A-rings satisfying, in addition to the preceding conditions, the

Splitting Property. For any K e St and positive element e in K, there is an extension K' of K in St such that e splits in K', i.e.

e=u1 with u, line elements in K'.

It follows by induction that any finite set of positive elements can be simultaneously split in a suitable extension. The splitting property will allow us to deduce general formulas from the simple case of line elements. For example, the property that AV(e) = 0 for e positive and

i > r = e(e) follows from the fact that A '(u) = 0 for all line elements u and

i> 1. More generally, for u e L we have directly from the assumptions ,t,(u) = 1 + ut,

and hence if e is split as above, then r

AI(e) = M1 + uit) i=1 r

i

1=1

where si is the i-th symmetric function. Since the coefficients A1(e) are given a priori as elements of the A-ring K, we see that the value of the symmetric function si(ul,... ,u,) is independent of the splitting of e as a sum of line elements in K'. For example, one sees from this formula that (1.2)

e(2 e) _ (s(e)}.

In other words, if Z is given a A-ring structure by AV(n) _ (n), then the i

augmentation a is a homomorphism of A-rings.

I-RINGS WITH POSITIVE STRUCTURE

[L§1]

5

Formulas for 1'(x y) and .11(11(x)) can best be expressed in terms of certain universal polynomials Pk and Pk,j as follows. Take independent variables U1, ... ,U,,, and Y 1 ,.. . ,Vn. Let Xi be the i-th elementary symmetric polynomial in U1,. .. ,U,, , and Y the i-th elementary symmetric polynomial in V1,.. . ,Vn. For m >_ k, n::?: k, let

be the polynomial of weight k in the variables Xi and in the variables Y; (where X. and Y, are assigned weight i), determined by the identity (A)

Y Pk(X I, ... ,Xk, Yl,... ,Yk)Tk = 1 1(1 + U; V T).

k?0

1, j

By setting some of the variables U; or V; equal to zero for i, j > k, one sees that the Pk are independent of the choice of m, n > k. Similarly define Pk,j(X1,...,Xkj)EZ[X1,...,Xkj]

of weight kj, by the identity for m >_ kj: (B)

Y_ Pk, j(X1, ...,Xkj)Tk =

k? 0

m

(1 + Ui, ... Ui.T

II

i, < ... < ij

n

Now if x =

ui, y =

vj, with ui, v, line elements, then

2(x'Y)=fl(1+uiv;t). From (A) this can be written (1.3)

Ak(x y) = Pk(A1(X), ... ,2k(x), 21(Y), ... ,)k(Y))

For example, if x is a line element, then or

At(XY) = A(y).

m

ui, then Aj(x) _

Similarly, if x =

ui,

uij, so

i,<---
i=1

2"().x)_

1J

ii<...
(1 +ui,.ui;t).

By (B) this can be written (1.4)

2k('10)) = Pk,j

... ,.l.kj(x)).

(x),..

6

[I, §1]

2-RINGS AND CHERN CLASSES

The identities (1.1)-(1.4) say that our A-rings are what Grothendieck calls special A-rings ([SGA 6], Exp. 0). This may be reinterpreted as follows. Given a commutative ring A, define A[[T]]+ = TA[[T]], and let

A(A) = 1 + A[[T]]+ be the set of power series in A with constant term 1. Define an addition in A(A) by the multiplication of power series; a product in A(A) by the formula

(1+Eapt`).(1+Eb,t')=1+ZPk(al,...,ak,bl,...,bk)tk; and A-operations by

P (l + Y_ a,t`) = 1 + >,

Pk.J(a1,...

,akj)tk.

One verifies easily that these definitions make A(A) into a special A-ring (cf. [AT] for a readable account). For any A-ring K, A,: K -> A(K)

is an additive homomorphism; K is special precisely when A, is a homomorphism of A-rings. Note that identities (1.1)-(1.4) hold for all elements of K, not only positive elements.

Remark. An element x in a 7-ring K is said to have d-dimension = n if A`(x) = 0 for all i > n, and A "(x) : 0. The ring K is called A-initedimensional if every element is a difference of two elements of finite .dimension. Since positive elements have finite dimension, our axioms imply that our Z-rings are all finite dimensional. Conversely, given a Afinite-dimensional special R-ring K, one can define E to be the elements of A-finite dimension. If one assumes that all one-dimensional elements are units, then E defines a positive structure in our sense. Let Y_ alt' be a power series in K[[t]] with ao = 1. The coefficients of the inverse series alt`)-1

Y_ bit` _ (Y,

can be determined recursively from the coefficients a= by the relation k

Ia;bk-i=0

i=o

for

k>0.

LI, §2]

7

AN ELEMENTARY EXTENSION OF A-RINGS

For e e E we define the series or(e) = R_,(e)-1 = Y_ o'(e)t'. =o

Then for each i we get a map 6: K .- K. If h is a homomorphism of K into some ring and cp(t) a K[[t]] is a power series, then we let h(cp(t)) be the power series obtained by applying h to all the coefficients of cp(t). In particular, we have e(6r(e))e( r(e)) = 1.

Lemma 1.1. Let e (e) = r + 1. Then

e(A-,(e)) = (1 - t)r+l

and

s(at(e)) -

1 (1-t)r+1I

So explicitly in terms of the coefficients, and

e(it'(e)) = Cr

Proof. Splitting e into

e(n'(e)) = I r J 3).

u,, with e(u) = 1, ..,(e) = jj (1 + u, t),

from which the formula for eQ_t(e)) is clear. Since a, is the inverse of A, , the formula for e(a,(e)) follows. The last formula follows from the identity 1 (1-or+1-

I\\

I

11

I §2. An Elementary Extension of % -Rings Given a A-ring K and a positive element e in K we construct a ring extension Ke of K as follows. Set s(e) = r + 1, r+1

(-1)'A'(e)Tr+1

Pe(T) _ =o

and let

K. = K[T]I(Pe(T)) = K[e],

A-RINGS AND CHERN CLASSES

8

[I, §2]

where a is the image of T mod pe(T); we call a the canonical generator. We have the defining relation r+1 (-1)'A1(e)er+1-I = 0.

Y_

i=o

In particular, for k >--_ r + 1, multiplying by powers of a and using Am(e) = 0

if m > r + 1, we get the relations k

0.

Y_ (-1)`A`(e)ek i=0

These relations translate into the single power series relation

YI

(_ 1)`,l`(e)ek - 1 tk.

k=0 (=O

Theorem 2.1. There is a unique A-ring structure on Ke, extending that on K, and satisfying ).t(e) = 1 + It. Proof. First define a A-ring structure on the polynomial ring K[T], such that c(T) = 1 and A1(T) = T, A '(T) = 0 for i > 1. From the fact that K is a special A-ring it follows readily that K[T] is also a special Aring. To show that this determines a A-ring structure on Ke, it must be verified that the ideal I = (pe(T)) is preserved by the A-operations. Set j = r + 1. Then

(-1)'pe(T) = 2'(e - T).

Using the identity (1.3) for products, one sees that it suffices to verify that Rkl'(e - T) e I = (A'(e - T))

for all k > 1. From the identity A, (e - T) = (1 + A1(e)t +

+ )'(e)t') (1 +

Tt)- 1

it follows that Ak(e

- T) = ± Tk_'A'(e - T) e I

for all k >_ j. Since Ak(E(x)) = Pk, j ().1(x), ...,A" (x)), it suffices to verify

[I, §2]

9

AN ELEMENTARY EXTENSION OF .t-RINGS

that each monomial appearing in the polynomial Pk, j(X1, ...,Xkj) contains some Xi with i > j. To see this, simply note that Pk,j(X1, ...,Xj-1, 0, ...,0)

is identically zero, as follows from the definition (B) of Pk, j in §1.

As seen from the proof, Ke is a special 2-ring. One may define a positive structure E. on Ke generated by E, ', and e - 8, i.e.

Ee = {y_ aijt'(e - ')jI i,j > 0, a,jeE}. The elements of Ee with augmentation 1 are of the form adi(e - ')j, with a a unit in E and j = 0 if r > 1. The equation pe(C) = 0 shows that e is a unit for all r, and that e - C is a unit if r = 1. Thus Ee defines a positive structure on K8, called the canonical positive structure. Theorem 2.1 may be used to construct an extension K' of K in which e splits. Let V1 = Ke, u1 _ t, e1 = e - ?. Let K(2 = Ke;) = K(1)[11],

and set u2 = e1, e2 = e1 - (1, and so on inductively. Then K' = K(r),

with e=u1+- +ur+1

Proposition 2.2. Let fe = f : Ke -> K be the K-linear functional such that

f (e') = al(e)

for

f (C') = 6'(e)

for all integers i > 0;

0 _< i << r.

Then:

f(C-")=0

for 15n
In fact, for n >- 1 we have the general relation (- 1)nrAr+ 1(e)n f-n

= [` nr

and (-1}'"fir+ 1(e)^f(' ^) is the coefficient of t"r in the power series A-r(ey-1

Proof. We apply f to the power series relation and use K-linearity to get

A(j)t') Y (E (- 1)`V.°(e)ok- i(e)tk = 1, k=0 t=0

\

A-RINGS AND CHERN CLASSES

10

[I, §21

so f(P) = oJ(e) for j > 0. This gives the values of f on positive powers of C.

Next we look at negative powers of C. For e` with n = 1 the stated relation is the equation for z°, namely

i=o

The general relation follows by induction, directly from the definition of

a A-ring. We apply f to the values obtained in the first part of the proposition. Then we find that (- 1)"rAr t 1(e)' f(t - n)

is the coefficient of t"r in the power series A _(ne)a,(e) =.At(e)"u,(e) = A_t(e)"-1.

Since A(e) = 0 for i > r + 1, the coefficient of t"r in this power series is 0 if

nr > (n - 1) (r + 1), which gives precisely n < r + 1, or n:< r. This concludes the proof.

Remark. The factor (_ 1)"r2r+ 1(e)" is a unit in K, so the second part of the proposition gives the values of f at negative powers of C. The functional

fe:Ke -,K such that fe(C`) = al(e) will be called the functional associated with the extension Ke of K. As we did above, if e is fixed throughout a discussion, we omit the subscript e and write simply f. Although we have no immediate use for it, we give immediately the following application.

Corollary 23. Let q = e - e. Then fe('t(q)) = 1.

[1, §3]

CHERN CLASSES AND THE SPLITTING PRINCIPLE

11

Proof We have f (A,(q)) = f (A (e) (At(e)) -1)

_ at(e)

(-1)`f(e')t'

= zt(&-t(e)

as was to be shown. Remark. For a 2-finite-dimensional A-ring K, the considerations of the preceding sections show that the following are equivalent:

(i) K is a special 2-ring. Every e in K with finite A-dimension splits in some I-ring extension of K. (iii) Every e in K with finite A-dimension splits in some special A-ring extension of K. (iv) For every e in K with finite A-dimension K. = K[e] is a special A-ring, with 1,(t') = 1 + et. (ii)

If a class R of A-finite-dimensional A-rings contains Ke for each K e R and each e c- K of finite A-dimension, then the splitting principle holds for

all K in R, and all K in R are special.

I §3. Chern Classes and the Splitting Principle The formalism of symmetric functions, Chern class homomorphisms, and the splitting principle (for instance as in this section) were used and

developed for the first time by Hirzebruch [Hi] for the proof of the Hirzebruch-Riemann-Roch theorem. Let A be a graded (commutative) ring,

A=QA` i=O

and

A+=QA', i=1

where A' is the i-th graded component of A. Let

A°(A)= {1 +alt+azt2 +..

12

A-RINGS AND CHERN CLASSES

[I, §31

be the group of formal power series with leading term 1 and a, e A'. Let K be a A-ring, and let c,: K -+ A°(A)

be a homomorphism of abelian groups. We write c,(x) = Y, c'(x)t'

with c'(x) e A'. The fact that c, is a homomorphism can be written :

ck(x + Y) = : c`(x)c'(y). i+j=k

We call c, a Chern class homomorphism with values in A, if in addition, it satisfies properties CC 1, CC 2, CC 3 below, and the splitting principle which follows.

We require the following conditions for line elements L: CC 1.

For u e L, c(u) = 0 for i > 1, that is c'(u) = 1 + c'(u)t.

CC 2. For u, v E L we have c'(uv) = c'(u) + c'(v).

In other words, c' : L -+A' is a homomorphism.

The i-th graded component c'(x) is called the i-th Chern class. The third condition mentioned above is then: CC 3.

For all e e E and all i >_ 1, c'(e) is nilpotent.

Remark. The formal theory of Chern classes actually does not require the nilpotence until Chapter III, §3 and §4, where an even stronger condition will be imposed. If we do not require nilpotence, then instead of the graded ring A we should take the ring oo

A = fl A' 1=0

which is the completion of A, and consists of all formal series Y_ a,, with

a,eA'.

[I, §3]

CHERN CLASSES AND THE SPLITTING PRINCIPLE

13

Under CC 3, given x a K it will follow that all ei(x) vanish for large i, and then c,(x) = Y, ci(x)ti

is called the Chern polynomial of x. The class c(x) = E ci(x)

in A is called the total Chern class of x. If we did not assume CC 3, then c(x) would lie in A. We then have a homomorphism c(x) = 1 + : c1(x)-

c: K -> 1 + A+,

We often write simply c: K -* A

for this homomorphism from the additive group of K to the multiplicative group of units of A. The variable t is convenient in order to keep track of the grading. Furthermore, representing the Chern class homomorphism as a polynomial c, exhibits better the formal analogy with A-rings and the power series .1,. Treating t as a variable also allows us to substitute special values, like t = - 1, so that for instance we get

c_I(u) = 1 - c'(u). Both notations, with and without the variable t, are useful for applications.

We shall also require a Splitting Principle. Given a finite set {ei} of positive elements of K,

there is a A-ring extension K' of K in which each ei splits, such that c extends to a homomorphism

c: K'->A' for some graded extension A' of A.

Let us note some consequences of this splitting principle. If e splits into

e=ul +-+u,n,

let ai = c1(ui). Then we get a factorization In

c,(e) = fl (1 + ait) i=I

14

A-RINGS AND CHERN CLASSES

[I, §3]

of the Chern polynomial into linear factors. In particular ci(e) = 0

for i > m.

In addition, ck(e) = sk(al,... ,a,,,) = Sk(Cl(1ll),... ,chum))

is the k-th elementary symmetric function of the first Chern classes of The at are called Chern roots for e. The equation shows that any symmetric polynomial in the Cherh roots can be written as a polynomial in the Chern classes of e, and that the resulting expression is independent of the splitting of e. n

If also f = Y_ vj, with bj = c1(vj) Chern roots for f, then j=1

(3.2)

c,(e f) = fl (1 + (ai + b j)t). i.j

In other words, the Chem roots for a product are pairwise sums of Chern roots for the factors. In particular, if f = v is a line element, one has the useful explicit formula Cafe - v) = E t'ct(v)m - `C`(e),

i=o

or

ck(e v) _

k

' (I)ci(e)c'(vt'. m .1j}

0

ctop(e) = Cm(e) = II ai i=1

the top Chern class of e; it is often convenient to omit the value e(e) = m from the notation. There are similar formulas for Chern classes of V e, since (3.3)

c,(2 e) =

[J (1 + (at, +

+ ai)t).

As in §1, these formulas may be expressed in terms of universal polynomials. Note, however, that the formulas for ck(e f) and ck(.Ve) depend

on the ranks s(e) and s(f) as well as on their Chern classes. To avoid

[I, §3]

CHERN CLASSES AND THE SPLITTING PRINCIPLE

15

this complication, we shall consider the restriction of c, to K = Ker(s), the elements of augmentation zero. In §1 we used universal polynomials to construct a A-ring A(A). One sees from the definition that the product and Ai take A°(A) into itself, so A°(A) becomes a 7-ring, but without unit. The identities (3.1)-(3.3) imply that ct: K -+ A°(A)

is a homomorphism of A-rings without unit.* We say that ct is a dhomomorphism in this case. Formulas (3.2) and (3.3) were deduced from the splitting principle. We shall see that, conversely, these identities imply the splitting principle, in an explicit form that will be useful later.

We shall now construct a graded ring extension of A in a way similar to the construction of K. from K in §2. First note that the polynomial ring A[W] (where W is a variable) has a unique grading extending that of A such that W has degree 1. Given an element c = 1 + Y_ ci t` E A°(A) and an integer m such that ci = 0 for i > m; let pc(W) be the polynomial m

P'(W) = Wm -

c1Wm-' +

... +. Cm = z (-1)ici Wm-i. i=0

Then p,(W) is homogeneous, and the factor ring A,: =

A[w],

defines a graded ring extension of A. The element w = W mod(pc(W)) is called the canonical generator of A,. For later use, we define the associated functional

gg:Ar *A to be the A-linear homomorphism such that 9`(w')

0 for0Sj<m-1, 1

for j = m - 1.

* One can also construct a d-ring structure on Z x A°(A), so that K -+ Z x A°(A),

x --. (c(x), c,(x))

is a homomorphism of A-rings with unit (cf. [SGA 6], p. 30 ff.).

16

d-RINGS AND CHERN CLASSES

[I, §4]

Theorem 3.1. Suppose c,: K -+ A°A is a A-homomorphism, and e is a

positive element in K such that e(e) = m and ci(e) = 0 for i > m. Then c, extends uniquely to a A-homomorphism c,: K. -, A°(Ac(e)),

such that if 8 and w are the canonical generators for K,, and A,(,), respectively, then c,(C) = 1 + wt.

Proof. Let K[T] be the A-ring extension defined in the proof of

Theorem 2.1, and extend c, to a homomorphism

c,: K[T] - A°(A[W]) by setting c,(T) = 1 + WT. It is straightforward to verify that this extension is also a A-homomorphism. To conclude the proof, we must show that for k >_ 1, ck(pe(T)) e J, where J is the ideal in A[W] generated by pc(e)(W). Equivalently, we must show that ck(Am(e - T)) E J.

Since c, is a A-homomorphism, we get c,(Am(e - T)) = Am(c,(e - T)) = hm(a),

where a = c,(e)/(1 + Wt). Therefore a = 1 + Z aktk, with

ak = (-

W)k-m(cm(e) - cm-1(e)W + ... + (- 1)myym)

for all k ? m. Therefore a, e J for all k ? m. Since Am(a) = 1 + Y Pk m(ai, ...,akm)tk,

and, as we saw in the proof of Theorem 2.1, each monomial in Pk, m(a) is divisible by some ai for i > m, it follows that Pk, m(a) e J, as required.

I §4. Chern Character and Todd Classes The splitting principle allows us to go further, by using systematically the factorization of the Chern polynomial c,(e) in linear factors 1 + alt. Let Cp(t) e Z[[t]]

or

lp(t) E Q[[t]]

[I, §41

CHERN CHARACTER AND TODD CLASSES

17

be a power series with integer coefficients, or rational coefficients if A is also a Q-algebra. To each such power series we can associate an additive homomorphism chq,: K -+ A

as follows. We first define ch, on E, by setting m

chor(e)

=

Y_ gq(at) i=1

Since we assume that the first Chern classes a; are nilpotent, the evaluation of the power series cp(a) is defined, and is a polynomial in at for each i. Furthermore, the value on the right-hand side is independent of the choice of the splitting. To see this, note that if W1,. .. ,W,, are new independent variables, then m

ao

t=i

p (Wit) _

Hj(sl, ... ,sj)tj, t=o

where If, is a polynomial of weight j with rational coefficients, and sj is the j-th elementary function of W1..... ,W.. We call Hj the associated Hirzebrnch polynomials. Then chor(e)

= Z Hj(c'(e),...,cj(e)).

It follows immediately that for e, e' e E we have ch.,(e + e') = chor(e) + chq,(e'),

so ch, is a homomorphism on the semigroup of elements of E. For any element x = e - e' of K we define ch1,(x) = chq,(e) - chq,(e').

It is trivially verified that this is well defined, i.e. independent of the representation of x as a difference of elements in E, and that ch., is a homomorphism of K into A. Explicitly,

ch,(x) = E Hj(cl(x), ... ,cj(x)).

The most important example is the Chern character written without subscript

ch:K -A

18

A-RINGS AND CHERN CLASSES

[I, §4]

such that ch = ch.,, where 9 is the exponential power series

cp(t)=exp(t)=Y_

Ck !

In this case, A must be a Q-algebra, or we tensor A with Q. Then by definition, if e = Y_ ui and ai = c' (ui), we have i=I M

ao

ch(e) _ E Y

it, k=o

ak kt

Proposition 4.1. The Chern character ch: K -. A is a ring homomorphism.

Proof. It suffices to verify this for products of elements in E. Say e = > ui and e' _ Y v;. Then ee' _ Z uiv;, and

ch(ee') _

exp(cl(uivj))

Y exp(c'(ui) + cl(v3)) i, f

= ch(e) ch(e')

as desired. Of course, we also have ch(1) = 1, as follows directly from the defini-

tion of c'(1) = 0. For the Chern character ch, the first few Hirzebruch polynomials Hi as mentioned above can be calculated to be: H1(s1) = s,. H2(11 S2) = 2 (s - 2s2), 1

H3(s1, s2, 33) = 3 (si - 3s,s2 + 3s3), H4(s1, s2, s3, s4) =

4!

(si - 4sis2 + 4s,s3

+2S2 - 4s4).

Remark on notation. The Chern classes are usually denoted by c, instead of c. To lay down the general formalism we thought it better to preserve the upper numbering, in order not to break the notational analogy, say of the power series c, with A,. However, we now see that this upper numbering is extremely disagreeable if we wish to substitute the

[I, §4]

CHERN CHARACTER AND TODD CLASSES

19

Chern classes in the Hirzebruch polynomials, because we have to take

ordinary powers. Hence in practice, one may revert to the lower numbering and write for instance ch(e) = e(e) + cl + -(ci - 2cz) + ....

We can perform a similar construction multiplicatively. Let (p(t) E 1 + tZ[[t]]

(P(t) E 1 + tQ[[t]]

or

be a power series with constant term 1 and integer coefficients, or rational coefficients if A is a Q-algebra. Then we define the corresponding Todd homomorphism on positive elements by m

td,(e) = fl (p(ai). i=1

If W1,. .. , Wm are independent variables, then we can write m

fl w(wt) _ Y_ Q;(s, ... s;)tj, where Q;(s,, ... ,sf) is a polynomial of weight j with integer (resp. rational) coefficients in the elementary symmetric functions s1,.. . ,sm of W1,... , W.. Again we call Q; the associated Hirzebruch polynomials. Then oD

td,,(e) = Y Q,(c'(e), ... ,c'(e)) J=o

is independent of the splitting of e. Thus

td,:K-31+A+ is a homomorphism from the additive group of K into the multiplicative group of units of A, and in fact those units which are of the form 1 + b with b nilpotent. If (p = fi is the power series /3(t) _ e` te`

1

,

where e = exp(t).

20

x-RINGS AND CHERN CLASSES

[I, §5]

then we write td(e) instead of tde(e), and call this simply "the" Todd homomorphism, determined by the original data of a Chern class homomorphism. In this case, A must be a Q-algebra, or we tensor A with Q. The first few Hirzebruch polynomials QJ can be calculated to be: Q1(S1) = z si, Q2(s1, S2) = 12 (s + S2),

_

Q301, S2, S3) - 214 S1S21

Q4(S1, S21 S3, S4) _ - 720 (S1 - 4s s2 - 3S2 - S1S3 +S4)-

Generalizations. 1. Even without the assumption that the c'(x) are

nilpotent, one can define a homomorphism

ch4,,,K - A[[t]] by ch,p,1(x) _ Y HJ(c1(x),... ,e'(x))t'. For the ordinary Chem character, ch,: K -> A[[t]]

is a ring homomorphism, as is

ch:K-*A, where A is the completion of A.

2. For Todd classes of positive elements e it is not necessary to assume that the constant term of tp is 1. One may define td, r(e) = (p(0)81e) + Y, Q;(e1(e)...... cJ(e))tj. j?. 1

This td0,, will take sums of positive elements to products. If tp is a polynomial, or if the c(e) are nilpotent, then tdm(e) e A. If (p(0) is a unit in A, then td,, extends to a homomorphism on all of K. 3. With a systematic use of symmetric functions and Hirzebruch polynomials, one may avoid any explicit use of a splitting principle.

I §5. Involutions We shall be concerned with .1-rings K which have an involution, by which we mean a homomorphism x"x" from the ring K to itself, satisfying

x"" = x,

s(x") = s(x),

and

u" = u-1 for ueL.

[I, §5]

INVOLUTIONS

21

We assume also that any positive element can be split in some extension K' to which the involution extends.

Lemma 5.1. Let e e E and 8(e) = m. Then for all i with 0 < i < m we have hi(e) = Am-i(eV)Am(e).

Proof. By definition, using a splitting, we get m

A(e)ti = 11 (1 + uit)

.1.,(e)

i=1

m(/

11 (uit)(ui It-1 + 1) i=1 m

m

i=1

1=1

m

m

= n ui tm fJ (I+ ui" t- 1) Tj ui.tm>,Ai(ev)t-i i=O

i1=11

l

m

= Am(e) ` /l.(e v )tm - i, i=O

which concludes the proof.

Conversely, if the formula of Lemma 5.1 is valid for e, then the involution v extends to an involution of Ke, with (" = C-1. This follows

from the equation (-1)m-iA.m-i(e")(P-1)i =

If c,: K -+ A°(A) is a Chern class homomorphism, then

C1 (U') = c'(u-1) _ - CL(u)

for a line element. From the splitting principle it follows that (5.1)

Ci(x") _ (-1)ici(x)

for all x e K. It follows that (5.2)

ch(x") = -ch(x).

22

A-RINGS AND CHERN CLASSES

[I, §5]

Another simple formula which follows easily from the splitting principle is Proposition 5.2. For a positive element e,

td(e") = td(e) exp(-cl(e)). Our main interest, however, lies in the next formula, which embodies a Riemann-Roch relation as will be seen in Chapter II, Theorem 2.1. Proposition 5.3. For a positive element e we have td(e) ch(A-,(e")) = c`°P(e),

where ch(2-1(e"))Y(-1)ich %(e"). Or in other words, ch(.l_1(ev)) = c`P(e) td(e)-1

Proof'. By definition, splitting e = I ui, with a; = c1(ui), we have i=j

td(e) _ fl Y(ai) = i

11 i

Also,

..,(e")=fl(1+uY t), i

whence

ch ),(e") _ fj (1 + ch(ui")t) i

=n(1+e-°"t) and therefore

ch A- (e') = r[ (1 - e-°") i

= fl (ea" - 1)/e°". i

Multiplying, we get td(e) ch(,1 -I(e")) = n ai = cm(e) i=1

This proves the proposition.

23

ADAMS OPERATIONS

[I, §6]

I §6. Adams Operations We return to a single 2-ring K. We define the Adams power series and the Adams operations Ii': K K by the formula

00

11,(x) = E(x) - t d log 2-,(x) = E '(x)t' j=o

Proposition 6.1. (i) (ii) (iii)

If u e L, then ii'(u) = u' for all j. For all j, the map O' is a ring homomorphism. Oi(O'(x)) = /''(x) for all x e K and all i, j.

Proof. The first assertion is immediate. For the second, it suffices to prove the homomorphic property for elements of E. The fact that 0j is additive is immediate, and that it is also a multiplicative homomorphism follows by splitting an element of E as usual, and by using the first assertion. The third statement is then clear since the desired relation is true on elements x = u in L. This concludes the proof.

Since Ii' is a ring homomorphism like the Chem character, we may call it an Adams character rather than Adams operation. We can also write : go

d

at log A (x) =

j=1

m

If e e E is a positive element, and e =

i=I

ui is a splitting, then

.1,(e) = rj (1 + uit) so

dtlogA,(e)-Y1+ut-Y(-1)j-1(ui+.+ Therefore, if N j is the (Hirzebruch-Newton) polynomial with integer coefficients such that WI 1 +

+ WJm = Nj(Sl,...,sm),

where s ... ,Sm are the elementary symmetric functions of W1, ... ,Wm, then i,1i'(e) = Nj(A'(e),... ,A"(e)).

[I, §6]

2-RINGS AND CHERN CLASSES

24

Let cp(t) be a polynomial, say with integer coefficients, and constant term equal to 1. In the present context there is a Todd homomorphism

td*:E - K from the additive monoid of positive elements to the multiplicative monoid of elements of K, by the same method as before. From a splitting of e we let td,,(e) = ]j cp(u;).

The value is independent of the splitting, and is a universal polynomial determined by cp alone. If cp(u) is a unit for each line in element u, then td, extends to a homomorphism from the additive group K to the multiplicative group K* (see Generalization 2 of §4). Let j be an integer ? 1. We let 0' = tdq,, where cp;(t) is the polynomial

cpj(t)+t+

+t'-1

1-t

Thus by definition, FEI

0'(e}=

1).

The classes 8'(e) are known as "Bott's cannibalistic classes". If it happens that j is a unit in K, then 0'(e) is a unit, and 0' extends to all of K. The next result is an analogue of Proposition 5.3, and will be interpreted as a Riemann-Roch theorem in Chapter 11, Theorem 3.1. Proposition 6.2. For a positive element e we have >j(A-I (e)) =

(e)0'(e).

Proof. Using the splitting, we get

O'(1-(e))fl(I -uj) -u;)fj(1 A- je)01 (e),

as was to be shown.

+uz+...+uj-')

[1, §6]

ADAMS OPERATIONS

25

The following proposition also follows immediately from the definitions.

proposition 6.3. Let c: K - A be a Chern class homomorphism. Then for all integers j > 1 and k > 1 we have chk *j(x) = jk chk(x),

where chic is the k-th graded component of ch.

On may also define t/r' for j < 0 by the formula .(x) = 0 -'(x V),

so that Proposition 6.1 continues to hold for this extended family of operations. We shall not need OJ for negative j, however. For a discussion of Adams operations on representation rings of finite groups, see [Ke] and [Kr].

CHAPTER II

Riemann-Roch Formalism

This chapter deals with the axiomatization of the functorial properties of the Grothendieck group K(X). The covariant and contravariant functorial properties of the K-functor, and another related graded ring functor A(X), are such that to prove the Riemann-Roch formula it suffices to do so for morphisms which generate the category. In geometry, there are two types of morphisms to which one reduces the proof: regular imbeddings;

projections from a projective bundle P(E).

In Chapter IV we describe the geometry of these morphisms. The regular imbeddings are local complete intersections. Among these are the elementary imbeddings which are the zero sections of a vector bundle. It turns out that any regular imbedding has a deformation to an elementary imbedding into the normal bundle. In Chapter V, we derive basic functorial properties of such morphisms on the K-group. They have simple algebraic formulations, and it turns out that these simple algebraic

properties suffice to give a proof of the Riemann-Roch formula. For example, in Chapter V, Proposition 4.3, we show that for a regular section f of a vector bundle E, if we let e = [E] be its class in the K-group, then

fK(1) ='1-i(e"),

where e" is the class of the dual bundle. We take this, and the analogous formula on the graded ring functor A, as the abstract definition of an elementary imbedding in the present Chapter II, §2. The essential part of the proof of Riemann-Roch for such a morphism, depending only on this property, was given in Proposition 5.3 of Chapter I. Similarly, Chapter V, Theorem 2.3 and Corollary 2.4 give the basic

structure of the K-algebra for a projective bundle. This structure was axiomatized in Chapter I, §2, and the Riemann-Roch formula using only these axioms is then proved in the present Chapter II, Theorem 2.2.

RIEMANN-ROCH FUNCTORS

27

Therefore readers may profitably read simultaneously Chapter V and Chapters I and II.

For a projective variety X, the ring A(X) can be taken to be the Chow ring of cycles modulo rational equivalence, tensored with Q. This requires more algebraic geometry, for which we refer to [F 2]. In

[SGA 6], Grothendieck showed how one could define a filtration in K(X) and how the associated graded algebra (tensored with Q) could be used instead of the Chow ring. We have taken this graded ring for A(X)

for the main statement of the Grothendieck Riemann-Roch theorem given in Chapter V, Theorem 4.3, complemented by the more geometric comments of Chapter VI, §5, especially Propositions 5.4 and 5.5 which relate the Grothendieck filtration to filtration by codimension. However, the axiomatization of Chapter II, §1 and §2, provides the algebraic formalism for other situations. Again, readers should compare immediately these two parts of the book, and the discussions of Chapter VI (giving other geometric contexts) to get a better feeling both for the underlying algebra, and the geometric applications which motivated it. Despite the fact that the algebraic formalism of the first three chapters originated in the theory of vector bundles, it exists independently of that

theory, and is applicable to the theory of group representations. An algebraist who wishes to disregard topology or vector bundles may therefore still understand the first three chapters without having to go through the algebraic geometry of Chapters IV and V. The fundamental reason why the general algebra was placed first was to exhibit clearly its independence from any of the multiple contexts in which it may be applied. For the context of group representations, we refer the reader to various papers of the Bibliography by Atiyah-Hirzebruch, Evens-Kahn, Grothendieck, Knopfmacher, Thomas.

II §1. Riemann-Roch Functors It is now convenient to view the objects we have defined so far in a functorial setting. We start with a category CY. We shall be concerned with functors on E which are simultaneously contravariant and covariant. Such a functor H assigns to each object X in (E; a ring H(X), and to each morphism f : X -* Y in E homomorphisms* f H: H(Y) -+ H(X)

and

fH: H(X) -' H(Y)

* Homomorphisms like f' and fH are usually denoted f* and f*. The more explicit notation is useful for Riemann-Roch, where several such functors are considered simultaneously.

28

RLEMANN-ROCH FORMALISM

[II, §1]

satisfying the following conditions:

F 1. X i-- H(X) is a contravariant functor fom C to rings via f'. F 2. X i--H(X) is a covariant functor from t to abelian groups via fH. F 3.

The projection formula holds, that is for all morphisms f : X -+ Y, and all x e H(X), y e H(Y) we have fH(x - MY)) = fM(x) - Y.

An important special case of the projection formula is the formula (1.1)

.fa(fH(Y)) = fH(1)Y

By a Riemann-Roch functor we mean a triple (K, p, A), where K and A are functors satisfying F 1 to F 3, and

p:K -A is a morphism of contravariant functors, i.e. for each X, px: K(X) -> A(X) is a ring homomorphism, and f APY (Y) = Px(f K(Y))

for all f : X -. Y, y e K(Y). We shall call p the Riemann-Roch character. In special cases it may

bear other names such as Chern character or Adams character, to emphasize the special features as they arise. These special cases will be dealt with in subsequent sections. We shall say that Riemann-Roch holds for a morphism f if, for some element T f e A(X), Pr fx(x) = fA(T f - Px(x))

for all x e K(X). That is, the diagram

K(X) sf p fKJ

A(X) hA

K(Y) P - A(Y) is commutative. As we have done, it is customary to omit the subscripts, writing p in place of px or pY. The factor rf measures the extent to which p fails to be covariantly functorial. We call T f the Riemann-Rock multiplier, or simply the multiplier. When precision is necessary we say that Riemann-Rock holds for f

RIEMANN-ROCH FUNCTORS

29

with respect to (K, P, A) with multiplier r f, if the preceding diagram is commutative.

Next we give some general criteria for Riemann-Roch to hold. Theorem 1.1. Let f : X --> Y and g: Y -* Z be morphisms. Assume that

Riemann-Roch holds for f and g with multipliers rf and Ty. Then Riemann-Roch holds for g o f with multiplier

rg,f = Proof. The routine is as follows: by R-R for g

PZ(9KJK(x)) = 9A(ry'PYIK(x)) = 9A(r, ' fA(r f ' PX(x)))

by R-R for f

= 9AfA(f A(TS)' rf ' PX(x))

by projection formula,

thus proving the theorem.

The next criterion will apply to certain types of imbeddings, first in the abstract context of Chapter II, Theorem 2.1, and then to geometric situations like Chapter V, Proposition 4.3. Theorem 1.2. If fK: K(Y) -+ K(X) is surjective, and there is an element r in A(Y) such that Pr(.fK(1)) = fA(1)r,

then Riemann-Roch holds for f with multiplier r f = f A(r).

Proof. Given x E K(X), let x = f K(y) with y e K(Y). Then PfK(x) = P.fK.fK(y)

= p(fK(1)y)

by projection formula

= P(fK(1))P(y)

= fA(1)rp(y)

by assumption

= fA(f A(rp(y)))

by projection formula

= JA(J A(t).f AP(y)) _ .fA(.f A(T )Pf K(y))

= fA(rf P(x)),

as required.

30

RIEMANN-ROCH FORMALISM

The next notion is an abstract version of the main properties of deformations which will be constructed in Chapter IV, §5. Let

f:X-*Y be a morphism. We shall say that f admits a basic deformation to a morphism f': X Y' with respect to (K, p, A) if there exist morphisms as shown in the following diagram: Y'

Y

and a finite number of inorphisms h,: C,, - M with integers m e Z satisfying the following conditions: BD 1.

For each x r: K(X) there exists some z e K(M) such that and

fK(x) = gK(z)

.f K(x) = g'K(z).

BD 2.

9A(1) = g'(1) + Y m,,hvA(1).

BD 3.

For each z e K(M) as in BD 1 and all v, hK(z) = 0.

BD 4.

g is a section of it, and it o g'- f' = f.

Theorem 1.3. Let f : X -> Y be a morphism which admits a basic deformation to a morphism f' for which Riemann-Roch holds. Then Riemann-Roch holds for f with multiplier Tr = T f..

Proof. Given x e K(X), choose z in K(M) as in BD 1. Then by BD 1

9AP.fK(x) = 9A PAZ) = 9A9AP(z)

by contravariance of p

= 9A(1)P(z)

by projection formula

= 9A(1)P(z) + Y

by BD 2

9A9AP(z) + Y_ m,,hVAhV'P(z)

= 9A Pg,K(z) + Y m,,

Ph5(z)

by projection formula by contravariance of p

= gA pg,K(z)

by BD 3

= 9APf' (x)

by BD 1.

RIEMANN-ROCH FUNCTORS

[II1 §11

31

We now apply itA. Since g is a section of it, PfK(x) = itA gA PfK(x)

= nAg'APf' (x)

by the preceding steps

=

by R-R for f'

= fA(Tf'(P(x)))

by HD 4.

This concludes the proof.

The reader may easily verify the following proposition.

Proposition 1A. If (K, p, L) and (L, a, A) are Riemann-Roch functors, then (K, up, A) is also a Riemann-Roch functor. If Riemann-Roch holds for f with respect to (K, p, L) (resp. (L, a, A)) with multiplier r f (resp. V f ), then Riemann-Roch holds for (K, up, A) with multiplier v f 6(T f).

A Riemann-Roch functor can be obtained in the context of Chern classes as follows. A Chern class functor on E is a triple (K, C, A), with

K, A functors satisfying F 1 to F 3 and for each X in E a Chern class homomorphism cx: K(X) -+ 1 + A(X)+

satisfying the following conditions: CCF 1.

Each K(X) is a I-ring with involution, and f" is a homomorphism of A-rings with involution.

CCF 2.

Each A(X) is a graded ring, and fA is a graded ring homomorphism of degree 0.

CCF 3. For f : X -+ Y, y e K(Y), we have

fAC(y) = C(fK(y))

Since fA and f K are ring homomorphisms, when A is a Q-algebra it follows that we also have the functorial rules fA ch(y) = ch(f K(y))

and

fA td(y) = td(f K(y))

We conclude :

If X i-' (K(X), c x, A(X)) is a Chern class functor, then

X is a Riemann-Roch functor.

chX, QA(X))

32

RIEMANN-ROCH FORMALISM

[II, §2]

On the other hand, we get a Riemann-Roch functor in a somewhat simpler situation as follows.

Let K be a functor from E to A-rings, satisfying F 1, F 2, F 3 and CCF 1. Then for each j > 0 the Adams character >/,'x = i' : K(X) -> K(X)

commutes with f", and therefore (K, fir', K) is a Riemann-Rochfunctor.

Such functors K will be called-ring functors and will be studied in §3.

H §2. Grothendieck-Riemann-Roch for Elementary Imbeddings and Projections We say that a morphism f : X -+ Y is an elementary imbedding with respect to the Chern class functor (K, c, A) if

fK:K(Y)-K(X) is surjective, and there is a positive element q in K(Y) such that

f"(1) = 2_'(q)

and

fA(l) = c`0 (q")

Note that fl is surjective whenever f is a section, i.e. there is a morphism a from Y to X with n o f = idx. The element q is called a principal element for the imbedding. Consider the associated Riemann-Roch functor (K, ch, A), assuming A is a Q-algebra. Theorem 2.1. Riemann-Roch holds for elementary imbeddings, with multiplier

if =

td(f"q")-'.

Proof. This follows immediately from Theorem 1.2, and Proposition 5.3 of Chapter I.

Next we shall consider a "dual" situation. A morphism f : X - Y will be called an elementary projection with respect to (K, c, A) if the corresponding map f1: K(X) -. K(Y)

[II, §2]

GROTHENDIECK-RIEMANN-ROCH

33

is isomorphic to the functional

fe:Ke -K associated with some positive element e of K = K(Y), and furthermore, letting c = c(e), if fA: A(X) --> A(Y)

is isomorphic to the functional

g,:A,->A. These functionals were defined in §1 and §3 of Chapter I, respectively. By this we mean that there are two commutative diagrams

A(X) -SA(Y),

K(X) --> K(Y),,

If,

fx

/9.

\fA

K(Y)

A(Y)

and that the top arrows are K(Y) (resp. A(Y)-) isomorphisms, viewing

K(X) as a K(Y)-algebra via fK, A(X) as an A(Y)-algebra via fA. Furthermore, under the identifications given by these isomorphisms, we require that CV) = w.

where ( and w are the canonical generators of K. and A.. Theorem 2.2. Riemann-Roch holds for elementary projections f, with multiplier

i f = td(te").

Proof. Let e(e) = r + 1. Since Ke is generated as a K-algebra by the

elements Ck, - r < k < 0, and fe and g, are linear over K and A, it suffices to show that ch fe(ek) = ge(td(Ce") . ch(Ck))

On the left, we have I

ch feyk) _ fO

if k = 0,

if -rSk<0.

On the right, we have gc(td(Le") exp(kw)).

34

[II, §2]

RIEMANN-ROCH FORMALISM

The formula to be proved is therefore a formal identity. Let x1,.. . ,xr+1 be Chern roots for e. Then w - xi are Chern roots for :Pe", so +1

td(ee") _ fl fl(w - xi), i=1

where j3 is the defining power series for the Todd class. The assertion therefore follows from the following purely algebraic lemma.

Lemma 2.3. Let x1i ... ,xr

be independent variables, and consider the

ring r+1

Q[[x1, ... ,xr+1]][T] fl (T - xi). i=1

The class of the power series r+1 ekT

II f(T - xi)

i=1

in this ring is represented by a uniquely determined polynomial in T of degree < r with coefficients in Q[[x1, ... xr+ 1]]. Then the coefficient o f Tr in this polynomial i s 0 f o r -r < k < 0 and is 1 for k = 0.

Proof. The following proof is due to Roger Howe. We first begin by standard considerations concerning polynomials.

Let R be a ring without divisor of 0. Let g(T)=Td-ad-,Td_1-...-ao

by a polynomial in R[T], and consider the equation

Let x be a root of g(T). Then by using the Euclidean algorithm or induction, we see that for any integer n > d there is a relation

T _ -a0.n+al.n+...+ad-1.nT d-1 with coefficients ail a R. If F(T) E R[T] is a polynomial, then we can write

F(x) = ao(F) + a1(F)x +

+ ad_

1(F)xd-1,

[II, §2]

35

GROTHENDIECK-RIEMANN-ROCH

where the coefficients a,(F) lie in R and depend linearly on F. Suppose that the equation has d roots and that there is a factorization

g(T) _ fl (T - x) i=1

Substituting x, for x with i = 1.....d and using Cramer's rule on the resulting system of linear equations yields Aa; (F) = W (F),

where A is the Vandermonde determinant, and Aj(F) is obtained by replacing the j-th column by `(F(x1 ), ... ,F(xd)), so

1

x, ... F(x,) ... xl 1 x2- 1 x2 ... F(x2) ...

1

xd

1

AJ(F) =

If A

...

F(xd)

... xd-1

0 then as (F) = A3 (F)/A.

If F(T) is a power series in R[[T]] and if R is a complete local ring, with x1,...,xd in the maximal ideal, and x = x, for some i, then we can evaluate F(x) because the series converges. The above formula for the coefficients a, (F) remains valid.

Let x1,... ,xd be independent variables, and let A be the ring

Q[[x1,...,xd]][T] u(T - xi). /i=1

Substituting some x, for T induces a natural homomorphism (pi of A onto

Q[[x1,...,xd]] = R. and the map (pH (tp,(z), ... ,(pd(z)) gives an imbedding of A into the product of R with itself d times. Now we let F(T) be the power series of Lemma 2.3, that is

d (T -

F(T) = eIT II i=,

eT-xi -

d

1

fl J3(T = ekT ,=,

- xi).

Under the substitution of some xJ for T it becomes a power series in xx and x2 - x,, and thus converges in Q[[x...... xd]].

36

[II, §2]

RIEMANN-ROCH FORMALISM

Then we obtain d

mod fl (T - x;),

F(T) _- ao(F) + ... + ad_ 1(F)Td-1

I=1

with ao(F),...,ad(F) e Q[[x1,...,xd]], and these coefficients are given by the formula with the determinants as above. This follows for the power series simply by taking limits of polynomials formally converging to the power series.

We now come to the heart of the proof which computes the last coefficient using the expression in terms of the determinants. Let A _ V(x1,... ,xd) where V denotes Vandermonde. We have

V(x 1, ... ,xd)ad- 1(F) =

Furthermore, xi- x"

Fx =ekxj xj n*j

We use the inductive relation of the Vandermonde determinants V (x 1 , ... ,xd) = V (x 1 i ... ,x j, ... ,xd)(- 1y - j fl (xj - xn)

n*j

We expand the determinant for ad_1(F) according to the last column, to get d

1

e('`+a-1)xjn

ad-1(F) _ Y

n,j exj - ex

j=1

We use the inductive relation, and replace x; by ex', which we denote by y1 for typographical reasons. We then get Yi+d-1 1

YI

...

1

Y2

..

I

Yd

V(Yv.. ,Yd)ad-1(F) =

Yi

Y2 2

Y2+d-1

Yd+d-1 yd-2

If k 0 then two columns on the right are the same, so the determinant is equal to 0. If k = 0 then we get the Vandermonde determinant on the

right, so ad_1(F) = 1. This concludes the proof of the lemma.

[II, §3]

ADAMS RIEMANN-ROCH

37

Summary. The results of this and the preceding section imply that, to

prove Riemann-Roch for a morphism f with respect to (K, ch, A) it suffices to factor f into a composite p o i, where p is an elementary projection, and i admits a basic deformation to an elementary imbedding.

II §3. Adams Riemann-Roch for Elementary Imbeddings and Projections In certain contexts of a Riemann-Roch theorem, the rings K and A are

the same. Or we may start with a functor K and obtain from it in a natural way various Riemann-Roch functors as we shall see in Chapter III, §4 involving both cases when K = A and K - A affecting each other. Thus we have to make an appropriate definition. We shall say that a

functor K from a category E to A-rings with involutions is a I-ring functor if it satisfies axioms F 1-F 3 and CCF 1, that is to each morphism f : X - Y in (E there are homomorphisms of abelian groups

fK:K(Y)->K(X)

and

fK:K(X)->K(Y)

such that: X F-+ K(X) is a contravariant functor of A-rings with involution via f K;

X H K(X) is a covariant functor of abelian groups via fK; the projection formula holds, that is for all morphisms f : X - Y, fK(xf K(Y)) = A WY,

all x e K(X), y e K(Y).

Except in Chapter VI, in this book our functors are both covariant and contravariant. In a context where singly variant functors occur as well, one might add the qualification that the above three properties define a doubly variant 2-ring functor.

Throughout this section we let K be a A-ring functor as above, so that we have Riemann-Roch functors (K, a/i', K) with integers j >-- 0 as men-

tioned at the end of §1. In all A-rings arising in the sequel (K(X), K(X)e, etc.,) if u is a line element we assume that I - u is nilpotent. A morphism f : X -,, Y is called an elementary imbedding with respect

to K if f K : K(Y) -' K(X)

38

RIEMANN-ROCH FORMALISM

[II, §3]

is surjective, and there is a positive element e c K(Y) such that fx(1) = A-1(e)

Remark. The surjectivity in practice comes from the fact that f is a section of a morphism Y - X. The additional property of a section plays no role here, but will play a role in Chapter VI, §1 and §2. Theorem 3.1. Riemann-Roch holds for elementary imbeddings, with respect to (K, O', K), with multiplier

Proof. This follows from Theorem 1.2; and Chapter I, Proposition 6.2.

A morphism f : X -> Y is called an elementary projection with respect to K if the corresponding map

fx: K(X) -' K(Y) is isomorphic to the functional fe:Ke-+ K

associated with some positive element e in K = K(Y). Theorem 3.2. Let f be an elementary projection. If j is invertible in

K = K(Y) then 9'(ee") is invertible in K. = K(X), and Riemann-Roch holds for f with respect to (K, O', K), with multiplier rf=j6'(et")-1.

Proof. We shall reduce the theorem to a formal identity of power series, similar to that of Theorem 2.2, over any ring where j is invertible. However, there is an alternative proof as follows. Since the identity is formal, one can verify it when K is replaced by Q ® K. In this case,

we shall prove in Theorem 4.3 of Chapter III (applied to the element q = - et" + 1) that Theorem 3.2 is actually a consequence of Theorem 2.2.

As to the power series proof, let us begin with the invertibility of O. If u is a line element then

1+u++uj-1

[II, §3]

39

ADAMS RIEMANN-ROCH

is invertible, because we can write u = (u - 1) + 1 and use the nilpotence of u - 1 and the geometric series to do the inversion. Let

i=1

be a splitting of e. We have by definition d

6'(e)= F1 (1+Ui+-+ i=1

so 8j(e) is invertible, and also O (et ") is invertible. Recall qd(u) = u'

for any line element u.

We must show that the following diagram commutes: Ke j9V(ee') 'O'' Ke

K Oj

rK

i.e. show that j.fe(0

(ee-1)-tqIi(x))

= JrVJx))

for

xeKe.

But fe is K-linear, and fJ is a ring homomorphism, so it suffices to prove this commutativity relation for the elements x = t- 0 < n < d - 1, which form a basis of K. over K. Recall that

0

if 1
The desired commutativity amounts to proving

{iIi 0

if n = 0,

if n=1,...,d-1.

At this point it is useful to adopt a notation which is used in the theory of formal groups, but the reader does not need to know any part of this theory. We begin with some general comments. Let A be a commutative ring and I an ideal such that every element

of I is nilpotent, or A is complete in the I-adic topology. We also

RIEMANN-ROCH FORMALISM

40

[II, §3]

assume that j is invertible in A. If a e I then 1 + a is invertible in A. For a, b e I we define

a[+]h=(1+a)(l+b)-1

a[-]b=(1+a)(1+b)-I-1.

and

The law of addition [+] is associative and commutative, and makes I into a group, where 0 is the additive zero element. This group is called the formal multiplicative group. For our purposes, we don't need to

know anything besides the definition of [+] and [-]. We also use the notation of "variables", so if we denote by Z the "variable" of this group, then we can write

Z[+]Z'=(1 +Z)(1 +Z')- 1. For any positive integer j we have the operator [j] which is defined by iteration :

[j]Z=Z[+]Z[+]...[+]Z

takenjtimes.

Then [j]Z - jZ mod Z2. We let a; = u; - 1, and for n = 0, ... ,d - 1 we let Z)"j

(1 + Then

d

Z[+]a;

1= 1 [.1] (Z [+] ai)

is symmetric in the a;, and is in fact a power series with

coefficients in K[[Z]] since we assumed j invertible in K. Furthermore

The formula for F. is of course motivated by the formal expression 9i(eC-I)-I

- F.'1

u-?'-' - 1

which does not make sense, but which is useful nevertheless. The Riemann-Roch formula is being reduced to the following result on formal power series.

Lemma 3.3. Let R be a commutative ring in which j is invertible. Let be defined by the al, ...,ad, Z be independent variables. Let above product, so that e R[[al, ... ,ad, Z]].

[II, §3]

41

ADAMS RIEMANN-ROCH

There exist unique elements bo"),... ,bd") E R[[s..... Sd]] (where s1, ... ,sd are the elementary symmetric functions of a1, ...,ad) such that d

F"(Z)

b0 (n) +

+ bd°! 1Zd-I

mod f Z[+]a1. i=1

and we have d-1

E(-1)"b(")=1/T if n=0, 0 if n = 1,...,d-1.

"=0

Indeed, the leading coefficient of fl Z [+] a; is fl (1 + a;) and is therefore invertible in R[[a1, ...,ad]]. The division algorithm applies to give the desired congruence and the uniqueness of the coefficients bo), ...,bd"? 1

We specialize to the case when a, = u; - 1 in K. We view t-1 - 1 as the generic root of fl Z [+] a, in

K[Z]/fl Z[+]a,. Since fe(e-1) = 0, substituting 8-1 - 1 for Z, and then - 1 for (-1 - 1 amounts to substituting - 1 for Z in the polynomial bo) + ... + b(dn)- 1Zd-1.

This accomplishes the desired reduction to the formal power series relation of Lemma 3.3. Furthermore, it suffices to prove the relation of Lemma 3.3 when R = Z[1/j] because in fact the coefficients bo"°, ...,bd! 1 lie in the image of Z[1/j] [[a,,. -.,ad]] and are determined by universal formulas, which we shall make explicit with the Vandermonde determinants, as in Lemma 2.3. So we let R = Z[1/fl.

The polynomial fl Z [+] a; has d roots

z, _ [-1]a, in R[[a1,...,ad]]. Let

A = V(z1, ...,zd) = Vandermonde determinant;

A,,(F") = determinant obtained by replacing the v-th column in the Vandermonde determinant by

...,F"(zd)), so

1

z1

F"(z1)

1

Z2

F"(z2)

... ...

1

zd

Fn(zd)

...

O"(F,,) _

zd1-1

zZ-1

zd -1

42

[II, §3]

RIHMANN-ROCH FORMALISM

Then Abv" =

and d-1

d-1 1)"b( n)

A V=0

_

v=0

d-1

(-1)°Av(F") d

1)k+1Fe(Zk)Mkv(Zl,

... Zd)r

v=0 k=1

where Mkv(zl, ...,Zd) is obtained by deleting the k-th row and v-th column from Vandermonde, that is the kv-minor of the Vandermonde determinant. We invert the order of summation and use the next lemma. Lemma 3.4. d-1

Mkv(Z 1, ... ,Zd) = V(z1, ... ,Zk, ... ,Z,) i#k

V-0

(1 + Z).

Proof. This is a special case of the Jacobi-Trudy identities, cf. [F 2], Lemma A.9.3, p. 422 which contains a short proof of the general identities. One can evaluate each term in the sum, namely Mkv(Z1,.,Zd) = V(z1,...,Zk,...,Zd)Sd-1-u(Z1,...,Zk,...,Zd),

where sd-1- , is the elementary symmetric function. The index p is used to index consecutively the variables with Zk omitted, sop = 1, ... ,d - 1.

The general identities give the value of a perturbation of the Vandermonde determinant, whereby the powers of the variables are increased by certain amounts, denoted by J; in the above reference. Here the amount

is 1 for all the powers from the v-th column onward. This shows how the lemma is a special case of the Jacobi-Trudy identities. Going on with the proof, since [j]Z

jZ mod Z2 we find

F"(zk) = F"([- 1]ak) _ (1 + zk)"'

a

a

1 i#k [11 (ai C-] ak)

_ (1 + zk)"3 1i#k

u, - uk

j Uk

uk

ut - Up

where ui = 1 + ai. But 1 + z; = u; 1, so by Lemma 3.4: A

a-1

(-1)vbv")

V-0 d

_

(-1)k+1uk-"j V(z1,...,Zk,...,z)-

ui - uk

t4

j

43

AN INTEGRAL RIEMANN-ROCH FORMULA

[II, §4]

We now use the recursive product for Vandermonde: V(T1,...,Td) =

(Tk - T).

V(T1,...,Tk,...>Td)(-1)d-k

i#k

On the one hand, we have Zk - zi = Cu, - uk)/Ukui, so that V(Z1,...,Zk,...,Zd) _ (- 1)d-kV(Z1,...,Zd)

u1

u k

i* Ui - Uk

On the other hand,

_

fl _ i#k U,

1)d-1+d-k

V(ui,.,uk,...,ue)

V(U11 ...IUD

uk

Therefore

lybw

_ 1 V(Z11 ...Zd) rd

L J J ul,...,ud k=1 .

1

(-')d -kni Uk

Uk

-11 V(uI/ ... ,uk,1 ... >Ud)

If n = 1,. .. ,d - 1 then this last expression is 0 because it is the expansion of a determinant with two equal columns. If n = 0, then the sum on the right is the expansion of V(uil, ...,ud) according to the last column, so we find the value le 3

thereby proving Lemma 3.3 and also Theorem 3.2.

II §4. An Integral Riemann-Roch Formula Although the formalism developed in §1 and §2 was based on having a ring homomorphism p from K to A, some of the same ideas can be used in other contexts. We illustrate this by a "Grothendieck-Riemann-Roch theorem without denominators", which can be used to compute Chern classes, and not just the Chern character (which requires denominators). Such a formula was first given by Grothendieck, and proved more generally by Jouanolou [J], cf. [BFM 1], [F 2]. First we establish systematically another general formula for the Chern classes. If r

e = Y_ ui i=1

44

[11, §4]

RIEMANN-ROCH FORMALISM

is a splitting of a positive element e, and a, = cl(uj), then c,(e) = fl (1 + a1t) 1=1

is a splitting of the Chern polynomial c,. We shall be concerned with the Chern polynomials of various combinations of positive elements. As we saw in §1, c,(a. (e)) =

11

ki<...
(1 + (ak, + ... +

We wish to compute the Chern class of a combination c(l-1(q v )e)

instead of ch(J -1(q v )e) which would introduce denominators. For the moment, let a1,... ,ar and b1,. .. ,bd be independent variables. We define the power series Qr.d(al.... ,a,; b1,...,bd)

to be the power series with integer coefficients given by the formula r

d

Qr.d(a,b)= 11 n !=1

fl

(1+ai-bk,-..._bk,)c-1N

This power series depends only on the integers d and r and has constant term equal to 1. It is symmetric in a1,...,a, and also in b1i...,bd. Therefore it can be expressed uniquely as a power series with integral coefficients in the symmetric functions s 1(a), ... ,sr(a)

and

s l (b), ... ,sd(b).

In addition Q(a, b) - 1

is divisible by the product b1 . . bd.

Proof. It suffices to show that Q - 1 is divisible by b1, since Q - 1 is symmetric in b1,. .. ,bd. If we set b 1 = 0 then each term in the product

with a given value of j and k1 > 1 cancels a term with k1 = 1 and j replaced by j + 1. This proves our assertion.

[II, §41

AN INTEGRAL RIEMANN-ROCH FORMULA

45

In light of the divisibility, there exists a unique power series P'pd(a, b) ("sp" for "split") with integral coefficients, such that Qr.d(a, b) = 1 + b1... bdP;°d(a, b).

Again, P;?d(a, b) is symmetric in a,,...,a, and also in b1, ... bd. Therefore there exists a unique power series P,.d with integer coefficients such that Prpd(a, b) ='r,d(sI(a),...,S (a); s1(b),...,sd(b))

In the context of A-rings and a Chern class homomorphism c, we can

now substitute first Chern classes for the variables a, b. If e, q are positive elements of augmentation r, d, respectively, we shall use the notation Pr,d(e, q) = P,,d(ct(e),...,c'(e); c1(q),...,cd(q)),

where c` are the Chern classes. We recall that in the context of Chern classes, if b3 = c1(vj), then c`°''(q) = b 1... bd,

The definition of Pr,d has been made in such a way that by applying the formula for c,(AVe) we immediately find:

Proposition 4.1. Let c he a Chern class homomorphism. For positive elements e, q of augmentations r, d respectively, we have c(A - I(q ')e) = 1 + c1OQ(q)P,. d(e, q).

This formula now looks formally similar to the formula in Proposition 5.3, which led to a Riemann-Roch theorem for elementary imbeddings.

Hence we are led to make the appropriate definitions in the present context. Let (K, c, A) be a Chern class functor. Let f : X -- Y be a morphism. We shall say that Integral Riemann-Roch holds for f if there exists a positive element q e K(X) such that for any positive element e in K(X) we have c(fx(e)) = 1 + fA(Pr, d(e, q)),

where d = e(q) and r = e(e). We call q a Riemann-Roch element for f. Theorem 4.2. Integral Riemann-Roch holds if f is an elementary imbedding.

46

RTEMANN-ROCH FORMALISM

[II, §4]

Proof. This follows from Proposition 4.1, exactly as Theorem 2.1 followed from Proposition 5.2 of Chapter I. Theorem 4.3. Let f be a morphism which admits a basic deformation to a morphism f' for which Integral Riemann-Roch holds. Then Integral Riemann-Roch holds for f with the same Riemann-Roch element q.

Proof. Identical with the proof of Theorem 1.3, replacing p by c; that proof did not require p to be a ring homomorphism.

CHAPTER III

Grothendieck Filtration and Graded K

The object of the first two sections is to construct from a A-ring K a graded ring Gr K, with a Chern class homomorphism satisfying the properties of Chapter I, §3.

III §1. The y-Filtration We let K be a A-ring as in Chapter I, §1. Define the operations

y':K->K by the series Y:(x) = A:/(1 -:)(x) = Y_ A`(x)(1

t

- t) =

ylx)t`.

Since t/(1 - t) = s is another parameter generating the power series ring

K[[t]] = K[[s]], we see that the y` also define a A-ring structure on K: that is, for all positive integers k we have y°(x) = 1, yl(x) = x and k

yk(x + Y) _

Y:(x)'l,k

i=o

`(Y)

In addition, it follows immediately from the definition that if u e L, then

7,(u - 1) = 1 + (u - 1)t

and so

y`(u - 1) = 0

for

y'(1-u)=Y_ (1-u)'t`

and so

y'(1-u)=(1-u)'

for i>=0.

i > 1;

48

GROTHENDIECK FILTRATION AND GRADED K

[III, §1]

Proposition I.I. Let e be a positive element of K with E(e) = m. Then m

m)tm-i = Y .1(e)(t

yi(e -

(a) 1=0

-

I)m-i

i=a

ym(e - m) = A _ 1(e") - Am(e)

(b)

Proof. From the definition of y, using a new variable v, we get

Me - m) =

Y

v = t-1 and multiplying by tm yields (a). For (b), set t = 0 in (a) and use Lemma 5.1 of Chapter I: m

ym(e - m) ° Y

)(e)(--1)m-

i=0 Y_ (_ 1)m-ir-i(ev)A.m(e)

_)_

i(e")1m(e).

This proves the proposition.

Next we introduce the Grothendieck y-filtration. We let

F1=F1K=Keys Then for n ? 1 we let F" = F"K = Z-module generated by the elements yr`(x1) . . . yrk(xk) with

x,,...,xkEF1

and

>ri?n.

It is immediately verified that this defines a filtration, and F" is an ideal for each n, because xyr'(x,) ... yrk(xk) _ (X - E (x))y,,(x) ... yrk(xk) + e (x)(...

and the first term on the right-hand side belongs to F"+1 It is convenient to have the filtration defined for all integers, so we let

F"=K

for nSO.

[III, §1]

49

THE y-FILTRATION

Since the augmentation gives a homomorphism of K onto Z, we have a natural isomorphism F°/F1 = K/Ker E

Z.

We note that in case K is generated by line elements, then Fl is generated over Z by elements u - 1 with u c L, and

F'= (Fl)' for all i >- 1. Indeed (u1 - 1)...(u" - 1) = yl(ul - 1)...yl(u" - 1)

so (F1)" c F". Conversely, it suffices to prove that y'(x) e (FI)' for all i >_ 1, and x e K. From the values y'(u - 1) and y-(l - u) which we derived previously, the desired inclusion follows at once.

We turn next to proving a Graded Splitting Property. Given a positive element e E K, there exists a t-ring extension K' (with involution if relevant) such that F"K' n K = F"K

for all integers n > 0. As in Chapter I, §2, we consider the extension

Ke = K[t], where &I is the generic root of the equation e(e)

Y- (- 1)( i(e)tfe(e)-i = 0.

i-0

From Proposition 1.1(a), setting t = 1 - :', we see that e - 1 is the generic root of the equation e(e)

i=o

(-1)iy'(e - E(e))(e - 1)`(e)-` = 0.

For present purposes it is convenient to let

s(e)=r+1.

50

GROTHENDIECK FILTRATION AND GRADED K

[III, §1]

By Chapter I, Theorem 2.1, Ke is a A-ring extension of K, so Ke has a y-filtLation F"Ke. We write F" = F"K. Recall that F° = K, if n <- 0. Theorem 1.2. For all integers k > 0 we have r

FkKe = Y Fk-i(e - V. i=o

Proof. Let x = ? - 1. Define 00

Rk=

i=0

Note first that the Rk form a ring filtration of Ke, i.e.,

c R,+k We need the following (*)

If y, z e K, and k is a positive integer such that yi(y) a Ri and yi(z) a Ri for all 1 < i < k, then yk(y z) e Rk.

The statement follows from the existence of universal polynomials Pk of weight k such that Yk(y . Z) = Pk(Y1(y),... ,Yk(y), Y1(z), ... ,Yk(z))

Next we claim that (1.1)

Rk = FkKe.

That Rk is contained in FkKe follows from the fact that ylx = x and yix = 0 for i > 1. For the reverse inclusion it suffices to show that if y e F'K, then yky a R, Writing y = aixi, a, a K, then e(ao) = e(y) = 0, so it suffices to show that for a a K, i > 0, we have yk(axi) a Rk.

This follows, by induction on i, from («). From the equation for x we have xr+1EFr+1+Fr.x+...+Fl.x.

[III, §1]

THE y-FILTRATION

51

It follows by induction on j that (**)

for all j > r. For if this holds for j, then x'+1EF;.x+F'-1.x2+...+Fi-r,xr+1

and F'-'xr+1 c F'-' (F'+ 1 + F' x +

+ F1 xr), so

+F'.x+...+F;-r+1.x'.

x'+1EF;+1 Finally we have the equalities (1.2)

k+r+1

r

i=0

i=0

Rk = Y F*-i.xi =

ZFk-i.xi r

The first equality follows from the equation KQ = Y F° x'. The second j-0

follows from (**), since for i > r, Fk - i . xf

Fk - i(Fi + F'-' - x + ... + F'-'- Xr) Fk + Fk - 1 . x + ... + Fk-,. X,.

The theorem follows from (1.1) and (1.2).

Corollary 1.3. Let fe: KQ -+ K be the functional such that ff(e) = 6i(e) for all i >_ 0. Then for all k,

ffe(F°Ke) e F. Proof. Immediate from Theorem 1.2 and the K-linearity of f. It follows from Theorem 1.2 that Fk = Fk(KQ) n K.

The graded splitting principle then follows as in the argument in Chapter I, §2 by constructing a chain of elementary extensions

KcK(1) =Kee...eK1' =K' so that e splits in K', and FkK' n K = Kk.

52

[III, §1]

GROTHENDIECK FILTRATION AND GRADED K

In the applications, the elements of F' will be nilpotent. In fact, something much stronger will be proved in Chapter V, Corollary 3.10, namely that F' = 0 for i sufficiently large. Here we give another proof of nilpotency, but the rest of this section will not be used any further in the book.

A line element u e L will be called ample for K if, given x c- K there is an integer n(x) such that for all n > n(x),

u"x=e - m for some positive e and some integer m. (To see where this terminology comes from, see Chapter V, Lemma 3.1.) Lemma 1.4. If u is ample for K, then for any v E L, v - 1 is nilpotent.

Proof For n ? n(- v- L) we may write

-v-'u" =e-m for eeE, m>0. Let w=vu-". Then mw - 1 =(e+v-lu")w- 1 =ew lies in E, so for a suitable positive integer k we have k

0 = Ak(mw - 1) = E (-1)k `d`(mw) i=0

(-

-I (mw)

1(w)'

w)'".

Thus 1 - w is nilpotent. The same argument, with v = 1, w = u-" shows that 1 - u-" is nilpotent for sufficiently large n. Therefore

1-v=(1-w)-v(1-u-") is also nilpotent.

Proposition 1.5. Assume that for each positive a in K there is an exten-

sion K' satisfying the graded splitting property for e, and having an ample line element. Then every element of FLK is nilpotent.

Proof. Immediate from Lemma 1.4 and the splitting property. We give an application of the splitting property.

[III, §1]

THE y-FILTRATION

53

Lemma 1.6. Given an element x e FK, there exists an extension K' of

K such that x can be written as a linear combination with integer coefficients

(u1 - 1)mi ... (iik - I)mk

with line elements ui and positive integers mi such that

Emi>n. Proof. This is a version for one element of a fact we have already noticed that F` = (F')i if K is generated by line elements. One could also apply Zorn's lemma to splitting extensions to get a huge extension K' which satisfies this property. Theorem 1.7. Let L be the multiplicative group of line elements. Then the map u H u - 1 induces an isomorphism

L = Gr1(K) = F'K/F2K. Proof The map is obviously a homomorphism into Gr'(K). We shall construct an inverse. As usual, let E be the set of positive elements. Let

det:E -*L be the map such that

det(e) = Ar(e)

if e(e) = r.

If e = e' + e" then det(e' + e") = det(e') det(e") from the addition formula for 7'(e' + e"), combined with the fact that 2`(e') = 0 if i > F(e) and similarly for e". Hence det is a homomorphism of E into L which extends to a homomorphism of K into L.

This map det is trivial on F2K. To see this, let x e F2K. By the splitting principle, in some extension of K we can write x as a linear combination with integer coefficients of elements (ul - 1)m, ... (Uk - 1 )mk

with line elements u, and positive integers in, such that Y_ mi > 2. Such an element contains some factor

(u - 1)(v - 1) = uv - v - u + 1, and for any line element w, it is immediate that

w(u-1)(v-1)=wuv-wv-wu+w

54

GROTHENDIECK FILTRATION AND GRADED K

[III, §2]

lies in the kernel of det, so FZK c Ker det as asserted. Thus det induces a homomorphism

det: K/F2K , L. Let g: L -p Gr'(K) be the homomorphism u i--* u - 1 mod FZK. Since det(u - 1) = det u = z, it follows that det o g = id. Conversely, to show that g o det = id on Gr'(K), we use the splitting principle. Let x e F'K so s(x) = 0. We can write

x=Yniui=>n,(u;- 1) with nl e Z and line elements ui in an extension of K. Then

det(x) = fl u and g o det(x) = x mod FZK. This concludes the proof.

For the interpretation in the geometric context, see the end of Chapter V, §3.

III §2. Graded K and Chern Classes Associated with the filtration F" on K, we have the associated graded ring

Gr(K) = Q Fk/Fk+1 k=0

When no confusion is likely we write G for Gr(K), and Gk = Grk K for the k-th graded piece Fk/Fk+' For a positive element e in K define the i-th Chern Class to be c'(e) = y'(e - c(e))

mod F'+1

so c'(e) e G'. If e(e) = m, and

with ul a L, then we put a1 = c'(u1) = u, - 1 mod P. Therefore ct(e) _

c'(e)t' =

m

[1 (1 + a, t). r=0

[III, §2]

GRADED K AND CHERN CLASSES

55

With the present definition of Chern classes, we see that the isomorphism L -+ Gr'(K) of Theorem 1.7 is given by the first Chern class u

c'(u)=u-1 mod F2.

The fact that yt defines a A-ring structure on K implies that the map defined on E by ct: ei--*Y c`(e)t`

and extended by additivity to K is a Chern class homomorphism in the sense of Chapter I, §3, provided c`(e) is nilpotent for i > 0. The splitting principle follows from the graded splitting principle of §1. Proposition 1.5 (using an ample element) or better Corollary 3.10 of Chapter V can be used to verify the axiom that the c'(e) are nilpotent. If we had not assumed CC 3 and had taken values of Chern classes in A in Chapter I, §3 we would not need nilpotency. As it is:

For the rest of this section, we assume that all elements of Gr' are nilpotent for k > 1. The same assumption is also made for Gr' K(X) when K is a A-ring functor below.

A homomorphism f K: K -* K' of A-rings maps Fk`K to F`K', so induces a homomorphism

fG=Gr(fK):GrK->GrK' of graded rings. This satisfies J'G(c`(x)) = c'(f K(x))

for

x e K.

Suppose fK: K' K is a K-linear homomorphism via f K. We say that fK has graded degree d for some integer d if

.fK(F"K') c F* "K

for all integers k. Then JK induces a graded homomorphism

of degree d. Note that d may be negative. This map is Gr(K)-linear, i.e. we have the projection formula fG(f G(Y)x) _ yfG(x)

[III, §2]

GROTHENDIECK FILTRATION AND GRADED K

56

The above discussion tells us how we shall obtain Riemann-Roch functors in practice:

Let K be a A-ring functor satisfying the above nilpotency condition. Then with respect to all morphisms which have a graded degree, (K, c, Gr K) is a Chern class functor and (K, ch, QGr K) is a Riemann-Roch functor as in Chapter 11, §1. Here QGr K denotes Q OO z Gr K.

With these considerations, we may apply the Riemann-Roch formal-

ism of Chapter II, taking the graded ring A to be Gr K. We consider first elementary imbeddings, then elementary projections. Proposition 2.1. (i)

Let K be a 2-ring and q e K a positive element. Let d = e(q). Then c10P(gv) _

(ii)

1(q)

mod F"+I

Let K be a A-ring functor and let f': X -+ Y be a morphism such that fK: K(Y) -> K(X) is surjective and

fK(l) = 1-I(q) for some positive element q a K(Y). Let d = e(q). graded degree d, and

Then f has

fc(1) = c`°P(qv).

Proof. By Proposition 1.1(b) we have y"(qv - d) = A_1(q)2"(qv)

Since l"(qv) is a line element, it follows that A"(qv) = 1 mod F', so c`°P(q v) = y"(q v - d) = A- (q) mod

F"+'.

This proves the first assertion. Since f K: K(Y) --> K(X) is surjective and commutes with a and y`, it follows from the definition of y-filtration that f' maps F'K(Y) onto FmK(X) for all m. Given x e FmK(X), write x = .f K(y),

y e FmK(Y)

[III, §2]

57

GRADED K AND CHERN CLASSES

Then fx(x) = fx(fKY) = Y Tx(1) = y.A_1(q)

is in F"+dK(Y), so f has graded degree d. The value ff(l) comes from the first part of the proposition. The conditions of Chapter II, §2 are therefore satisfied, and we have the

Corollary 2.2. With the assumptions of the proposition, f is an elementary imbedding with respect to the Chern class functor (K, c, Gr K). In particular, Riemann-Roch (K, ch, QGr K), with multiplier

zf =

holds

for

f

with

respect

to

td(fxq")-1.

For a positive element e in an arbitrary a-ring K, consider the extension Ke of K. In Theorem 1.2, let f K: K -> Ke be the natural inclusion. (In the applications, fl will arise from a i-ring functor.) In light of

Theorem 1.2, fK induces an injective homomorphism on the graded rings, which we denote fG: Gr(K) -> Gr(K¢).

On the other hand, by Corollary 1.3, the functional fe: Ke -+ K maps FkKe to F°-'K, where s(e) = r + 1. Therefore 1e induces a homomorphism on the graded rings, which we denote

f0:GrKe-*GrK, lowering degrees by r. Let:

w e Gr' K¢ denote the class oft - 1 mod F2Ke; r+1

pe(W) = E

(-1)`c`(e)Wr+1-:

i=O

If e splits into Y, u;, then p,(W) = fl (W - a,), with a; = cl(u). Proposition 23. Let c = c(e). There is a canonical isomorphism Or K¢

(Gr K),, = (Gr K) [W]/(p,,(W))

such that w corresponds to W mod(p,,(W)). The homomorphism

fG: Gr(Ke) - Gr(K)

58

[III, §3]

GROTFIENDIECK FILTRATION AND GRADED K

has the property that

(0

fo(w')

1

if 05j
In other words fo is the functional g, discussed in Chapter I, §3. Proof. Proposition 1.1 shows that w is a root of p,(W). The isomorphism is then a consequence of Theorem 1.2. By Corollary 1.3 the K-linear map fe maps FkKe into Fk_rK, so

if j < r.

f&(w') = 0

For j = r,

.fl(x'r) = E(f((j - 1)9) (/ 1

1_i r i=o

+ i\ (r) i

by Lemma 1.1 of Chapter 1.

z

But this expression is the coefficient of tr in the expansion of 1/(1 - t), so is equal to 1, as one sees from the expressions (1-t)r=L(_l)r-i(i

)

tr-i

and

I

(r

(1-t)r+l=jY=0 j

J) ti

This concludes the proof.

Corollary 2.4. Let K be a .l-ring functor. Let f : X -. Y be a morphism for which there exists a positive element en K(Y) such that K(X) is isomorphic to K(Y)e as a K(Y)-algebra via f K, and such that fK corresponds to fe. Then f is an elementary projection with respect to the Chern class functor (K, c, Gr K).

From Chapter II, Theorem 2.2 it follows that Riemann-Roch holds for f with respect to (K, ch, QGr K), with multiplier

if = td(8e1).

III §3. Adams Operations and the Filtration Let K be a .-ring. We want to see the effect of the Adams operations O' on the graded ring Gr(K). Our goal is Theorem 3.5, which will combine properties of Adams and Chern characters.

[III, §3]

ADAMS OPERATIONS AND THE FILTRATION

59

Recall Proposition 6.3 of Chapter I: Chk(ii'x) = Jk chk(X).

This tells us that if the Chern character is to give an isomorphism between QK and QGr K, then, if j > 2, the eigenspace of ii-' corresponding to eigenvalue jk should map isomorphically onto QGrk K. The y-filtration F" of K induces a filtration

QF"=Q©ZF" of

QK=QOzK. Proposition 3.1. Let j >_ 1. Let n be an integer > 0. If x e F" then

i'(x) - j"x mod F"+1 Hence Gr" K is an eigenspace for Gr i/i' with eigenvalue j".

Conversely, let j ? 2, and let x e QK. If

4li(x) = jx mod QFn+1 then x e QF".

Proof. For n = 0 the first assertion is immediate. Using the addition formula for the y's, one sees easily that it suffices to prove this first assertion for elements of the form x = y"(e) where e is positive. Using the assumption that there exists a 7-ring extension K' of K which splits e and such that F"K' n K = F"K,

we see that we may assume e split. Again using the addition formula for the y's, it suffices to prove the first assertion for elements of the form x = (ut - 1)...(u" - 1),

where u1,...,u" are line elements. But then n

/ fr(x) = fl(u -1)= i=1

n i=1

n

11(ui-1)11(1+ui+...+u-1) i=1

60

GROTHENDIECK FILTRATION AND GRADED K

and 1 + u; +

+ ul-1

[1I1, §31

j mod F1, so the first part of the proposition

follows.

As to the second, suppose jiJ(x) = j"x mod QFrs+1 Let m be the largest integer such that x e QFm, and suppose m < n. We have 0'(x) __ j"x

ir'(x) = jmx mod

and

mod QF"+1

QFm+'.

Hence

(j"- jm)xEQFm+1

which contradicts the definition of in. This concludes the proof of the proposition.

We now let

V=QK=Q®K,

so V is a vector space over Q. For each j > 2 and each integer m ? 0

we let:

Vj(m) = eigenspace for the operator /i' with eigenvalue jm.

Proposition 3.2. Assume that 17d+1 = 0 for some integer d. Then the space V'(m) is independent of j, and so can be denoted V(m), and a

QK = Q V(m). m=o

Proof. By Proposition 3.1 we have for any integer k ? 2 and m ? 0:

fl(0'-j")(,A k-km)=0,

nom

and in the product, we actually have a finite product since we can take n < d. Hence V(m) c Vk(m), so we have equality by symmetry. Again by Proposition 3.1, d

fl (f' -j")=0 n=0

on K,

and hence the left-hand side is also the 0 operator on V. Therefore there is a decomposition of the identity a

id = L F1 (01

-jm)/(jn - jm).

n=o ,n*n

The image of the m-th projection is V(m). This concludes the proof.

[III, §3]

ADAMS OPERATIONS AND THE FILTRATION

61

Remark. Since '(x) = x for all x, that is i' is the identity, it follows that the eigenspace V(m) is also an eigenspace for 0I with eigenvalue 1.

The following corollary merely gives a convenient reformulation of Proposition 3.2.

Corollary 3.3. For m > 0 we have a direct sum decomposition QFm = V(m) c QFm+I

We shall use this decomposition to get an isomorphism ch: QK -I, QGr K.

Assume that F`K = 0 for i sufficiently large. Define a map g:

by defining it separately on each component, and for x e QGrm, let g(x) = unique element x in V(m) such that x - g(x) mod QFm+ I

The existence and uniqueness of g(x) follows at once from the decomposition of Corollary 3.3. Since g is well defined, it follows easily that g is a ring homomorphism.

Proposition 3.4. If x = u - 1 mod F2 with u c L, then

g(x) = log(' + (u - 1))

Y, (-1)"-I

(u - 1)" n

Proof. It is immediate that the right-hand side mod FZ is equal to u - 1 in PIP = GrI. Since i/i' is a ring homomorphism, we can apply

alit term by term to get the eigenspace property for the expression on the right-hand side, as desired.

Theorem 3.5. Assume that F`K = 0 for i sufficiently large. Then the maps

ch:QK ->QGrK

and

g:QGrK ->QK

are inverse ring isomorphisms. In fact, for each integer m > 0, ch induces a Q-vector space isomorphism ch : V(m)

2

QGrm K.

GROTIIENDIECK FILTRATION AND GRADED K

62

[III, §4]

Proof. We may pass to an extension which splits a given element. In that case, it suffices to prove that the two maps are inverse to each other on line elements u and u - 1 mod F2. In this case the assertion is obvious from the definitions of ch and g. For x e QK write ch(x) = X chm(x) >o

with chm(x) e QGr' K. For example, ch°(x) = e(x).

Proposition 3.6. If ch(x) = 0 for all i < m, then

chm(x) = x mod QF" 'K. Proof. Let x be the image of x in QGrm K. Then g(x) - x e QFm+ I, and

ch'(g(z) - x) = 0 so

for

i < m,

ch' g(x) = chm(x), as was to be proved.

In a geometric context, the condition that F'K = 0 for i sufficiently large will be proved in Chapter V, Corollary 3.10.

111 §4. An Equivalence Between Adams and Grothendieck Riemann-Roch Theorems In this section we let K be a A..-ring functor. We suppose that for each X, there exists an integer d such that F'K(X) = 0 for i > d. Since we work with rational coefficients, we write K(X) for QK(X) and G(X) for QGr K(X).

It will be convenient to introduce the characters cpi: G(X) - G(X),

which are multiplication by jk on the k-th graded piece GkX. Each TJ is a ring homomorphism, and cp' o q)' _ gyp"'. If f : X - Y is a morphism, then f Gyp, = q)'f G, while if fG : G(X) -> G(Y) raises degrees by d, then pfG(x) = fG(j4 p'x)

[III, §4]

ADAMS AND GROTHENDIECK RIEMANN-ROCH THEOREMS

63

for x e G(X). (This trivial formula may be regarded as a Riemann-Roch formula for f with respect to (G, (p', G), with multiplier jd.) Proposition 6.3 of Chapter I reads (P' ch(x) = ch t/i'(x)

for x e K(X). Similarly (pi td(x) = td i/i'(x).

Theorem 4.1. Fix j >_ 2. Let f : X - Y be a morphism, let

teI+G+(X), and let d be a fixed integer. Then the following are equivalent:

(1) fK has degree d, and Riemann-Roch holds for f with respect to (K, ch, G), with multiplier T. (2)

Riemann-Roch holds for f with respect to (K, ur', K), with multiplier 0 e K(X) defined by ch(B) = jdr-IQ'(r).

0.

Proof. Note that ch is an isomorphism, so the equation in (2) defines Similarly let z e K(X) be defined by ch(z) = r-1

We shall use Theorem 3.5 as a matter of course, without further explicit reference.

Step 1. Suppose (2) holds. Then fK(z V(m)) c V(m + d). where V(m) denotes the eigenspace of ' with eigenvalue jm. To see this, if x c V(m), then

'.fK(z'x) _ f(0 ' /i'(zx))

by (2)

= .fK(ch - l(jd,r - 1(p'(r)) 1k' ch- 1(r - 1) O'(x))

= .fdz {{K(..i

fir' ch-1(e) i' ch-1(T 1)' 4G'(x))

= JK(JdZ'!mx) = jd+mfK(z'x), as required.

Step 2. (2)

(1). By Step 1, since FkK(X) = Q V(m), m?k

64

GROTHENDIECK FILTRATION AND GRADED K

[III, §4]

it follows that fK(F"K(X)) c Fk+aK(Y). To finish, we must verify that, for any y e K(X), Ch 1x(y) =

Let x = z

y, with z as above. The required formula is equivalent to

showing

ch fK(z x) = ff(ch(x)) It suffices to verify this for x e V(m), since K(X) is a sum of such spaces.

Then fK(z x) is in V(m + d) by Step 1. But then ch(x) e G-X is represented by x mod F"'K(X), and ch fK(z x) is represented by fK(z-x)modFt+a+'K(Y) (cf. Proposition 3.6). And

x - z x mod F'"+'K(X) since s(z) = 1. Hence fK(x) = fK(z x) mod F'"+a+'K(Y), and ch fK(z x) is represented by fK(x). Since x represents ch(x), fG(ch(x)) is also represented by fK(x) mod F' +a+'K(Y), which completes the proof that (2) (1).

Step 3. (1) showing that

(2).

Since ch is an isomorphism, (2) is equivalent to ch /'fK(x) = ch fK(e-i'x)

for all x e K(X). Now

ch fK(e 'x) = fc(r - ch(6 . ('x)) by (1) - i p (z) - ch(i'x)) = fG(z jati

=

ch(x)))

= P'fG(ti - ch(x)) = gyp' ch(fK(x))

by (1)

= ch( 'fK(x)),

as required. This concludes the proof of Theorem 4.1. To apply this theorem we will need elements i and 0 related as in (2). Such are provided by the following lemma. Lemma 4.2. Let K be a A-ring. Let q e K, with a (q) = d e Z. Then for any j >_ 2, we have

ch(e'(q))

where r = td(q")-'

=jai

- t gyp' (i),

[III, §4]

ADAMS AND GROTHENDIECK RIEMANN-ROCH THEOREMS

65

Proof. Since both sides are homomorphic in q; by splitting it suffices to verify the formula when q is a line element. In that case, let a = c'(q). Then

ch(9J(q)) = 1 + e° + ... + e(1- ')a,

td(q ) Tj td(q') _

a

1 - ea'

-e 1 - e'°'

and the lemma follows immediately.

From Lemma 4.2 and Theorem 4.1, we obtain: Theorem 4.3. Let f : X -> Y be a morphism, let q n K(X), and d = e(q). The following are equivalent.

(1) f has graded degree d and Riemann-Rock holds for f with respect to (K, ch, G) with multiplier td(q")-' (2)

For some j > 2, Riemann-Roch holds for f with respect to (K, OJ, K) with multiplier 6J(q).

(3)

Same as (2), for all j > 1.

Remark. We shall use Adams Riemann-Roch in Chapter V, §6 to show that certain morphisms have graded degree.

CHAPTER IV

Local Complete Intersections

We now switch from abstract algebra to algebraic geometry. This chapter describes in detail the basic category with which we shall deal in the context of algebraic geometry, namely regular morphisms. By

this we mean morphisms which can be factored into a local complete intersection imbedding, and the projection from a projective bundle. Of course, it must be proved that such morphisms form a category. We study the basic geometric objects associated with such morphisms, namely the normal and tangent sheaves. Such sheaves are related by exact sequences, which will be interpreted in K-theory in Chapter V. It is also natural to consider blow ups as part of the theory of projective bundles, and we give a concrete realization of the deformation of a regular imbedding to the normal bundle satisfying the axioms of Chapter II, §1 concerning basic deformations. In this chapter, we use Koszul complexes in connection with regular sequences and regular imbeddings. In the next chapter, we shall use Koszul complexes to calculate K-groups explicitly.

IV §1. Vector Bundles and Projective Bundles We first recall the basic notion Proj(5"), where .0 is a sheaf of graded Ox-algebras on a scheme X (cf. [H], II). Assume

>o

91 is a coherent sheaf of OX-modules, and 9' is locally generated by 91 as an algebra over Ox. Then

P = Proj (.'),

p: P - X

is a scheme over X, equipped with a canonical invertible sheaf Op(l) on P.

Locally X is Spec(A), and Y corresponds to a finitely generated

[IV, §1]

VECTOR BUNDLES AND PROJECTIVE BUNDLES

67

graded A-algebra S. Taking independent variables T0,...,T, corresponding to generators for S1, we have S = A[T0,...,T.]/I with some homogeneous ideal I. In this case P is the subscheme of P'A defined by the ideal I, and O(l) is the restriction of the canonical invertible sheaf on P. In general P can be patched together from such local descriptions.

A graded sheaf M of .9'-modules determines a sheaf of Op modules on

P, denoted .,t"'. For example Op(d) = Y(d)-.

where .9"(d) is the translated module whose k-th graded piece is 9k+a A surjection .9" -> 9' of graded OX algebras determines a closed imbedding

is P' = Proj (91') c* Proj (') = P, with i*Op(1) = Op.(l), and p o i = p'. By a locally free sheaf I on X we shall always mean that tin has finite

rank in addition to being locally free. For such 9, we let P1 = P(ct) = Proj (Sym I),

p: P(I) -> X

be the associated projective bundle. The natural action

Sym(9) ®I - SYm(I)(1) corresponds to a surjection of p*I' onto Op(l). Letting tr be the kernel, this gives the universal, or tautological, exact sequence

0_-.

--p*4'-,Op(l)->0

on P. If rank(f) = r + 1, then *' is locally free of rank r, and we call .;' the universal hyperplane sheaf on P(I). For another description of .*', see Proposition 3.13. The above sequence is universal in the following sense. If f: Z -> X is

a morphism and . is an invertible sheaf on Z, and

Ps -+9'

68

LOCAL COMPLETE INTERSECTIONS

[IV, §1]

a surjection, then there is a unique morphism g: Z -* P(s) with p o g = f, and an isomorphism of g*0p(1) with 2, so that the diagram is

g*p*g

g*tP(1)

;zu

f*e

commutes. In particular, any surjection of 9 onto an invertible sheaf 2 on X determines a section X -* P of p.

Given 9, the above considerations apply to the locally free sheaf 19 O Ox. We shall now globalize to P(4 ED Ox) the simple concept of a hyperplane and its complement in projective space. Let fir: P(S (D Ox) - X

be the corresponding projective bundle, and let 0-+ 2->0*(9(D Ox)-+0(1)-+ 0

be the universal exact sequence on P(B Q+ 0x). We call 2 the universal hyperplane sheaf on P(4 1) Ox). The projection of Q Ox -> 0x on the second factor determines a canonical section

of 0, called the zero section. This imbedding f will be our main example of the axiomatic notion of elementary imbedding introduced in Chapter II. Since 9 is the kernel of the projection from 9 p Ox to Ox, we have

f*2=9. The other projection 9 Q+ Ox -* 4' determines a closed imbedding

is P((r) i P(8 ® Ox) called the hyperplane at infinity. The vector bundle associated with of is defined to be n: V(4') -* X

where

V(.C) = Spec(Sym 4').

The surjection Sym(4') -* Ox sending 8' to 0 determines the zero section

g:X->V(8') of it.

[IV, §1]

VECTOR BUNDLES AND PROJECTIVE BUNDLES

69

It should be remarked that d' is the sheaf of sections of the dual of the bundle V(s). The natural open imbedding j: V( ,O,) - P(1: O Ox)

gives a decomposition of P(f Q+ Ox) into a disjoint union P(c;° O+ Ox) = V(s) U P(9),

after we identify V(s) and P(s) with their images under j and i respectively.

Locally, we describe the natural imbedding j as follows. Suppose X = Spec(A),

= E"",

where E is a free A-module. Then P(e Q+ Ox) = Proj (Sym(E

A)) = Proj (S[T]),

where S = Sym(E), and T is an indeterminate. Now i(P(.e)) is the subscheme defined by T = 0, and the complement is one of the basic affine open sets covering P(e Q Ox), namely Spec(S[T](T), where S[T]ET} is the subalgebra of S[T]T consisting of quotients of degree 0. Since

S[T]M

S = Sym(E),

this proves the first assertion locally. The compatibility of the morphisms then follows from the definitions.

We may summarize this in the diagram

V(.')r

'

' P(e O Ox)

4 PY)

x with the following commuting properties:

°j=ir,

j°g=f,P.

We may call P(' O+ Ox) the projective completion of V(').

70

[IV, §21

LOCAL COMPLETE INTERSECTIONS

IV §2. The Koszu] Complex and Regular Imbeddings We start this section with general facts about Koszul complexes in commutative algebra. Such complexes give explicit resolutions, and he at the base of what follows. We then translate this commutative algebra in the context of sheaves and give the applications to regular imbeddings.

Let A be a ring and E a finitely generated free module over A, of rank n. Let

a^ ... E

d' -A

be a homomorphism. Let I be the image of dl, so I is an ideal of A, and A/I is the cokernel of d,. Then we may form the Koszul complex

0AE

A"-'E

A1Ed-A l

- p,

where dP is defined by the formula P

(-1)f-ldl(tj)tl A...AtjA...A tP.

d,(tl A...At,)= Y_

j=1

We shall determine conditions under which the Koszul complex is exact (except for the last term), and so gives a resolution of All. Note that we have not excluded the possibility that I is the unit ideal. Suppose that I is generated by n elements, I = (a1,... ,a") such that if e1,.. . ,e" is a basis of E then a, = d1e;.

In terms of this basis, we let KP = APE = free module with basis {ej, A

A e;o},

it < ... < ip.

Then the boundary dP:KP_+Kp_1

is given by the formula P(e

la`'ei' A ... A elf A ... A e,P.

A ... A eip) j=1

Note that Ko = A. In terms of the choice of a basis, or of the sequence a1,...,a", the Koszul complex is denoted by K(a)

or

K(al,...,a").

[IV, §2]

THE KOSZUI, COMPLEX AND REGULAR IMBEDDINGS

71

One may also construct K(a) inductively: K(a1,...,an) = K(a,,...,an-1)© K(a.)

We say that (a) _ (a1,.. . ,an) is a regular sequence if 10 A, if a, is not

a divisor of 0 in A, and if the image of a; in A/(a1,...,a1_ 1) is not a divisor of zero. By the augmented Koszul complex, we shall mean the complex

-*Ko-*All -*0. where we stick All at the end. Proposition 2.1:

If a1,...,an is a regular sequence, then the augmented Koszul complex is exact, and so gives a resolution of All. (b) If A is local and Noetherian, and a1, ... ,an are in the maximal ideal of A, and the augmented Koszul complex is exact, then a1,.. . ,an is a regular sequence. is the unit ideal, then the Koszul complex K(a) is (c) If I = (a1,.. . (a)

exact.

For a proof of (a), cf. [L], XVI, 10.4. Since that reference does not include a proof of (b) and (c), we do it here. We go back to the notation of [L], XVI, proof of Lemma 10.3. If C is a complex, and x an element of A, then there is an exact sequence for p > 0: -'HP+1(C)-*HPt1(C)-

HP(C) -a H,(C)

HP+1(COO K(x))

H,,(C (D K(x)).

This exact sequence exists independently of any further assumptions on C. The map from HP(C) to HP(C) is multiplication by (-1)Px, Then (a) is proved immediately from this sequence and induction.

We now prove (b). Assume that A is local Noetherian, and that a1,. .. ,an lie in the maximal ideal. Let

C = K(a,,...,an_1)

and

x = an.

We use the end of the exact sequence H1(C (D K(x)) -> H0(C) - H0(C),

so the right arrow is multiplication by x. Since H1(C 0 K(x)) is assumed to be 0, it follows that multiplication by x is injective on H0(C),

72

LOCAL COMPLETE INTERSECTIONS

[IV, §2]

Hence a is not divisor of 0 in that factor ring. Furthermore, under the assumption that

which is

Hp+1(C O K(x)) = HP(C (& K(x)) = 0

for p > 1, the exact sequence implies that multiplication by x is an isomorphism on H,(C). Since x lies in the maximal ideal of a Noetherian ring, it follows that HP(C) = 0 by Nakayama's lemma. The proof of (b) then follows by induction.

As to (c), we use the same type of technique. To prove that the Koszul complex is exact, it suffices to do so when we localize at each prime ideal of A, so we may assume that A is local. Under the assumption that I is the unit ideal, some element in the sequence is a unit. After reordering the sequence, say x = a is a unit. In the long exact sequence, the map HP(C) --* HP(C) is which is an isomorphism. Therefore HP+1(C O K(x)) = 0, thus proving that the Koszul complex is exact. This concludes the proof of Proposition 2.1. The next theorem belongs to commutative algebra and will be applied to the geometric study of regular imbeddings and blow ups.

Let A[X] = A[X1,...,XJ, and let Q be the ideal of A[X] generated by

{a;Xj-ajX;},

I
Consider the canonical homomorphisms of graded A-algebras:

A[X]/Q

'

SYmA(I)

-) Q+ 00 1m,

M=D

where O(Xi) = a; e I = Symi,(1).

Theorem 2.2 (Micali). ( are isomorphisms.

If a...... a is a regular sequence then 0 and

Proof. By construction ifr and rp are surjective maps of graded algebras. It therefore suffices to show that if f is a homogeneous polynomial

in A[X] such that f (a, , ...,a.) = 0,

then f e Q. The proof will use the following lemma.

[IV, §2]

73

THE KOSZUL COMPLEX AND REGULAR IMBEDDINGS n

n

aif eQ, then

Lemma 2.3. If f1,...,fneA[X], and

Xifi eQ.

Proof. Let Ak = A/(a1i ...,ak). By assumption there are f , j e A[X],

i < j, with 21 ak fk = F, fi, j(a,Xj - ajX), i<j

k=1 or

>akhk=0,

(*)

k=1

where

hk=fk- Y- fkjXj+Y- fik'i. k<j

i
We show by descending induction on k that there are gi, k e A[X], i < k, with hk + L. ajgk,j = E, aigi,it j>k

i
and k-( (

1 Y- ail hi + Y ajgi.j J - 0. i

=1

\

j>k

For k = n, since a is a non-zero-divisor on Ai_1[X], from (*) we find gi, n e A[X], i < n, with hn = L, aigi,n i
and

n-1 i=1

ai(hi + angi,n) = 0.

Inductively, if (*)k+1 and (*)k hold, since ak+1 is a non-zero-divisor in Ak[X], there are gi, k e A[X], i < k, with (*)k

hk + L ajgk.j = L, aigi.k j>k

i
74

LOCAL COMPLETE INTERSECTIONS

[IV, §2]

k-1

E ai(hi + Y ajg,,j} = 0. j>k

i=1

This completes the inductive step. Now using the definition of hk, and then (*)k:

L Xkfk = L Xkhk + L fk,jXkXj - L fi,kXiXk

k=1

k<j

k

_

i
Xkhk k=1

E gi,ka1Xk - E 9k,jajXk j>k

i
Y gi,j(ai.Xj - aXi) i<j

which is in Q, as desired.

Now we conclude the proof of the theorem. Let f be homogeneous of degree m in A[X] with f(a1,...,an) = 0. We may write

f = Z(Xi-ai)fi, i=1

for some fi E A[X], deg fi < m - 1. Write m-1

f=>j=obj.i, where b, i is homogeneous of degree j. Equating homogeneous terms of degree 0, 1,...,m in (**), we obtain

[Laibo,i - 0, n

i=1

zXibk-l,i- Y-aibk,i=0,

i=1

i=1

n

f = Y_Xibm-,,i. i=1

0
[IV, §2]

THE KOSZUL COMPLEX AND REGULAR IMBEDDINGS

75

Now by the lemma, from (**)o it follows that > X; b0, i c- Q. Applying

the lemma and

(**)k

inductively,

we have Y-Xi bk, i e Q for k=

1, 2,...,m - 1. Then by (**)m, f aQ, as required. Corollary 2.4. If a1, ...,aq is a regular sequence, then the canonical homomorphisms

A/I[X1i...,Xj --> SymA,I(I/IZ) - O+ Im/I.+i m=0

are isomorphisms.

Corollary 2.5. The canonical homomorphism A[T2, ...

T2 - a2,.. . ,a1 T - an) --, A[a2/al, ... ,an/al]

which sends T to a;/al, is an isomorphism.

The first corollary follows from the theorem by tensoring with A/I. The second follows from the theorem by inverting the image of X1 and setting T = X i/X 1.

For later use we insert the following lemma. Lemma 2.6. Let I be a proper ideal in a Noetherian local ring A which is generated by a regular sequence. Then any minimal set of generators for I forms a regular sequence. Proof. If a ...,an is a regular sequence generating I, any minimal

sequence of generators of I must have the form b...... b., with n

Aja,,

bi = i=1

.li; e A, and A = (2ij) an invertible matrix. Then A determines an isomorphism of K(a) with K(b), which, by Proposition 2.1(b), concludes the proof.

The Koszul complex globalizes as follows. Let e be a locally free sheaf on X of rank n, and let

If

'

(9X

76

[IV, §2]

LOCAL COMPLETE INTERSECTIONS

be a homomorphism of S to the structure sheaf of X. We can form the Koszul complex

Me

0

d^

'

'0,

SOX

where dp is defined by "contraction", namely dP(tl A ... A tP) = j=1

l- 1)j- Ldl(tj)tl A ... A ti n ... A tn.

If

d'

8

OX

-0

is exact, then the Koszul complex is exact, because locally on X, it is just the same as the one constructed with a free module and we can apply Proposition 2.1(c). Hence we may say that the Koszul complex is the Koszul resolution of OX determined by dl.

On the other hand, let s be a section of S. Then s determines a homomorphism

dl = s": e" - OX. The image of s" is a sheaf of ideals, which defines a closed subscheme of X denoted by Z(s) and called the zero scheme of s. We then obtain the Koszul complex K(s): 0

)A"8

An-Lev_

)...

-,A1S"--+ 0,

0

in which dp is now defined by dn(tI A ... A tp) =

(-1)j-Ltj(s)tl

j=1

A ... A tj A ... A tP.

For x e X the stalk 9x is a free module over the local ring 0,,, x of X

at x. Taking a basis for 8, we can represent s by a sequence al, ...,a of elements of 0, . Then the stalk of K(s) at x is isomorphic to K(a). If x o Z(s), then the Koszul complex is exact at x by Proposition 2.1(c); and by Proposition 2.1(a) and (b) for x e Z(s), it follows that the following conditions are equivalent, and define what we mean by a regular section s :

The Koszul complex K(s) is a resolution of OZ(s)

In the above local representation, the sequence (al, a point of Z(s).

is regular at

[IV, §31

REGULAR IMBEDDINGS AND MORPHISMS

77

In this case, we call the following exact sequence the Koszu] resolution of OZcs determined by s:

An-I9v- ...

0------

A'

Lx->OZ(s)

0.

Next we give an important example of a regular section.

Given a locally free sheaf ° on X, consider the projective bundle i/r: P(9 O Ox) -> X with its universal exact sequence

0- 2->W(D09P- 19p(l)-*0. The dual of the first map gives a homomorphism from Op = Op to 2", which is a section s of 2". We call s the canonical section of 2". Proposition 2.7. The canonical section s of 2" is regular, and its zero-

scheme Z(s) is f(X), where f is the zero section imbedding of X in P(e Oa Ox).

Proof.

The assertions are local on X, so we may assume X =

Spec(A), and a is free with basis T,, ... ,T,,, so

P(t4O+ (9x) = Proj(A[T0,...,T])

The zero-scheme Z(s) is disjoint from the hyperplane P(s) = Z(T0) at infinity. On the complement V(s) = Spec A[TI, ...,T,], 7r = 0 1 V(dS), 2 restricts to n* f', and s is the tautological section of 7*gff, which define whose local equations are the regular sequence the zero section of V(.s'), as required.

IV §3. Regular Imbeddings and Morphisms in this section all schemes are Noetherian. Let is X -+ Y be a closed im-

bedding, and let I be the ideal sheaf defining X in Y. The conormal sheaf Wx1r to X in Y is the coherent sheaf of Ox-modules defined by ,exlr =

j,/j,2.

We say that i is a regular imbedding if every point of X has an affine neighborhood Spec(A) in Y such that the ideal of X in A is generated by a regular sequence.

78

[IV, §3]

LOCAL COMPLETE INTERSECTIONS

Proposition 3.1. Let is X -> Y be a closed imbedding. The following are equivalent: (i) (ii)

i is a regular imbedding.

Each point of X has a neighborhood U in Y such that there is a regular section of a locally free sheaf on U whose zero-scheme is

XnU. (iii)

For each x e X the ideal .x of X in 0,,y is generated by a regular sequence.

(iv)

For each x r= X the ideal J,, 6, r in the completion Ox, r is generated by a regular sequence.

Proof. The implications (i) a (ii)

(iii) are immediate from the defini-

tions. For (iii) . (i), choose an affine neighborhood U = Spec(A) of x such that there are elements a,,... a in the ideal I of X in A which give a regular sequence of generators for .F . Shrinking U, one may assume a ... ,a generate 1. Consider the Koszul complex 0.

Sitice this complex is exact at x, it is exact in a neighborhood of x, for example since the support of the homology is closed and does not contain x. The equivalence (iii) a (iv) follows from the fact that, for 0 = 0x,,., 0 is flat over 0. Therefore if a,, ...,a. is a minimal set of generators for .1, K(a) is a resolution of 0/3 0 if and only if

is a resolution. The proof concludes by Lemma 2.6, noting that a minimal set of generators for J,, is also a minimal set of generators for .51. Proposition 3.2. (a)

if i: X -+ Y is a regular imbedding, then the conormal sheaf W,jy is

locally free. (b) If X is the zero scheme of a regular section of a locally free sheaf

on Y, then

Wxjr = i*ev. Proof. (a) follows from Corollary 2.4 which implies that I/I2 is free over A/I. For (b), consider the Koszul complex Az 80 '

d2

._

"

-0

0.

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79

Since the image of d2 is contained in .f&", tensoring by O/J gives the required isomorphism

Corollary 3.3. If 9 is a locally free sheaf on a scheme X, then the zero section f : X -*P(W1 ®OX)

is a regular imbedding, with conormal sheaf 9.

Proof. This follows from Proposition 2.7, and the fact that f *9 = e. Proposition 3.4. If is X -+ Y and j: Y -+ Z are regular imbeddings, then j o is X -+ Z is a regular imbedding, and there is an exact sequence 0-1i*,iY/Z---I' WvX/Z---l,eXfY--+ 0.

Proof. If a1,...,am is a regular sequence generating an ideal I in a ring A, and b1,. .. ,bn are elements in A whose images in A/I form a

regular sequence, it follows immediately from the definition that

al, ...,am, bl, ...,b is a regular sequence. This proves that the composite of regular imbeddings is regular. For any closed imbeddings X c Y c Z one has an exact sequence of sheaves '9Y/Z ©6,

OZ-'WX/z"6xlY->0

on X. With regular sequences locally generating the ideals as above, one checks easily that this sequence is also exact on the left.

Remark. Given closed imbeddings i: X -+ Y, j: Y - Z, there are also partial converses to this proposition: (i)

If j o i and j are regular, and

(ii)

is exact and locally split, then i is regular. 1f j o i and i are regular, then there is a neighborhood U of X in Z such that the imbedding of Y n U in U is regular.

For proofs the reader may consult [EGA], IV.19.1, or [SGA 6], VII.1.

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Proposition 3.5. Let is X - Y be a closed imbedding, and let f: Y' be a flat morphism. Form the fibre square :

(a)

X,

i'

. Y'

X

i

)Y

Y

If i is a regular imbedding, then i' is a regular imbedding, and ex /r = g *rpx/r

(b) If f is surjective, and i' is a regular imbedding, then i is a regular imbedding.

Proof. Recall that f is flat if, for all y' a Y', letting y = f (y'), 0 = OY,Y, and br' = O,,.,Y., 0' is a flat 0-module. This means that if .,t is an exact complex of 0-modules, then V ©, 0' is also exact; in fact, since 0 - 0' is a local homomorphism, the converse is also true, i.e. 0' is faithfully flat over 0. Applying this when .J' is a Koszul complex yields the proposition.

Let X and Y be schemes which are regularly imbedded in a scheme Z. We say that X and Y meet regularly if at each x e X n Y, whenever a1,...,a,,, (resp. b1,...,bn) is a regular sequence defining X (resp. Y) in Z near x, then al, ...,a,,,, b1, ...,b is a regular sequence defining X n Y in Z near x. Equivalently, if s and t are regular sections of locally free sheaves of and whose zeroschemes are X and Y near x, then s E t is a regular section of Gyp 9 whose zero-scheme is X n Y. "

Proposition 3.6. If X and Y meet regularly in Z, then the inclusions i, j, k of X n Y in X, Y, Z are regular imbeddings, and ,OXnY/Z = 'XnY/X

r XnY/Y = i*'Y/Z©f*'XJ2

Proof. This is essentially the same as the proof of Proposition 3.4, so will be omitted.

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81

Recall that a morphism f : X -+ Y is etale if it is flat, and for all x E X, y = f (x), the induced homomorphism &.,x

"q, Y

of completions is an isomorphism. A morphism f : X -+ Y is smooth

for each x e X there are neighborhoods U of x, V of f (x), with f(U) c V, so that the restriction of f to U factors: if,

U g ). AV

P----4 V

with g etale and p the projection of a trivial vector bundle. We refer to [AK], VII for a readable account of basic properties of smooth morphisms.

A simple example of a smooth morphism is the projection morphism for a projective bundle or vector bundle. Such a morphism is locally isomorphic to a projection A', -,. V. In fact, these bundle projections are

the only smooth morphisms that are necessary for our treatment of Riemann-Roch, but we include general statements for completeness. Recall also that for a morphism f : X - Y, the cotangent sheaf f the conormal sheaf to the diagonal imbedding of X in X x.X.

y

is

Proposition 3.7. (a) (b)

If f : X -> Y is smooth, then L2' Y is a locally free sheaf on X. If f : X -3 Y and g : Y - Z are smooth, then g a f : X -+ Z is smooth, and there is an exact sequence 0 -p f *O'YJZ

S2X/Z

nz/Y - 0.

(c) If f : X - Y is smooth, and g: Y' --> Y is any morphism, form the

fibre square

X f--+ Y Then f' is smooth, and Ox',ly. = g'*S2X/r.

Proof. We note only that the assertions are evident for the case of bundle projections, and refer to [AK] for the general case.

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Lemma 3.8. If f : X -> Y is smooth, and is Y -+ X is a section of f, i.e. f o i = idY, then i is a regular imbedding, and ,eY/z = i*i2x/Y.

Proof. When f is a projection Spec(A) = V

A;, = Spec (A[T,,

a section i is determined by the choice of a ...,a a A, mapping T, to a;. Then the ideal of Y in X is generated by T, - al, ...,T - a,,, which form a regular sequence in A[TL,... ,T ]. One sees directly in this case that the canonical homomorphism from WY/, to i*S2z/Y is an isomorphism.

When f is etale, a section i of f must be a local isomorphism, so the lemma holds in this case. The general case follows easily from these two cases, for locally f is a composite X

s

,AY

P

)Y

of an etale g and a projection p. Form the fibre square

Zh,X g'z

AY

We have seen that g o i is a regular imbedding. The section i determines a morphism j: Y -+ Z with h -j = i, g -j = idY. Since q is etale, j is a local isomorphism. Since g is flat, h is a regular imbedding (Proposition 3.5). Therefore the composite i = h -j is a regular imbedding. To check that the canonical map from 1YJx to i*t2X1Y is an isomorphism, one may localize and complete, so one is reduced to the first case.

Remark. In our applications, we shall deal with projective bundles, and only the first case will be relevant. This is the reason why we gave it first, with explicit coordinates. However, it is worth pointing out that once the section i has been proved to be regular as above, then the isomorphism 0Y/, _. i*S2X/Y

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83

can be seen directly as follows. We work locally, with X = Spec(A), Y = Spec(B), A = B/I where I is the ideal of X in B and I is generated by a regular sequence. Then we have a map I , OXI /y/ic4 , = ally © B/I such that

bF+db

mod ISlz/y

The rule for the derivative of a product shows that I2 is contained in the kernel. Given b e B there exists a e A such that b a mod I and db = d(b - a). Hence our map is surjective, and is a homomorphism of B/1modules, that is A-modules. Both of these modules are free of the same rank, and hence the map is an isomorphism 1/I2xOXI/ i11 X/Y as desired.

Proposition 3.9. Consider a commutative triangle

P

with f a smooth morphism, and i and j closed imbeddings. Then i is a regular imbedding if and only if j is a regular imbedding, in which case there is an exact sequence 0 -

%/r

Proof. Form the fibre square

WX/P 11*S46

0.

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and let h: X -* Q be the section of g determined by j. Then g is smooth (Proposition 3.7(c)), so h is a regular imbedding by the lemma, with Wx/Q

(i)

h*nQ/x =^'ly-

Suppose i is a regular imbedding. Then i' is regular by Proposition 3.5, with qfQIP = g*rexly.

Therefore h*WQ,p = h*g*lex1y = Wx/r

(ii)

By Proposition 3.4 the composite j = i' o h is then regular, with the sequence 0 -+ h*WQ p -+'x1

'X/Q i 0

exact. Combining this with (i) and (ii) gives the exact sequence asserted in the proposition. Now assume that j is a regular imbedding. It remains to show that i must be regular. Factoring f locally as usual, it suffices to prove this

when f is etale or a projection AY -+ Y. When f is etale, h is a local isomorphism, so i' is regular, and Proposition 3.5(b) implies i is regular. For a projection AY -* Y, we may assume Y = Spec(A), with A local, so that h is the restriction of a section Y - AY of f, given by Ti -+ ai as in the proof of Lemma 3.8. If bl,...,b,,, is a minimal set of generators for the ideal of X in A, then (b1, ...,b,,,, T1 - al, ...,T,, - a.) Since is a minimal set of generators for the ideal of X in A[T,,... this sequence is regular in A[T1,... ,T,,,], it follows that b1, ... ,bis a

regular sequence in A, as required.

Corollary 3.10. Let f : X -. Y be a morphism which admits two factorizations

X `-+ P P -Y,

X J Q 9 .Y

with i and j closed imbeddings, and p and q smooth. Then i is regular if and only if j is regular.

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85

Proof. Compare the two factorizations with the diagonal:

X-b,Px,Q

Y.

Since p' and q' are smooth, the proposition implies that the regularity of i or j is equivalent to the regularity of S. Proposition 3.11. Consider a commutative triangle

X

Y

r\1° Z

with f and g smooth, and i a closed imbedding. Then i is a regular imbedding, and there is an exact sequence Q

T'X/Y

I*n,lz-*r' -40.

Proof. Form the fibre square

X --f --+ Z and let j be the section of p corresponding to i. Since p is smooth, j is regular (Lemma 3.8). Since q is smooth, it follows from the preceding proposition that i is regular, and Dar*XIY - WX,w -i*QW/Y -i0

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[IV, §3]

is exact. Since

x,w and 01X/Z' 1

the proposition follows.

Remark. Related to the last result is the general fact: if is X - Y is a closed imbedding of regular schemes, then i is a regular imbedding.

This follows from the fact that if A is a regular local ring, and I is ideal in A such that A/I is regular, then I is generated by a regular sequence (d [Ma], 17.F).

We shall say that a morphism f : X -> Y is a regular morphism if f factors into poi:

X 1-

where 9 is a locally free sheaf on Y, p is the projection, and i is a regular imbedding. It is a consequence of Corollary 3.10 that if f is factored into any closed imbedding j followed by any smooth morphism q then j must be a regular imbedding, but we do not need this fact. Our regular morphisms are what are often called projective local complete intersection morphisms.

In case X is a scheme over a field k, X is called a local complete intersection if the structure morphism from X to Spec(k) is regular in the above sense. Note that such X need not be a regular scheme, although we shall see that local complete intersections do share several properties

of non-singular varieties. Note also that fibres of a regular morphism need not be regular, or even local complete intersections. In order to see that regular morphisms form a category, we need an additional assumption, which will be valid for all schemes X considered in the next chapter: (*)x Any coherent sheaf on X is the image of a locally free sheaf.

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87

Proposition 3.12. If f: X - Y and g: Y -+ Z are regular morphisms, and (*)Q holds for all projective bundles Q over Z, then g -f is also a regular morphism.

Proof. Let f: X -> Y be a regular morphism and j: Y -> Q a closed

imbedding. We claim first that granting (*)Q, there is a locally free sheaf g on Q such that f factors into a regular imbedding

followed by the bundle projection from P(j*.') to Y. To verify this, let f = pl ° it be any factorization of f into a regular imbedding it of X in P(411) for some locally free sheaf 91 on Y, with pt the projection. Choose a surjection

for some locally free sheaf d' on Q. This surjection determines a closed imbedding of P(4'1) in P(j*e) which is regular by Proposition 3.11. By

Proposition 3.4, the composite imbedding of X in P(j*4) is regular, which proves the claim.

Now if f and g are regular morphisms, by the claim just proved we may find a commutative diagram (which one might call the staircase decomposition of g - f)

Xr

z

' P(j*.-)r

P(I)

where the vertical maps are bundle projections and the horizontal maps are regular imbeddings. Then g o f is the composite of j' o i and q o p'. To conclude the proof, it suffices to show that q o p' can be factored as a regular imbedding followed by a projective bundle projection. In other words, we must show that if is locally free on Z, Y = PJF, of is locally free on Y, then the composite of the two projective bundle projections

f : X = Pu4-. Y

and

g: Y= P.F -.Z

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is a regular morphism. Indeed, for n sufficiently large, g*(.9 ® Op,,(n)) is locally free on Z, and 9*9*(e (3 aPF(n)) - if ®(OPF(n)

is surjective. This is a standard fact, but we shall reproduce a proof in Chapter V, §2, R 4 and Proposition 2.2. This determines a closed imbedding

X

P(.9

dP.,F(n)) `* P(9*(' ® OP,,(n)),

as required.

Proposition 3.13. Let 41 be a locally free sheaf on a scheme Y, P = P(s) the associated projective bundle, f: P -+ Y the projection, and U

0

+

f *4'-° -> op(1)

,0

the universal exact sequence. Then f is smooth, and OP(- 1),

ill Ply =

so

ve,;:t SZp,y(1).

Proof. We have seen that f is smooth. Consider the diagonal P Pi

P

b

,P xyP

Y

If

Pz

P

Note that fop, = f o P2. The composite

Pig PhU 'Pif*.`=P2*f*.'

P*V

P2*01)

is a section of the locally free sheaf ,Y-(p W, p2* (90)) = p1 *--Y 0 p2* 00).

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Looking locally, one sees that this section is a regular section, whose zero-scheme is precisely b(P). It follows from Proposition 3.2 that reP,PXyP = b*(p .*' ©P2*0(1))v

or equivalently ply

which proves the proposition.

Remark. As D. Laksov has pointed out, the same argument applies to general Grassmann bundles. If G = Grassd(f°)

is the Grassmann bundle of rank d quotients of E, f: G -* Y the projection, and 0-+ 9 ->f*0'-+.2 ->0 the universal exact sequence on G, then f is smooth, and

Indeed, as .-t

above, G is the zero-scheme of a regular

section of

z(pif, pi ,2) on G x y G.

A regular imbedding i: X -+ Y has codimension d at x e X if X is locally defined by a regular sequence of d elements near x; equivalently, `'xir is locally free of rank d is a neighborhood of x. Since the rank of `Bxir is constant on connected components of X, the codimension is constant when X is connected. A smooth morphism f: X -+ Y has relative dimension n at x c- X if f factors locally near x into an etale morphism followed by a projection AV V; equivalently, cI , is locally free of rank n in a neighborhood of x. When X is connected, the relative dimension is constant. A regular morphism f: X -, Y has codimension d if f factors into

X t - P p>Y, where p is smooth (proper) morphism of some relative dimension r, and

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i is a regular imbedding of codimension d + r. It follows from the proof of Corollary 3.10 that d is independent of the choice of factorization. We mention another general fact, which has important applications for residues and duality but will not be used in this text. Proposition 3.14. Given a commutative triangle

X' >Y

assume that p is smooth of relative dimension n, that i is a closed imbedding, with X locally defined in Y by n equations, and that f is finite. Then i is a regular imbedding, and f is flat, so f*9x is locally free on Z. Proof Let x e X, and let A, B, C be the local rings- of X, Y, Z at x, i(x), f (x). Let b1,.. . ,b be a sequence of elements generating the ideal of

X in B. Let A, = B/(b1,...,b), so A0 = B and A. = A. Let k be the residue field of C. Since p is smooth, B ®ck is a regular ring, in particular Cohen-Macaulay. Since dim(A (Dc k) = dim(B ®c k) - n

it follows that the images of b1, ...,b, in B ®c k form a regular sequence. Consider the exact sequences

A; -Ai

Aa+1

0,

where cp; is multiplication by b,+1. By the local criterion for flatness ([Mat], 20.E), from the injectivity of cp; ®c k follows the injectivity of gyp,

and the flatness of its cokernel Ai+1. This implies that b,,. .,b. is a regular sequence, and A = A. is flat over C. Corollary 3.15. In the situation of Proposition 3.14, for any base change Z'-a Z the induced imbedding

XxZZ'->YxZZ' is a regular imbedding. In particular, a base change of a finite flat regular morphism is finite flat regular.

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91

IV §4. Blowing Up Y be a closed imbedding, and let 5 be the ideal sheaf of X in The blow up of Y along X, denoted B1XY, or B, is the projective scheme over Y constructed from the graded sheaf of OY-algebra OJm: Let is X Y.

B1 XY = Proj ( Q .fm). m?0

Let q: BI XY -+ Y be the structure morphism, and let E be the exceptional divisor, i.e the inverse image of the scheme X:

E = cp-'(X). The fibre square E

B1XY

will be called the blow up diagram of the imbedding i. Let 0(1) be the canonical invertible sheaf on BI1Y (§1). Lemma 4.1. (a)

As a scheme over X via q/, E = P r o j ( Q fm/fm+ 11 m>0

/J

The imbedding j of E in B1XY is determined by the surjection

O'Xm-, Ojm/jcm+1 of graded algebras.

(b)

E is a Cartier divisor on B1XY,, with ideal sheaf

0(-E) = (9(1). (c)

If f : Z -* Y is a morphism such that f -'(X) is a Cartier divisor on Z, then there is a unique g: Z -> B1XY so that f = (p o g.

Proof. For any (p: Proj (5) - Y, and X c Y, (p -1(X) = Proj (° 0 r Ox).

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Since .gym ®r O,r = fm ®r 0,.1.f = .gym/jPn+ 1, this proves (a). For (b), the

ideal sheaf of E in B is defined by the graded sheaf of ideals

O jm+1 = Y(1)

mz0

in .9, which defines 0(1). To prove (c), we may assume Y = Spec(A), X is defined by an ideal

I = (a,,. _,a ). Then we get a surjection

by 7 - ' a , inducing a closed imbedding of B1xY in P". By the universal property of projective bundles, f factors uniquely through g: Z -> P°r- 1, by T --> f *(ai). One checks easily that this g factors through Bl Y We shall require the case when i is a regular imbedding, so that we may take al,... ,an to be a regular sequence. By Theorem 2.2, BIxY is the subscheme of PA 1 defined by the equations.

a;T -a1T =0,

1 _i<j<-n.

The lemma is particularly evident in this case. A consequence of (c) is the

Lemma 4.2. If i imbeds X as a Cartier divisor on Y, then q:BIXY -* Y is an isomorphism.

Proposition 43. (a)

If is X -+ Y is a regular imbedding, with conormal sheaf Wxly, then E = P(`L xrr).

(b) If X is the zero-scheme of a section of a locally free sheaf d',

(c)

there is a canonical imbedding of B1xY into P(4") over Y, which is a regular imbedding if i is a regular imbedding. Let is X -+ Y be a regular imbedding. If the ideal sheaf is the image of a locally free sheaf (in particular, if condition (*)x at the end of §3 is satisfied) then the morphism Blx(Y) -+ Y is a regular morphism.

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93

Proof If i is regular, Corollary 2.4 yields Sym0.(5/52) x O fm/ frn + m20

which proves (a). A section s of it ' determines a homomorphism s" from

if" to Oy whose image is the ideal sheaf of the zero-scheme X. This induces a surjection of graded algebras

Sym(') - 0 sm -+ 0 m20

which determines the imbedding of B1XY in P(d"). If i is a regular imbedding, locally on Y we may write X as the zeroscheme of a regular section t of a locally free sheaf F. Localizing further

if necessary, we may assumes' factors into

e" Ua.FV `toy for some surjective homomorphism u. This gives

Bl Y C P(.F') c P(01"). The first of these inclusions is a regular imbedding by Corollary 2.5, the second is clearly regular (Proposition 3.11), so the composite is regular (Proposition 3.4). This proves (b). Since a surjective homomorphism 9" -+ 5 -' 0 is equivalent to a section of 9 whose zero scheme is X, (c) follows from (b).

Let E be closed subscheme of a scheme B. Assume E is a Cartier divisor on B. Let V be a closed subscheme of B which contains E as a subscheme:

EcVcB. The residual scheme to E in V on B is defined to be the subscheme R of B whose ideal sheaf 5(R) is related to the ideal sheaves .f(E) and 5(V) of E and V by the equation

5(R) f(E) = 5(V)

or

5(R) = 5(V)5(E) -1.

Since 5(E) is invertible, this determines 5(R) uniquely. Note that R is a subscheme of V; local equations for R in B are obtained by dividing local equations for V by a local equation for E. Let X Y and Y---. Z be closed imbeddings. Let B = BlZ be the blow up of Z along X, with exceptional divisor E, and let Y = BlxY be

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the blow up of Y along X, with exceptional divisor X. We have the blow up diagrams :

X, Y

E--->B and

X>Z There is a unique closed imbedding of f = B1XY in B = B1XZ such that the diagram Lemma 4.4.

commutes, and X = E n Y. Proof. If f is the ideal sheaf of X on Z, then Sd r is the ideal sheaf of X on Y, so there is a surjection of graded (l? -algebras

(@ .f' &Z 0, -' Q (.f 0y), -. 0.

This corresponds to an imbedding of f in rp-'(Y). The other assertions are special cases of Lemma 4.1(c).

One calls Y = B1XY the proper transform of Y in B = B1XZ. It is a closed subscheme of the total transform cp-1(Y). Note that

Eccp-1(Y)cB. Theorem 4.5. If X -> Y and Y

Z are regular imbeddings, then Y is the residual scheme to E in (p-1(Y) on B. In addition the imbedding of Y in B is regular, with conormal sheaf W Y,13 = rl *wy/Z ® &Y(X )

Proof. Assume that Z = Spec(A), and Y is defined by a regular sequence a1,. .. ,ad in A, and X by a regular sequence al,... ,a , n > d. As we saw in the proof of Lemma 4.1, B is the subscheme of P,-' defined by the equations

aiT=a, T,

1:5i<j_<_n.

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95

Similarly, Y is the subscheme of r,-' defined by equations

d
and

T=O,

1<_iSd,

where a; is the image of a, in A/(a1,... ,aa). Therefore the proper transform Y is the subscheme of B defined by equations T1=...=Td=O,

al =...=aa=0.

The total tranform (p-'(Y) is defined by equations

Let U. c B be the affine open set where Tk 0 0, so Uk = Spec(A[tl,... ,tk, ... ,tn]/({ak ti - a;

where t; = T/Tk, by Corollary 2.5. On Uk the exceptional divisor E is defined by one equation ak = 0. We must show that the ideals defining (P '(Y), f, and E, on Uk are related by the equation I ((p -1(Y)) = I(Y) I (E).

If k < d, then Y is disjoint from Uk, and I((p-1(Y)) = (al, ...,ad) _ (ak) = I(E) since a; = t;ak. Similarly, if k > d, I((p- 1(Y)) = (al,...,aa) = (tlak,

... ,td(Zk)

_ (t

Since t1, ...,t, form a regular sequence in the coordinate ring of Uk, k > d, it follows that the imbedding of k in B is regular. It remains to verify the asserted relation between conormal sheaves. Starting with the residual relation

5(Y)OB = 5(r) 5(E), one deduces a surjection of sheaves on Y:

rl*(f(Y)/5(Y)2)

/s V)2 ® g*5(E),

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[IV, §51

where g is the inclusion of Y in B. Since both sides are locally free of the same rank, this surjection is an isomorphism. Since E r Y = X, 1*(Jx,Y & TYle o or(-9),

as required.

IV §5. Deformation to the Normal Bundle Let f : X -+ Y be a regular imbedding of codimension d, with conormal sheaf

Let (5.1)

f':X-->Y'=P(`,I,©dx)

be the zero section (see §1). We shall describe a deformation of f to f'. This "linearization" of f will be a concrete realization of the basic deformation considered in Chapter II, §1; we will construct a diagram Y'

Y \0 X

M

with 7t o g = id, and n o g'- f' = f. First consider the projective line over Y: p: PY = Proj (&1[To, T1]) -, Y.

There are two canonical sections of p: (5.3)

so: Y --*PY

and

sue: Y -PY,

determined by To i-* 1, T, H 0 and To -- 0, T, H 1 respectively. These sec-

tions each imbed Y as a Cartier divisor on PY , with trivial conormal sheaf (Corollary 3.3). Let Y(0) = so(Y), Y(oo) = s.(Y), and X(0) = so(f(X)),

X(oo) = sX (X)).

(IV, §51

DEFORMATION TO THE NORMAL BUNDLE

97

Define M to be the blow up of P,'. along X(oo), that is (5.4)

and let gyp: M -

M = BlXC.XPY).

P,'.' be the canonical morphism. Since

PX n s o(Y) = X(oo),

it follows from Proposition 3.6 that the conormal sheaf to X = X(oo) in

P' is given by Ox

WX/Pr = WX,Y

Therefore Y' = P(' (5 Ox) is the exceptional divisor for the blow up M - PI'. , which determines an imbedding g': Y' -> M

(5.5)

making Y' a Cartier divisor on M. We get the blow up diagram :

P(Wxlr (D Ox) 9 ' M

X-------------------- > PY

S,or

We define x: M -), Y to be the composite (5.6)

M I P,'.

'

Y.

From the definition, -go g' o f' = f. Since (p is an isomorphism over the complement of s,,,)(X), and so(Y) is disjoint from s,,(Y), the section so determines a section

g: Y-+M

(5.7)

of it, which makes Y a Cartier divisor on M. This completes the construction of the basic diagram. From the composite

S`Pj'. we obtain a closed imbedding of Y = B1XY in Blx(Pr'.), i.e. (5.8)

h: Y = BlXY - M.

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LOCAL COMPLETE INTERSECTIONS

[IV, §51

By Theorem 4.5, h is a regular imbedding of codimension 1, i.e. Y is a Cartier divisor on M; and k is the residual scheme to Y' in qo-1(Y(oo)), i.e.

f(co-1(Y(oo))) _

Since these are ideal sheaves of Cartier divisors, the equality of ideal sheaves implies the equality (5.9)

+ Y'

c0-(Y(00))

of Cartier divisors on M. Since W*(Y(0)) = Y,

(5.10)

imbedded in M by g, and O(Y(0)) (5.11)

(9(Y(oo))

(p*t(1), we deduce:

(9(Y) x O(Y + Y') = t(13) © O(Y').

It also follows from Theorem 4.5 that the exceptional divisor P('O+ Ox) intersects B1xY regularly in the scheme P(W): Y' n Y = P(W).

(5.12)

Note that P(s) is imbedded in Y = BlY as the exceptional divisor, and in Y' = P(c' (1 Ox) as the hyperplane at infinity. The imbedding of X in Y induces an imbedding of PX in PY P. From the composite

X

Sm. Pz

' Pr

we obtain an imbedding of BlxPX in BlxPY (Lemma 4.4). Since X is a Cartier divisor on PX, B1xP,1C is just Px1 (Lemma 4.2), so we have a closed imbedding (5.13)

F: P1 -+ M

which is regular by Proposition 4.5.

[IV, §5]

DEFORMATION TO THE NORMAL BUNDLE

99

We may summarize the results of this section in the deformation diagram

X r, Y'+Y 91h PX F )M

Is

s'1

W

eT SOI

A,

J

)Y-

-*P1 ISO

idY

Y

Each square in this diagram is a fibre square; the vertical maps are

imbeddings of Cartier divisors. Note that f'(X) is contained in Y' and is disjoint from P.

Remark. If f : X - Y is a closed imbedding which is not necessarily regular, the same constructions yield a deformation diagram with exactly the same properties. The only difference is that Y' = P(16 G P) must be replaced by the projective completion of the normal cone to X in Y, i.e.

Y' = Proj(b[T]),

where 9 = 6) J'"/Jii+1, 5 the ideal sheaf of X in Y, and T is a varim?0

able of degree 1. In this case f and F are closed imbeddings which may also not be regular. There are no essential changes in the proof, although Theorem 4.5 as stated does not apply in this situation. In fact, if Y = Spec (A), I is the ideal of X, and one identifies the complement of Y(O) in Pll. with

AY = Spec A[ 7],

then the complement of (p-1(Y(0)) in M is Proj

(1,

T)"l,

00

from which it is easy to verify (5.9) and (5.12) (cf. [F 2], 5.1).

CHAPTER V

The K-Functor in Algebraic Geometry

In the previous chapter we described the geometry of regular morphisms. Here we describe one part of their homology, namely the K-functor on the category of locally free sheaves (which are always assumed to be of finite rank). An arbitrary locally free sheaf does not behave well under the direct image. Fortunately, the K-group generated by the locally free

sheaves is also generated by a subfamily of sheaves which do behave well, and which we call regular sheaves. We use a cohomological characterization for these, due to Mumford. These regular sheaves allow us first to determine K(P(.9)) for a pro-

jective bundle as an algebra over K of the base. Then we use them to analyze the filtration. Finally they are used to prove certain properties of the push-forward maps on the K-groups. Among other things, we are in a position to prove that for regular morphisms f, the push-forward map fK satisfies (9°f)K = 9K- fK

as well as the projection formula. All the work has then been done to obtain the Grothendieck Riemann-Roch theorem in a few lines. In this chapter, all schemes are assumed to be Noetherian, and connected (cf. Appendix). In particular, the rank of a locally free sheaf is constant.

Locally Free Resolutions

For the convenience of the reader, we shall now recall some facts about locally free resolutions. The proofs are obtained by replacing script let-

ters for sheaves by latin letters for modules, in which case they are proved in [L], IV, §3, §8, and XVI, Theorem 3.8. The proofs for modules use a fact which we formulate here for sheaves : A coherent sheaf is locally free if and only if it is flat.

THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

101

Concerning resolutions, we have the following statements. Let

0-,e'-49-). e" -0 be an exact sequence of coherent sheaves, and 9, 9" locally free. Then c9" is locally free.

In the reduction to the statement about modules, one merely has to look at the stalk at each point, which is a module over a local ring, and use the statement: If F is a coherent sheaf, Js the stalk at a point, and $ is free, then F is free in a neighborhood of x. From the above property of short exact sequences, we conclude: A long exact sequence

0- 4e.-

S"_1-'...ilea-0

of locally free sheaves can always be decomposed into short exact sequences

0- p-4

'

-1-.0,

where .*, and .3YD_, are the kernel and cokernel respectively, and are locally free.

The above property of short exact sequences has a generalization to longer sequences:

Let F be a coherent sheaf on X which admits a resolution of length <_ n by locally free sheaves

Let

be a resolution where .,. ...... `gin free.

are locally free. Then 2 is locally

Basically, the proof for modules comes from "dimension shifting", whereby .F has "dimension" S n implies .2 has "dimension" 0. In practice, the category of locally free sheaves also satisfies the property that given a coherent sheaf .F on X, there exists a locally free sheaf .0 and a surjection

6'->.F-40.

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THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

[V, §1]

This is true for instance when there exists an ample sheaf on X, so when X is quasi-projective over an affine scheme. Since the category of locally free sheaves is closed under direct sums, it satisfies the three conditions under which one can do the basic K-theory, and is called a K-family in [L], IV, §3, where some general facts of K-theory are proved using only the above three conditions. In this chapter we shall see how they apply in the geometric context.

V §1. The k-Ring K(X) Let X be a scheme. We let: !3X = category of locally free sheaves on X;

K(X) = Grothendieck group of Zx.

Thus K(X) is the free abelian group on isomorphism classes {&} of locally free sheaves e, modulo the subgroup generated by

{9} - {9'} - {cr} for each exact sequence (1.1)

0-+

of locally free sheaves on X. We shall write [G9] for the class in K(X) defined by .'. Warning : The map from isomorphism classes of locally free sheaves into K(X) is not necessarily injective. In Proposition 4.1 we shall see that K(X) can also be described as the Grothendieck group of a category of certain coherent sheaves. Tensor product induces a ring structure on K(X) by the formula 1.011-F] = [9 © F]. Exterior powers induce a.-operations:

A'[,9] = [A'8].

The rank function gives the augmentation e: e([.C]) = rank(19).

The positive elements E are the classes [9] of locally free sheaves.

[V, §1]

THE A-RING K(X)

103

To see that K(X) is a )-ring, we must verify that for an exact sequence (1.1), we have

[A ] _ Y [A1' ® Aye"]. i+j=k

To see this, let ffl"' be the image of the canonical homomorphism

nil ' ®^k-ig - Ake.

This gives a filtration A' = .y o = ' M ...

,Fk+' = 0 of Akce by

locally free subsheaves, with

^ie' ®^k-ieii _ 3'19,'+1 Note that the multiplicative group of line elements L is precisely the image of the natural map Pic(X) -> K(X),

where Pic(X) is the multiplicative group of isomorphism classes of invertible sheaves on X. In fact, we have an isomorphism Pic(X)

L.

Indeed, the natural map is injective because the det map S H A`Pe as in the proof of Theorem 1.7 of Chapter III is now seen to induce a homomorphism of K into Pic(X) rather than of K into L, and the composite map

Pic(X)-'K(X)

det

.Pic(X)

is the identity. Thus the isomorphism classes of invertible sheaves behave

better in this respect than isomorphism classes of locally free sheaves, which do not map injectively in K(X) because the Grothendieck relations identify different extensions. There is no room for extensions when dealing with invertible sheaves.

There is also an involution on K(X), defined by [e] V = [I V ], where 9 v = $°omox(G°, Ox) is the dual sheaf.

The association X -+K(X) is a contravariant functor. For any morph-

ism f: Y -+ X we have the pull-back f *9 which

is a locally free

104

THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

[V, §2]

0, module of the same rank as S. This pull-back preserves a short exact sequence, and hence there is a unique additive homomorphism

f': K(X) -> K(Y) such that f K[fl = [ f *f]. Since f* commutes with the tensor and alter-

nating products up to isomorphism, we see that fl is also a homomorphism of 2-rings with involution, and behaves functorially.

The A-rings K(X) satisfy all the properties stipulated in Chapter I. Only the (graded) splitting property and facts about the y-filtration F"K(X) are not evident from the definitions; these will be verified in the next two sections. Principally, we have to develop the covariant functorial properties of

X i-- K(X). In §2 we do what is necessary for the case of projections from a projective bundle, and in §4 we do what is necessary for regular embeddings. The combination of §2 and §4 in §5 will then show that our functor is a A-ring functor on the category of regular morphisms under one other mild assumption that every coherent sheaf is the image of a locally free sheaf.

V §2. Sheaves on Projective Bundles Throughout this section, we let .9 be a locally free sheaf of rank r + 1 on a scheme X. We let

P = P(of) = Proj (Sym e) f X be the associated projective bundle, and we let 0(1) = Op(1)

be the tautological invertible sheaf (Chapter IV, §1). For any sheaf of (9p-modules, and n e Z we let

fi(n) = F ®, 0(1)®n, where for n < 0, one defines 0(1)®" = ((Q(1)")6(-"). We let 13X be the category of locally free sheaves on X.

The main point of this section is to determine K(P) as an algebra over K(X), and to show that K(P) is isomorphic to the extension K(X)e described in Chapter II, with e = [9]. To determine the additive structure of K(P) over K(X) we follow Quillen's method, which is not only

[V, §2]

SHEAVES ON PROJECTIVE BUNDLES

105

simpler for K = K°, but also works for higher K groups, see [Q], §8, Theorem 3.1. The splitting property will be a consequence of the structure of K(P) over K(X). We begin by recalling some facts about direct images. Then we construct the canonical Koszul complex on P. Koszul complexes, in various forms, give relations in the K-groups. For example, the first relation for [(9.(l)] is precisely the relation satisfied by t in K(X)Q as in Chapter II.

Direct Images

For any coherent sheaf .F on P we have the direct image f*.F, which is coherent on X. We also have the higher direct image sheaves Rf*.F, which are coherent (9X-modules. They may be defined to be the sheaves associated to the presheaves

UF->H`(f-1(U),F). In particular, R°f*.F = f*A If X = Spec(A) is affine, then

R`f*.F = H`(P,.fl We record the following properties. R 1.

For any exact sequence

0-.F'->,F - F"-->0 there is a (functorial) long exact sequence

R 2.

The cohomology functor has dimension <_ r, that is

R f*(.F) = 0 for i > r. R 3.

(Projection Formula for Rf*). For 9 e'B we have R`f*(F

f*9)=Rf*(F)®9,

all

i>0.

R 4. (Serre's Theorem). For all F there is no such that

R`f*(.F(n)) = 0

for i> O

and

n >_ n°.

106

R 5.

THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

[V, §2]

For n >_ 0 and A' coherent on X, we have f*(O(n) (D f *A') = Sym"(,g) O 'ff

R6. Rf*((9(n)(& f*4t)=0 for 0
fori=r,n> -r; and all f coherent on X.

The proofs can be found in [H], III, except for R 2, R 5, R 6 which are easy consequences of [H], III. The properties are local on X, so we may assume X = Spec(A) where A is Noetherian. For R 2 we note that P = PA is covered by r + 1 affine open sets. By Leray's theorem [H], III, 4.5 the Cech cohomology with respect to this covering is the same as the ordinary cohomology for any coherent sheaf, and so vanishes in dimension > r. As for R 5 and R 6, they are special cases of the fact that the projection formula R 3 is valid for any coherent f on X when F = 0(n), that is

R7. Rf*(0(n) p f *d1) = Rf* (9(n) px .'

for

i >_ 0, n e Z.

The values of Rf* O(n) are given locally as the cohomology of (9(n). The

explicit computation of [H], III, 5.1 for the cohomology on projective space over an affine base without the extra A' works just as well with an .' to give the statements listed above. Remark. R 5 is valid for all n e Z if it is understood that Sym"(g) = 0 for n < 0. Then Rrf*(&(n)) is determined for all n e Z by the existence of a duality R'f (9(n) x R°f*0(-r - 1 - n) /fir+ig for all n e Z; and R 6 follows from R 5 and this duality. We do not need these further results, however. The Koszul Complex on P

We are going to construct a canonical resolution of Op. From the canonical surjection of f*(f onto 0(1), we get a surjection d

f*e0(9(-1)0Op -)' O. The sheaf f *O(-1) is locally free, and we can therefore construct the Koszul complex as in Chapter IV, §2. Since for any invertible sheaf 9 and any locally free sheaf .F we have an isomorphism AP(.F (9 9) : AP. Q 1®P,

[V, §2]

SHEAVES ON PROJECTIVE BUNDLES

107

and since f* commutes with M, we obtain the exact sequence which will be called the Koszul complex on P, or Koszul resolution (2.1)

0->

(-r- 1)-

For any coherent sheaf . on P we tensor the dual of the Koszul complex with .F to obtain an exact sequence (2.2)

f*A2g" ©.`N(2)

0-+ .y -a f*9" px

_+ ... -+ f*p,r+Ie" © . (r + 1) --,. 0

which we denote by Kos V (F). Then

.Fi-->Kos"(F) is exact on the category lp of locally free sheaves on P. Regular Sheaves

We shall now analyze K(P) by considering a subcategory of 'xip which generates K(P) and behaves particularly well under direct images. Following Castelnuovo and Mumford, we say that a coherent sheaf ." on P is regular if

R`f*-fl-i) = 0

for

i > 0.

Note that by R 6, if is coherent on X, then f *4 is regular, and more generally, (9(n) Q f *.,y is regular for n > 0. If 0--a-

-0

is a short exact sequence of coherent sheaves, and .F', .`H" are regular, it follows from the long exact sequence of R I that .F is regular. We let: SRp = category of regular locally free sheaves on P; K(%p) = Grothendieck group of %p.

Proposition 2.1. The inclusion of 91p in the category of all locally free sheaves Op induces an isomorphism

K(91p) =-> K(JJ) = K(P).

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[V, §21

Proof. We consider an auxiliary category. Let n be an integer > 0. We let:

91 = category of elements IF E13p such that

R`f*(.F(j))=0

for all

i>0 and j>_n-i.

In a short exact sequence of coherent sheaves as above, if .F', F" are in 91 then F is in 91 c %,,,1. In Proposition 2.2(1) we shall see that 9? = 91p for all n. Here we prove: The inclusion of %,, in %.+1 induces an isomorphism K(9In)

K(94n+1)

Proof. Let 91 e 91i+1. Then by the definitions and R 4, it follows that f *A°off " ®.F(p) is in 9? for p >_ 1. Then [-5F]

"E(-1)"[.f*A°gv ®F(p)]

is an additive map from K(9?,,,,) to because .F i-+ Kos"(F) is exact on 'X3p, and this map gives the inverse of the natural homomorphism induced by the inclusion. By R 4, any locally free sheaf is in 91 for sufficiently large n, whence we obtain the isomorphism K(P)

for all n. Since 910 is contained in 91p, the proposition follows.

Remark. For future use, we define another category: 91;, = category of elements F e 3p such that

R`f*(.fln+j))=0

for all

i>0 and j>0.

Then we have the same statement as for 9?n: The inclusion 91;,

3p induces an isomorphism K(9?;,)

The proof is the same as for 91,,.

K(P).

[V, §2]

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109

Proposition 2.2. (1) If .F is regular, then F(n) is regular for all n >--_ 0. In particular,

R`f*(F)=0 for i>0. (2)

If .F is regular, then the canonical homomorphism

f*f*.F is surjective, and if 9 is its kernel, then 9(l) is regular. (3) If .F is regular and in !BP then f*.F is in FSx.

Quillen [Q], §8 has given an elegant proof of this proposition. We include his proof, since we shall need some of the concepts later. Decompose the Koszul resolution into short exact sequences

O-

(2.3)

of locally free sheaves, with o=

&p

and

M; = f*A.+Id'(-r - 1).

We prove (1). Let F be regular in 23P. Tensoring the short exact sequence with F (p) gives the short exact sequence (2.4)

0-,XP ©F(p)--> f*Apg®.F -Xp-I ®F(p)-0.

It suffices to prove that F(1) is regular. We shall now prove by descending induction on p that XP (9 . (p + 1) is regular for all p. Let first p = r + 1 so M;, 1 = 0. For p = r, by the projection formula R 3, we have R`. f*(, OO

(r + 1)(- i)) = R`f*(f

=0

for

i>0.

This proves that .fit ® F(r + 1) is regular. For the inductive step, we use the long exact sequence R `f*(f *A19 (9 F(- i)) -> R f *(M -p -1 ®.F (p - i)) - Ri+ i f*(.)l ® 91(p - i)).

The term on the far right is 0 by induction. The term on the left is 0 by

projection formula R 3 and the hypothesis that .F is regular. This

I10

[V, §2]

THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

proves that -V-p _ 1 ®. (p) is regular. For p = 0 this shows that

,(1) is regular, thus proving (1).

We prove (2). We tensor the first short exact sequence of the Koszul (n) to get the exact sequence

resolution of Cep with

0-->.''1 ®F(n)-+ f*f, ®F(n - 1)-,F(n)-,0. In the proof of (1), we have just seen that Y, ®.5(2) is regular. By (1) and the definition of regularity, it follows that R if*(.(1®.F(n)) = 0 for n >- 1, and therefore

e ©.f*(F(n - 1)) - .f*Jfln) -> 0 is surjective for n >_ 1. We have a commutative diagram

d®Symn -1(.9) ®f* o - Symn(S) ®f

.

1).

By induction, the left vertical arrow can be assumed to be surjective, and

the bottom arrow is surjective so the right arrow is surjective. Taking direct sums, we get a surjection Sym(9) ®.f*.

-+ (D f*.F(n) ---), 0. n=0

The assertion of (2) is local on the base, so without loss of generality we

may assume that X is affine, X = Spec(A), and P = P. Then the direct sum on the right is usually denoted by W

O f* Wi (n) _ (I *

00

)" = O H°(P, .y (n)) '.

n=0

n=0

Cf. [H], II, §5, especially 5.15. Now we use a general fact which we state as follows.

Let R be a graded ring. We suppose that R° is Noetherian, R1 is a finitely generated R0-module, and R is finitely generated as R0-algebra by R1. We let X = Spec(R°)

and

P = Proj(R).

[V, §2]

SHEAVES ON PROJECTIVE BUNDLES

I11

We let f : P -> X be the structural morphism. Let M be a graded Rmodule. Then we have M- (projective tilde), which is a quasi-coherent sheaf on P. We define two graded modules M, M' to be quasi equal if

M = M', for all n sufficiently large. We say that M is quasi-finitely generated over R if M is quasi equal to a finitely generated graded module over R. The result we have in mind is then as follows. The association M H M"' is an equivalence of categories between the quasi finitely generated graded modules over R (modulo quasi-equality) and the category of coherent (9p-modules. Furthermore the functors

MHMand

.F ---> r* 9 = (@ H°(P, f(n)) n=0

are inverse to each other (up to isomorphism). Finally, if N is a finitely generated R0-module, and N- its corresponding coherent sheaf on X = Spec(R°) (affine tilde in this case), then there is a natural isomorphism

(R (DR. N)---'.f*(N-) = Op 00,, N-. Note that the tilde on the left is the projective tilde, and the tilde on the right is the affine tilde. It is really not the place here to reproduce a complete proof of the above elementary theorem. Cf. [H], II, Proposition 5.15 and Exercise 5.9 (where a reference to a field k is unnecessary), stemming from Serre's original Faisceaux Algebriques Coherents.

We then apply the theorem to the case when R = Sym(E), 9 = E"', to conclude the proof of the first assertion in (2), that

f*f*.F ->.F ->0 is exact.

As to the second assertion, let Y be the kernel. We look at the beginning of the long cohomology sequence:

R°f*(f*f*fl --> R°f*(S)

Rlf*(.t) - Rlf*(f*f*F).

The term on the far right is 0 by R 6. The term on the far left is just f*.F by R 5, and the arrow on the left is the identity. Hence R'f*(ft) = 0. For i >_ 2, we consider the short exact sequence 0

T(1 - i) --* f*f*.F(1 - i)-.F(1 - i)--0

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[V, §2]

THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

giving rise to the long exact sequence R`-'.f*(F(1 - i)) -> Rf*(_T(1 - i)) -+ R`f*(.f*.f*91(1 - i)).

The term on the left is 0 because ffl" is regular. The term on the right is 0 by R 6. This concludes the proof that _T(1) is regular, and thus concludes the proof of (2).

Finally we prove (3). We let .F be regular locally free. By (1) each term to the right of .F in the Koszul complex Kos V (.F) has vanishing higher direct images. Therefore applying f* to Kos V (F) yields an exact sequence

0-'.f*.F-A'9'(D f*.

(r+1)->0.

By the remarks in the introduction to this chapter, to show locally free, it suffices to show that f* F (i) is locally free for i = 1,... ,r + 1. By (1), F(i) is regular for i > 1. Moving to the right it suffices to show that f*.F(n) is locally free for sufficiently large n. But this is a general fact :

Let F be a locally free sheaf on P. Then f*.F(n) is locally free for sufficiently large n.

Proof. The statement is local in X, so we may assume that X is affine, and the statement is equivalent to the fact that f*fln) is a flat (9X module for large n. The scheme P is just projective space over the affine X. Then F (m) is generated by global sections for m large, so for m large there is a surjection from a finite sum of (9p onto F(m). Twisting back by - m, we obtain an exact sequence 0-+.F'->9--+.F-*0, where I is a direct sum of sheaves of type (9(- m). Twisting by n > m and tensoring with f *./P where .# is coherent of X, we get a short exact sequence

0-+.F'(n)4f*,# -* (n)OO f*.1-+F(n)OO .f*J-+ 0, whence the exact cohomology sequence

R`f*(g(n) © f*J() -> Rf*(F(n) (9 f*.() -> R'+1 f*(F'(n) OO f

Then the term on the right is 0 because the cohomology functor has dimension < r by R 2. The term on the left is 0 for n > no Let i > r.

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113

by R 6, so the term in the middle is 0. Then we can do a descending induction to prove that given .F there exists nl such that R`f*(.F(n) 0 f*.I1) = 0

for

1>0, n > nl

and all coherent f on X. Taking n > nl we obtain a commutative diagram with exact rows:

f*(. '(n)) OO ,ff - .f*(5(n)) ©,ff ---> .f*(F (n)) OO A -p 0

0 -f*(`F'(n) © f *-#) -' f*(9(n) (9 f *A') -p f*(F (n) © f *.') - 0 But R 5 implies that v is an isomorphism. It follows that w is surjective for n > n1. Then u is surjective for n > n2 by applying this result to .F'

instead of F The snake lemma implies that w is injective, so w is an isomorphism. Therefore the functor

A' H f*(F(n)) ®A' is an exact functor, so f*.F(n) is flat. This concludes the proof of (3), and also the proof of Proposition 2.2. Remark. This final general fact is an elementary result of algebraic geometry which is proved in the course of proving [H], III, 9.9, (i) implies (ii). No hypothesis about A being without divisors of zero is used in that part of the proof, which uses the Cech complex directly. We included another proof for the convenience of the reader, because it fitted the techniques used in this section.

The next result can be viewed as a sheaf version of the fact that K(91) is generated as a module over K(X) by 1, We use the preceding proposition to show :

Any regular sheaf .F on P has a canonical resolution (2.5)

0-'f*9;(F)(-r)-'f*°S.-i(ce)(-r+ 1)-)... -> f*51(F)(-1)->

where 9-;(.y) are sheaves on X, and the functors F H ,(.F) are exact.

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This is constructed inductively as follows. Let 9-0(,F) = f* S, and let fQ be the kernel of the canonical map from to .$. Then X0(1) is regular by (2), so we may define and °d°1 by

07) = f*(

0(1)),

and the exact sequence 0-3.

(1)-f*J'1(F)->t0(1)->0.

This gives an exact sequence

-0

0- * %1

with TI(2) regular. Inductively define 5,,(F), -°P by 5-1(.F) = f*(-Tl -I (P)), (2.7)

0.

One sees by induction that tp(p + 1) is regular, and that P and fP are exact functors of regular sheaves .

In addition,

R`f*(f1+p(P)) = 0

for all i >_ 0, p >_ 0. For i = 0 this follows by applying f* to the sequence defining °YP(p). sequence

For i > 0 it follows by induction on i and the exact

Rl-1f*(.i+p-1(P)) -

Rf*( j+P(P)) - Rf*(f

)(-i))

In particular, -Tr(r) is regular because Rf* = 0 for i > r.

Since

f* Yr(r) = 0, we get .r(r) = 0 by (2) of the proposition, so -Tr = 0 as desired.

Remark. As shown easily in Quillen [Q], §8, the sheaves P(.F) are uniquely determined by the following property. Let 9P be coherent sheaves on X such that there is a resolution

0 4 f*g,(-r)--->f*9r-1(-r+ 1)-..._, f*(( -1)- f*9p-+.F -40. Then VP x JP(.OV) for p = 0,. .. ,r. We won't need this property, which can be viewed as a sheaf-theoretic version of the fact proved in Theorem

2.3 that 1,form a basis of K(P(&)) over K(X).

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115

We state two other properties of the canonical resolution.

If .F is locally free as well as regular, then each .?T(F) is locally free on X and _TP is locally free on P.

This follows from (3) of Proposition 2.2 and induction on p. If we denote the canonical resolution by '(.y), and

0-)

'- F-+.F,'-,0

is an exact sequence of regular sheaves, then the sequence of canonical

resolutions

0-+ W(.F')-cf(F)-T(F")-+ 0 is also exact.

This follows from the construction and is left to the reader. Theorem 2.3. Let e = [S] in K(X). Then we have an isomorphism of

K(X)-algebras K(P(O))

K(X)e

which sends the canonical generator e on [0(1)].

Here K(X)e is the 2-ring extension of K(X) described in Chapter 1 §2, with generator ?' and relation r+1//

/

(-1)'2'(e)t°r+1-i = 0.

i=0

Proof. Let Co = [0(1)]. The Koszul resolution shows that 'satisfies the same relation as &I. Hence there is a unique homomorphism (p: K(X )e -> K(P)

mapping t' on z°o. If F is a regular locally free sheaf, then the sheaves are locally free, and the canonical resolution (2.5) shows that

P=0

By Proposition 2.1 such [.F] generate K(P), so p is surjective.

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Let 91' be the category of locally free sheaves F on P such that R `f* (.F(j))

for i> 0, j >_ 0.

By the remark after Proposition 2.1, K(91') = K(P). The map K(91')--' (D K(X) n=0

given by

([f*.], [f*F(1)],...,[f*flr)]) is well defined because the functor

.FHf*F(n) is exact on 91' for n > 0, and then .' is a homomorphism. Consider the composite

i K(X) -' K(X)Q -K(P) = K(91') Q K(X), n=0

n=0

where the first isomorphism takes ( an to E an e-'. The composite is n=0

n=0

given by a triangular matrix with 1's on the diagonal, since for a locally free sheaf sl; on X,

.s if j = i,

ff*s ©a(-) (9) 00) = to

if j < i.

Hence 9 is injective as well as surjective. This proves Theorem 2.3.

We identify K(P(E)) with K(91,) by Proposition 2.1. There exists a unique homomorphism

fK: K(P(9)) - K(X) such that

fK[F] = [f*F] for any regular locally free sheaf F on P(s). Indeed note by Proposition 2.2 that is a locally free sheaf on X, and that R`f*.` ' = 0 for i > 0, so f* is exact on 91.

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Corollary 2.4. Under the isomorphism K(P(4)) to the functional fe of Chapter I, §2.

K(X)e, fK corresponds

Proof. By construction of fK, and R 5, R 6, we have fK[d(n)] = [Sym" 9]

for n > 0.

To complete the proof we must verify that [Sym" 9] = v"e, i.e. that

(Y_

[Sym" e ]t"XrY (-1)n[Ang]tn

n=0

n=0

1.

We give two proofs for this. On the one hand, there is a complex 0

A"'.9 Ox Sym(g) -+ /arc ' OO Sym(.9) -+ ... -+ A0,9 OO Sym( ) -> 0x -+ 0

which is the Koszul complex over the symmetric algebra Sym(o'), with respect to the map which sends A' naturally in Sym(9): dl : A' O Sym(9) -+ A°B O Sym(g) = Sym( '). Locally, if T°, ... ,Tr is a basis of the free module E over the ring R, then

Sym(E) = R[T°,...,T,], and dl maps a basis for a free module of rank r + 1 over Sym(E) to the elements T°,...,T,. Thus locally, the above complex is the Koszul complex of a regular sequence (namely the sequence of variables T0,. .. ,T, in the ring R[T°,... ,T,]). Hence the Koszul complex is exact. Since dp maps APP® O Sym9(e') into

AP`9 ©Sym9+1(E),

we can decompose this complex into a direct sum corresponding to graded component, and hence we have an exact sequence 0->Ar+140 (D Sym"-r-1(49)-+ ...-+ A'S©Sym"-i((g)-,Sym"(.9)-+0

for integers n > 1, it being understood that Sym'(8) = 0 if j < 0. This last exact sequence gives precisely the desired relation in the K-group. Alternatively, we would use the splitting property which will be proved below, and which reduces the assertion to the case when S has rank 1, when it is obvious.

Corollary 2.5. F"K(P(S)) n K(X) = F"K(X).

Proof. Apply Theorem 1.2 of Chapter III.

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Remark. The set of positive elements in K(P(S)), i.e. the classes of locally free sheaves, may be larger than the set E. described formally in Chapter I, §2. Lemma 2.6 (Projection Formula). If x E K(PX), y E K(X), then fK(x fKy) = fK(x) - Y

Proof. By Proposition 2.1 and linearity we may take x = [F] with F regular and locally free on Pd, hnd y = [p], 9 locally free on X. Then 070 f *# is regular by R 6, and f*(.F (9 f *9)

f*(.F) ® 9

which is the required formula.

Theorem 2.7 (Splitting Property). Given a locally free sheaf 9 on X, there is a morphism f: X' -> X such that

f": k(X) - K(X') is injective,

F"K(X') r K(X) = F"K(X)

for all n, and

f K[g] = [3 ] +... + [_Tm] for some invertible sheaves 2t on X'.

Proof. First let f be a bundle projection P(s) -> X. Then we have the tautological exact sequence

so [f *S] = [.,Y] + [0(l)], and the rank of )r is one less than the rank of S. By induction, we take a sequence of such bundle projections to conclude the proof.

V §3. Grothendieck and Topological Filtrations In this section we assume X is a connected, Noetherian scheme with an ample invertible sheaf Y. Recall that this means that for any coherent

sheaf F on X there is an integer no = no(.F) such that for all n > no, F ® JT®" is generated by its global sections. For example, if X is quasiprojective over an affine Noetherian base scheme S, then X has an ample

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119

invertible sheaf; for any ample 2 on X, and sufficiently large n, there is a locally closed imbedding i of X in some PS such that 2®" = i*0(1) (cf. [H], II, 7.6). We now connect ample sheaves with ample elements as defined in Chapter III before Lemma 1.4. Lemma 3.1. If 2 is ample on X, then u = [2] is an ample line element for K(X).

Proof. Given x e K(X), we must show that

u"x=e - m for some positive integers n, m, and a positive e in K(X) (Chapter III, §1). Choose locally free sheaves eI, e2 with

x = [e1] - V2] For large n there is an m > 0 and a surjection

a ''

2 ® 2® _ 2

d®m_

) 0.

If e' is the kernel of a, then

as required.

As in Chapter III, §1 let F"K(X) denote the y-filtration on the A-ring K(X). We modify slightly a definition of H. Bass (also used in SGA 6 and Manin), and introduce another filtration, denoted F;O K(X). For this some notation will be useful.

In this section a complex f' on X will be a bounded complex of locally free sheaves

b-'"00.

0

We say that 9' represents an element x e K(X) if b

x = E (-1)`[19`]. i=a

The support of t, denoted 1,9'1, will be the set of points x e X at which the induced complex of vector spaces e'(x):

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[V, §3]

over the residue field K(x) = d x, x/"X, x is not exact; IS'1 is the union of the supports of the homology sheaves A`(t), so is a closed subset of X. If Z is an irreducible closed subset of a Noetherian scheme Y, recall that the codimension of Z in Y, denoted codim(Z, Y) is the greatest length of a chain of irreducible closed subsets.

Z=V09Vi

...9VdcY.

For an arbitrary closed Z c Y, codim(Z, Y) is defined to be the smallest of the codimensions of the irreducible components of Z in Y. With this definition, if Z c Y c X, then (3.1)

codim(Z, Y) + codim(Y, X) 5 codim(Z, X).

If Z = 0, codim(Z, Y) = + oo. The dimension of Y, dim(Y), is the maximum codimension of any non-empty closed subset. We define: Fn topK(X) = set of elements x e K(X) such that for any finite family of closed subsets { Y} of X, x can be represented by a com-

plex o' on X such that for any finite family of closed subsets { Ya} of X, x can be represented by a complex 8' on X such that codim(I S' j n Y., Y,) >_ n

for all a. We say that such 61' represents x with respect to { YQ} and n.

We may write F",o j, or simply Fn p, for F" top K(X), and call it the topological filtration.

Proposition 3.2. (a)

The F;QPK(X) define a ring filtration on K(X).

(b)

If dim(X) < d, then F 0 1K(X) = 0.

Proof. We show first that Ftop is an additive subgroup of K(X). Given x, y e Fn p, to show x - y e F,op, suppose closed subsets Y. are specified. Choose complexes of (resp..F) representing x (resp. y) with respect to { Y} and n. Then

e.Q+.F'[-1] represents x - y with respect to { Y} and n, where .F'[-1] denotes the

shift of .F':.F'[-1]k =k-I To finish the proof of (a), we must show that if x e F ',.P, y e F" P, then x y e F"". ". Given closed subsets Y., chose g' representing x with re-

spect to { Ya} and m. Then choose .$' representing y with respect to

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121

{ l o"I n Y} and n. It follows that

represents x y with respect to {Y,,} and m + n. The point here is that t' ®F' is exact where either if' or F' is exact, so IC

Iif 'I n I.F'I;

the condition on codimension follows from (3.1). For (b), take Y = X ; if x e F oP 1X, x is represented by a complex 9'

exact on all of X, so

x = E (-1)i[ei] = 0 by definition of K(X).

For 2' ample on X, and F any coherent sheaf on X, write as usual .Wi(n) for .F (D 2®", and write .fi(x) for the fiber of F over the residue field K(x).

Lemma 3.3. Given a surjection F -+ 9 of coherent sheaves on X, and a finite set S of points in X such that 9(x) 96 0 for all x e S, then for all sufficiently large n there is a section of F(n) whose image in 9(n)(x) is not zero for any x e S.

Proof. For large no there is a section f of 2®"0 such that the complement Xf of its zero-scheme is affine, and S c Xf (cf. [H], pp. 154-155, or [EGA], II, 4.5.4). On Xf, . -) QV corresponds to a surjection of mod-

ules, from which one sees that there is a t in I'(Xf, F) whose image in 9(x) is non-zero for x c S. For large m, f't extends to a section of F(mno). For all large p there are sections g of 2'®P that are not zero at x e S. Then g f't is a section of .f(n) as required, with n = mno + p. Lemma 3.4. Let off 1, ... ,9P be locally free sheaves of the same rank r on

X, and S a finite set of points of X. Then there are integers m1, ... ,m and homomorphisms (3.2)

Y®1"1 O ... O Y®m,. -) 'i

for i = 1, ... ,p, whose fibres at each x e S are isomorphisms. Proof. We do this for one 4' = !®i 1, noting that the integers mi which arise can be chosen uniformly for any finite collection of di's. By the preceding lemma, take a large n1 and a section s1 of S(n1) that is not zero at any x E S. Define !Y 1 by

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By Lemma 3.3 there is, if r > 1, for large n2 a section s2 of 4'1(n1 + n2) whose image in 91(n2) is not zero at any x E S. Define 92 by

q®n2$vX

S)©S21

f'(n1 +n2)- 92 - 0,

and take, if r > 2, s3 in 4'(n1 + n2 + n3) whose image in !N2(n3) is not zero at any x e S. Continuing in this way one arrives at y®(n2+...+n,,)$...$2®nr$OX S,e

@S,

(n1 +...+nr)

such that the induced map on fibres at x e S has rank r, so is an isomorphism. Tensoring this by P®m, m = - E n1, yields (3.2). Proposition 3.5. For any X with an ample invertible sheaf,

F1K(X) = F pK(X).

Proof. If x c F oPK(X), taking Y = X, we see that x is represented by a complex if' which is generically exact. Therefore s(x) = y (- 1)` rank(4") = 0, so x e F'K(X). Conversely if x e F'K(X), write x = [.91] - [4"2], with 91, '2 locally free of the same rank r. Given closed subsets Y,, of X, let S be the set of generic points of the irreducible components of the Y,,. Construct homo-

morphisms as in (3.2) of Lemma 3.4, for 41 and 4'2i each defines a complex 4'* with non-zero terms in degrees -1 and 0, whose support meets each Ya in codimension at least one. Then d-1 ED (f A - 11

represents x, with respect to { Y} and n = 1, so x e F' pK(X), as required.

Lemma 3.6. Let 9 be locally free on X of rank r + 1, f : P(4') - X the associated projective bundle. Then there is an element z in F,.pK(P(4')) such that fK(z) = 1

in K(X).

Proof. By Proposition 3.5, if e _ [OP(1)], 1 - e-1 is in F OPP, so

z=(1-e-1)'aF'0 P. Since fK(1) = 1, and fK(e 1) = 0 for 1 < i :5 r (Corollary 2.4), the lemma follows.

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123

Let f: P -+ X be a projective bundle. We shall say that a (bounded) complex 9' of locally free sheaves on P is regular if each off`, each Ker(d`),

each Im(d`), and each homology sheaf 0'(C) is regular in the sense of §2. Since the push-forward of a short exact sequence of regular sheaves is exact, and each f*9' is locally free, it follows that f*(e') is a bounded complex of locally free sheaves on X, with J1 i,

`(f*e') = f*

`W)

If a regular complex e' represents an element x e K(P), it follows from the definition of the push-forward fK that f* f' represents fK(x). Lemma 3.7. Let x e K(P) be represented by a complex 9' which is

exact on an open subset U of P. Then there is a regular complex

,

exact on U, which represents x. Proof. As in Proposition 2.1, using the canonical exact sequences (2.3), it follows that for any locally free sheaf F on P and any n°, there are locally free sheaves .F°, ... , N on P with 1)'[397']

in K(P),

.F' = f *(` ® ©(m;), 19` locally free on X, and mL >_ n°. Indeed, (2.3) gives such for .. and no = 1; given (*) for some . and n°,

applying (2.3) to each 0 gives (*) for ffl" and no + 1.

Now given 9' representing x, choose no so that for each of the sheaves .4 = d`, Ker(d`), Im(d), and `(" ), the sheaf d(n°) is regular. Then choose °, ..., ' so that (*) holds for .F = (9p, and this n°. By the projection formula R 3 of §2, each of the complexes 9' ©.F ` is regular; by (*), x is represented by the regular complex N

c e'Og`[i] i=O

which is exact wherever 9' is exact; as in Proposition 3.2, [i] denotes a shift of the preceding complex.

Remark. Using the same canonical resolutions (2.3), one sees in fact that any complex t' on P admits a homomorphism

to a regular complex I' which induces an isomorphism on homology sheaves.

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Lemma 3.8. Let f : P -> X be a projective bundle, and let x e K(X). If f K(x) is in F,"oP K(P), then x is in F"top K(X).

Proof. Choose z e

K(P) satisfying the conditions of Lemma 3.6.

Then f K(x) z is in Fto P'K(P), with JK(f"(x).z)

= x'fK(z) = x.

Let {Ya} be a finite set of closed subsets of X. By Lemma 3.7 we may represent f K(x) z by a regular complex 9' such that codim(I ''I n f -1(YQ),f -1(Ya)) >_ n + r

for all a. Then f* e' is a complex representing x, with and

I.f*t,I = .f(Ie*I)

codim(I f*('I n Y, 1) >_ n.

(The last inequality follows from the fact that for any closed Y c X, and

Z - f -'(Y), codim(f(Z), Y) >_ codim(Z, f -'Y) - r.

Indeed, letting A be the local ring of Y at an irreducible component of f (Z), this follows from the fact that dim A[T1, ...,T,] = dim(A) + r (cf. [Mat], 14.A).)

Theorem 3.9. For any X with an ample invertible sheaf, and all n, F"K(X) c F,.PK(X).

Proof. Given x e F"K(X), we may assume

x= with xi e F1K(X),

yk,(xI)..... ykm(xm)

ki >_ n. By the splitting principle there is a morph-

ism

f:X' -- X which is a composite of projective bundle projections, such that each f K(x;) can be written as a sum of differences u - v of classes of invertible sheaves. For such line elements u, v e K(X'),

yXu - v) = yt(u - 1)/y,(v - 1) = (1 + (u - 1)t)/(1 + (v - 1)t).

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125

Therefore yk(u - V)

= (-1)'-'(u - v)(V -

1)k-1

Now by Proposition 3.5, u - v and v - 1 are in F, PK(X'). Since Fjpp is a ring filtration, it follows that f K(x) is in F" topK(X'). By Lemma 3.8, x must be in F"topK(X), as required. Remark. In [SGA 6] a filtration K(X)" was defined by the condition that an element x is in K(X)" if for any one closed Y c X, x is represented by a complex off" whose support meets Y in codimension at least n. Hence clearly F,0PK(X) c K(X)".

In particular, Theorem 3.9 answers a question left open in [SGA 6],4V.3:, 6.10: the y-filtration F"K(X) is finer than the topological filtration K(X)".

All the statements and proofs of this section work equally well for the because it is functorial: filtration K(X)". We prefer the filtration If f : Y -+ X is a morphism, then

f K(Ffl

top

K(X)) c F00P K(Y).

To see this, let x e F" PK(X), and let { Y,} be a collection of closed subsets of Y. To show that f Kx is represented by a complex with respect to {Y,,} and n, we may assume each YQ is irreducible, by replacing each Y" by all its irreducible components. Then stratify X by locally closed subsets XP so that each YQ n f 1Xp -+ XR

is equidimensional (e.g. flat). Then if 4F represents x with respect to (X.} and n, f * t represents f Kx with respect to {Y,,) and n.

Corollary 3.10. If dim X < d, then F°+ 'K(X) = 0. Proof. Proposition 3.2(b) with Theorem 3.9. Corollary 3.11. The Chern character ch induces an isomorphism of QK(X) with QGr K(X).

Proof. Corollary 3.10 and Chapter III, Theorem 3.5.

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Remark 1. The first Chern class c1 determines an isomorphism c1: Pic(X) -> F1K(X)/F2K(X) = Gr' K(X),

where Pic(X) is the multiplicative group of isomorphisms classes of invertible sheaves. This comes from Theorem 1.7 of Chapter III, giving an isomorphism of L with Gr1 K(X), and the isomorphism of Pic(X) with L as observed in §1. Remark 2. More information relating the Grothendieck filtration with geometric filtrations will be given in Chapter VI, §5.

V §4. Resolutions and Regular Imbeddings In this section we assume that all schemes X under consideration satisfy the following axiom: (*)

Any coherent sheaf of (Ox modules is the image of a locally free sheaf

Any scheme with an ample invertible sheaf, e.g. any scheme quasiprojective over an affine scheme, satisfies this axiom (cf. [H], III), which suffices for most applications. In fact, any scheme which is quasi-projec-

tive over a divisorial (e.g. a locally factorial or regular) base scheme satisfies (*) ([B]). We let:

Gx = category of coherent sheaves on X which admit a finite locally free resolution, i.e. there exists a finite resolution (4.1)

0

with 9i locally free for all i.

Since locally free sheaves are in Sx, there is a canonical homomorphism

K(X) - K(Sx). Proposition 4.1. This homomorphism is an isomorphism

K(X) = K(3x).

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Its inverse is given by mapping a class [F] on the alternating sum [.F] F-+ Y (- 1)`[&i], i>o

where off. is a finite locally free resolution of F. Proof. It is a standard lemma in the theory of Grothendieck groups (cf.

[L], IV, 3.7) that the above alternating sum gives a well-defined

inverse isomorphism, because the following property is satisfied: admits a finite resolution of length n by locally free sheaves f and

if F

is an exact sequence with K a locally free sheaf for 0 < i < n, then .e is also locally free; cf. the remarks in the introduction to this chapter. If f : X -> Y is a (closed) regular imbedding, then Koszul complexes (Chapter IV, §3) give, locally on Y, a resolution off, Ox of length r by locally free sheaves on Y, where r is the codimension of X in Y Let .F be locally free on X. By (*) we can find an exact sequence ->...-+ (ffo- f*F-+ 0

with iii locally free on Y for i = 0,. .. ,r - 1. Let 9, be the kernel of the arrow furthest to the left. By the above remarks, it follows that if, is also locally free, and we obtain a locally free resolution

0->9,-- 9,-1-----+

0 -+f*.F--), 0.

Thus we have shown that for any locally free sheaf .F on X the direct image f*F admits a finite resolution by locally free sheaves on Y Since

f*.F is just the extension of F by zero outside X, the functor f* is an exact functor from the category !Bx of locally free sheaves on X to the category (By of finitely resolvable sheaves on Y. This induces a homomorphism

fK : K(X) - K(Y)Explicitly,

if 0

->

o -> f* -> 0 is a resolution of f*.F by

locally free sheaves Ci on Y, then

i=o

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Lemma 4.2 (Projection Formula). If f : X -> Y is a regular imbedding, then fK(x _

fKy)

= fK(x) Y

for x e K(X), y e K(Y).

Y.

Proof. Let x = [F], y = [9], with F, T locally free sheaves on X and If 9. is a resolution off* Xthen TO 9. is a resolution of

f*(f *g ©F) = 9 O f*, from which the formula follows. Proposition 4.3. Let if be a locally free sheaf of rank d on a scheme Y, s a regular section of .9, X = Z(s) the zero scheme of s, f the imbedding

of X in Y Then f is a regular imbedding of codimension d. Let e = [8] in K(Y). Then: fK(1) = A-1(e") in K(Y);

(a)

fG,K(1) = c`°"(e)

(b)

in Gr K(Y).

Proof. We have seen that f is regular, and that there is an exact Koszul resolution

0,A dg"

-+Ad-lev

- ...-- A19" -+ dp-+ flog->0

as in Chapter IV, §2. This proves (a), and (b) then follows from Chapter III, Proposition 2.1.

To make use of the deformation to the normal bundle, we shall also need the following two propositions. Proposition 4.4. Let A, B, C be effective Cartier divisors on a scheme

M. Assume: (i) (ii)

(9(A) = 0(B + C); B and C meet regularly in M.

Let D = B n C, and let a, b, c, d be the imbeddings of A, B, C, D in M. Then

(a) (b)

aK(l) = bK(l) + cK(1) - dK(1) in K(M); aGr K(1) = bGr K(1) + CG, K(1)

in Gr' K(M).

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129

Proof. By (ii), D is the zero-scheme of a regular section of (9(B) p+ a(C).

Using the preceding proposition and (i), we get dK(1) _ A-i(((B) E a(C))

= 1 - [0(-B) p 0(-C)] + [0(-B - C)] = 1 - [(9(-B)] + 1 - [0(-C)] - (1 - [(9(-A)]) = bK(l) + cK(1) - aK(l).

This proves (a). Formula (b) follows from (i) and Proposition 4.3, since ci((9(A)) = cl((9(B + C)) = cl((9(B)) + cl(C9(C))

Proposition 4.5. Let F: P --> M be a regular embedding, and let co : Y -+ M

be a morphism. Form the fiber square:

XfY P F0 M Assume that f is a regular imbedding of the same codimension as F. (This is true for instance if 9 is a regular imbedding and P, Y meet regularly in M, in which case X = P o Y) Then: (a)

(P'FK = ./KY'K.

If Z is a subscheme of Y which is disjoint from f(X), and h: Z -> M is the morphism induced by gyp, then

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[V, §4]

Proof. Let p e K(P). We may assume that p = [.F ] with some locally free sheaf ,F on P. Let

0-1, en-*...--ito->F*(.F)-+0 be a locally free resolution. We have qo*F*(.) = f*o*(.F) since F* and f* are the extension by 0 because F, f are closed imbeddings. Hence to prove the proposition, it suffices to show that the sequence (P*8. -). T*F*(

) -), 0

is exact. Taking q * locally amounts to taking the tensor product, and by abstract nonsense of basic homological algebra, the homology of the complex

0-+ p*g,, -.,...-+ p*eo-*tp*F*(:F)--*0 is independent of the choice of locally free resolution of F*(.F). Hence

the desired assertion is local on M, and we may assume that M = Spec(A) and X is defined by an ideal 1 = (al, ... ,a,) _ (a),

where (a) is a regular sequence. Also since F is locally free, we may assume that F = (9P, and 9 is the Koszul complex O

Kr(a)-+ ..._*Ko(a)-*Aft ->0.

Taking T* amounts to tensoring with the structure sheaf of Y, and locally on Y we obtain the Koszul complex

0-iK,(a)-....-K0(a)-B/I-+ 0, where, say, Y = Spec(B), 1 = IB = A/I Q B, and a, is the natural image of a; in B. But the assumption that f : X - Y is a regular imbedding of the same codimension as F implies that (a) is a minimal set of genera-

tors for the ideal of X in Y locally. By Lemma 2.6 of Chapter IV it follows that (a) is a regular sequence, thereby proving (a) of the proposition.

Assertion (b) follows from (a), or from the observation that, with .9. as above, h*9. is an exact complex on Z. This concludes the proof.

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131

Remark. Proposition 4.5 will be substantially generalized later in Chapter VI, Proposition 1.1 and Theorem 1.3. Classically, geometers work with an intersection product of classes of cycles on a scheme (or variety). The K-groups can be viewed as a substitute for cycle classes, and the product in K(X) can be viewed as a substitute for the intersection product. In SGA 6 and [Man], global intersection formulas are proved using resolutions and Tor (see also [L], Chapter XVI, Theorem 10.11 and Proposition 11.1), after Serre's local theory (Springer Lecture Notes 11, 1965). Here we shall give such a formula as an application of Proposition 4.5, illustrating the special case already mentioned in its statement. Corollary 4.6. Let Y, Z be closed subschemes of X, regularly imbedded and meeting regularly in X. Then

[0Y]10A = Proof. Let is Y -> X and j: Z -> X be the regular imbeddings of sub-

schemes, and form the fibre square as shown :

YnZ9 t Z Y Then we have:

[(9Y][(9Z] = jK(1)JK(1) = JKJJK(iK(1))

by projection formula

=jK(gKhK(1))

by Proposition 4.5

_ (jog)K(1)

because hK(1) = 1

= [0YnZ] This proves the corollary.

The rest of this section is devoted to the proof of two lemmas, which are needed to construct a more general push-forward map in the next section, and to verify compatibilities of push-forward homomorphisms for imbeddings and projections.

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Lemma 4.7. Let .9 be a locally associated projective bundle

[V, §41

free sheaf of rank r + 1 on X with

f:P(d)-X. Let s: X -+ P(') be a section of f i.e., f a s = idx. Then s is a regular imbedding of codimension r, and f K ° SK = idK(x)

Proof. We saw in Chapter IV, Lemma 3.8 that any section of a smooth morphism is regular, the present case of a bundle being particularly simple. To prove the last assertion, let .F be a locally free sheaf on X. Then s*F is a regular coherent sheaf on P(s). To see this, note that s*(F) ® (9(k) = s*(F ® s*(9(k))

Since the restriction off to s(X) is an isomorphism, R`f*(s*(F ® s*(V(k))) =

0,

F ®s*O(k),

i > 0, i = 0,

from which it follows that s*97 is regular. By the construction of §2, s*.F has a canonical resolution

a,s*.F->0. 0-f *J,(-r)->...-f*JWe claim the each 5 = p(s*.F) is a locally free sheaf on X. We prove this by induction on p, together with the assertion that if 2Pp is the sheaf defined by (2.7), then .f*(fp(m))

is locally free for m > p.

Since tp(m) is regular for m > p, (2.7) determines exact sequences (4.2)

0 ->f*(9P(p + i)) -f*(f*g;(i)) - f*(YP-I (p + i)) - 0

for i > 0. Note to start that o = f*s*(.F) = 3F is locally free, and

0 -'f*(Z(i)) -> f*(f*F(i)) - f*(s*.F(i)) -> 0.

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133

Since f*f*.F(i) = F ® Sym` e and f*s*.F(i) = .F(i) ® s*0(i) are locally free, so is the kernel, which completes the proof for p = 0. Assuming the result for p - 1, then p(s*F) = fk( !'p-I(p)) is locally free by induction. And f* f *S,(i) = Jp ® Sym` g is locally free, so f*(°p(p + i)) is locally free for i > 0 by (4.2). From this canonical resolution we have

SK[F] = Y- (-1)p[ *.91p]? -p p=0

with C = [0(1)], and .p = .9,(s*R') locally free sheaves on X.

By

Theorem 2.3 and Chapter I, Proposition 2.2, fK(e-p) = 0 for p = 1,..., r, and

[90] = [97], which concludes the proof. Lemma 4.8. Let f: P(s) -> Y be a projective bundle, and let is X -+ Y

be a regular imbedding. Form the fibre square

P(i*e) ' 1 P(s)

x Then j is a regular imbedding, and

IK°9K=fK°JK

Proof. The regularity of j follows from Chapter IV, Proposition 3.5. From the definitions we have immediately (4.3)

JK9K(x) =fKiK(x)

for all x e K(X). Also, if t' = [0(1)] is the canonical generator of

K(P(9)), then

jK(t) = U*0(1)]

THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

134

[V, §5]

is the canonical generator of P(i*9). By Theorem 2.3, K(P(i*d)) is generated by elements gK(x) jK(e") for n >_ 0; to prove the formula of the lemma it suffices to see that both sides agree on such elements. Note also that ixfx(e") = iK[Sym" g] = [Sym" i*f] = 9Kjx(e").

Using the projection formula together with (4.3) and (4.4) we have fKjx(9K(x)'JK(e")) _ fx(jx(9K(x))' e") = fK( KZK(X)' e") = 1K(X)' fK(f")

= iK(x ' iKfx(e")) = ix(x ' 9x(jxe")) = iK9x(9K(x).jK(e)), as required.

V §5. The K-Functor of Regular Morphisms All schemes considered in this section will be Noetherian, connected schemes satisfying the condition (*) of §4.

Recall from Chapter IV, §3 that a regular morphism f : X -> Y is one which can be factored into a regular imbedding and a projection from a projective bundle, f = p ° i. The purpose of this section is to show how X i-+ K(X) is a A-ring functor (as defined in Chapter II, §3) on the category of regular morphisms. The contravariant property is trivially sat-

isfied, and we have to deal with the covariance and the projection formula. For a regular morphism as above, we shall define the pushforward fK : K(X) -> K(Y).

Let pK and iK be the homomorphisms defined in §2 and §4, and define fK = PK ° 1K'

Proposition 5.1. (1) The homomorphism PK ° iK is independent of the factorization of f (2) If f : X -> Y and g : Y -,. Z are regular morphisms, then g ° f is a regular morphism, and (g ° f )x = 9K ° fK-

Proof. In Chapter IV, Proposition 3.12 we proved that g of is a regular morphism. We now consider several cases.

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135

Case 1. If f : X -+ Y and g : Y -+ Z are regular imbeddings, then g -f is a regular imbedding, and (9 °.f )K = 9K °.fK

That g c f is a regular imbedding was seen in Chapter IV, Proposition 3.4. If F is a locally free sheaf on X, and 9. is a resolution of f*F by locally free sheaves on Y, construct a double complex 21.. of locally free sheaves on Z: 0

1

0

0

I

I

9n,m

-qn-1,m )

-9o,m

-9n,0

-9n-1,0 ,

-90,0

9* 9n , 9* (00n -1 ) ... - 9* -fo - 9 *.f* F 1

I

I

0

so that the columns resolve the sheaves g*&i. By Lemma 5.4 of the appendix to this section, applied to the homomorphism from 21.. to g* w'., the associated total complex of 21.. resolves (g f)*.F, so

ii] (9°.f)K[F] _[-(-1)i+jp L i,j

_ Y(-1)` E(-1)' i j 1)`9K{-fi]

= 9K(fKCFD-

0

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136

Case 2. Let ', d' be locally free sheaves on Y, P = P(ct), P' = P(d'), p and p' the projections. Form the fibre square

P xy P'

Y

Then q and q' are projective bundle projections, and PK°q' = PK°qK

To see this, let i and e' be the canonical generators of K(P) and K(P'). Note that gxgK(t a) = [p,* Sym' r] = P1KPK(1°).

By Theorem 2.3 the classes gI(ea) qK(C'b) generate K(P x y P') over K(Y), so it suffices to show that pK o qK and PK ° qK agree on such classes.

Using the preceding equation, with the projection formula, p' ° q' (gK(e°)q,K((b)) = PK(gK(gK((a)) .,,b)

= pl (p,K(PK(ja)).

e b)

= PKV°) - P'K

(,,b).

By symmetry, this equals PK o gK(gK(Co) , qK(e,b)), which concludes the proof in this case. Case 3.

Suppose f : X - Y is a regular imbedding which factors

through a projective bundle

Y

Then i is a regular imbedding, and

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THE K-FUNCTOR OF REGULAR MORPHISMS

137

For the proof, let q : P(f *.e) -, X be the induced projective bundle, and let s be the section induced by is

P(f *g) I - P(') X

Y

By Chapter IV, Propositions 3.5 and 3.9, j, s, and i are regular imbeddings. By Case 1, lK = JK ° SK.

By Lemmas 4.7 and 4.8, qK ° SK = idK(x)

and

fK ° qK = PK °.1 K

Therefore, PK ° ZK = PK °JK ° SK = fK ° qK ° SK = Al

as asserted.

We can now prove (1) of the theorem. Let f = p ° i = p'- i' be two such factorizations of f through projective bundles P and P'. Form the commutative diagram

where j = (i, i') is the diagonal imbedding. Since q and q' are projective bundle projections, Case 3 implies that j is a regular imbedding, with iK = qK °jK

and

ix=qK°lx

By Case 2, PK°iK = PK°qK°jK = Pic°gic°JK = PK°tiK9

as required.

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[V, §5]

Next we prove the second part of the theorem. Let

X` Pe p Z be a factorization of g -f into a closed imbedding followed by a bundle projection. This determines a commutative diagram:

with i = g'-j. Then 9K° K= 9K°pK°jK

by (1)

by Lemma 4.8 by Case 3

by definition,

which concludes the proof.

Remark. A more conceptual but less elementary proof of the proposi-

tion can be given along the following lines. If f : X -+ Y is a proper regular morphism, and 9' is a bounded complex of locally free sheaves on X, one can show that the complex Rf*(t) in the derived category is homologically isomorphic to a bounded complex .F' of locally free sheaves on Y, and that the Euler characteristic

y (-1)`[.

`]

in K(Y)

is independent of choice of S. Then Y_ (-1 AK[e`] = Y_ (_ UP"F`l

This description is independent of factorization; the functoriality follows from the equation

R(9°f)=R(9)oR(f) in the derived category. This approach also generalizes to "perfect" morphisms; for details, see [SGA 6].

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THE K-FUNCTOR OF REGULAR MORPHISMS

139

Proposition 5.2 (Projection Formula). For a regular morphism f : X -+ Y,

xeK(X), yeK(Y), fK(x'fKY) =fK(x)'Y Proof. This follows from Lemmas 2.6 and 4.2 which proved the projection formula for each one of the cases of a projection from a projective bundle and a regular imbedding respectively. We can now summarize our results in the following theorem.

Theorem 5.3. On the category of regular morphisms, X H K(X) is a 2-ring functor.

Remark. When Y has an ample invertible sheaf 2', a projective morphism f : X --> Y admits a factorization into

i a closed imbedding, p the projection. To see this, factor f through P(ct) as usual, and take n and m so there is a surjection 0®(n+1)-, e(D Yom-0.

This induces a closed imbedding P(s) = P(4' x0

®

P(0®(n+1)) = P"Y

as required.

With this remark, it is only necessary to study trivial projections PY -+ Y For several Riemann-Roch theorems, this simplifies the computations considerably. Homological Appendix

We have used a basic lemma from homological algebra, which is usually proved using spectral sequences. For convenience of the reader, we include an elementary treatment here, following the general principle that double complexes can be used directly.

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By a double complex in some abelian category we mean a commutative diagram

1

Eli

1

a'.'

...

E,-j,; lai-l'i

E,-i.i-i

-' ...

whose columns and rows are all complexes, denoted E. and E.; respectively. We assume the complexes are bounded below, i.e. E;j = 0 for i < N, j < N, some N. The associated total complex Tot (E ..) is the complex whose n-th term is

Tot(E.. ),, = (@ Eii and whose n-th boundary d is the sum of homomorphisms (atj, (-1)13 ): Eii -' Ei,i- i G+ Et- I,i.

A homomorphism (p..: E.. -- F.. of double complexes induces a homomorphism of complexes Tot ((p..): Tot (E.. ) -+ Tot (F., ),

as well as homomorphisms of column complexes

and similarly for the rows.

Recall that a homomorphism T.: E. -+ F. of complexes is called a homology isomorphism (or quasi-isomorphism) if the induced homomorphisms H1(cp.): H;(E.) -> Hi(F.)

are all isomorphisms.

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ADAMS RIEMANN-ROCH FOR IMBEDDINGS

141

Lemma 5.4. Let T..: E.. -> F.. be a homomorphism of double complexes such that each homomorphism ipi. is a homology isomorphism. Then Tot(T..) is a homology isomorphism.

Proof. For a double complex E.., let E..(r) denote the truncation of E.. obtained by omitting all columns E.. of E.. with i > r. Then E..(r) is a subcomplex of E.., with quotient double complex denoted E..(r). From a homomorphism cp.. one has a commutative diagram

0 --> E..(r) --, E..

E..(r)

F..

F..(r)

0

F..(r)

.

0

0

of double complexes, with exact rows.

Since, for a given n, Tot(E..) and Tot(E..(r)) have the same n-th homology, for r sufficiently large, it suffices to prove the lemma in case E.. and F.. have only a finite number of non-zero columns. We prove

this by induction on the number of columns i for which Ei. or Fi. is non-zero. If this number is one, the assertion is trivial, since the total complex is the same as the non-zero column. Otherwise one may choose an integer r so that the complexes E..(r) and F..(r), as well as the quo-

tients E.. (r) and F.. (r), have fewer non-zero columns. By induction Tot((p..(r)) and Tot(E..(r)) are homology isomorphisms. From the above diagram of double complexes one has a corresponding diagram of total

complexes, also with exact rows. From the long exact homology sequences, and the Five Lemma, it follows that each is an isomorphism, as required. For the next two sections §6, §7, we work under the following conditions.

We fix an affine Noetherian base scheme S. Let (I be the category whose objects are connected schemes X which are quasi-projective over S, and whose morphisms are regular morphisms, i.e. projective local complete intersection morphisms. Any X in (E satisfies (*) of V, §4, namely a coherent sheaf on X is the image of a locally free sheaf.

V §6. Adams Riemann-Roch for Imbeddings Under the stated conditions, by Theorem 5.3, and referring back to Chapter II, §3 we have the Riemann-Roch functors (K, ,/ii, K)

with integers j >_ 0,

where ' is the Adams character.

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[V, §6]

Lemma 6.1. Let f : X -+ Y be a regular imbedding. Let I"= P(Wx,r O+ (9x), and let f be the zero section of X in T. Then the deformation to the normal bundle constructed in Chapter IV, §5 makes f' a basic deformation of f with respect to the Riemann-Roch functor (K, i/i', K) (j >_ 0), in the sense of Chapter II, §1.

Proof. We have to verify the four BD properties. Property BD 4 of the definition of a basic deformation is valid by construction; BD 2 follows from Proposition 4.4(a). To prove BD 1 and BD 3, given x c K(X), let

x" = prK(x) e K(Pz),

where pr: PX -+ X is the projection. Let y = FK(z)

in K(M).

Then BD 1 and BD 3 follow from Proposition 4.5. This proves the lemma.

Lemma 6.2. Let 9 be a locally free sheaf on X and let

f:X->P(9(D OX) be the zero section imbedding. Then f is an elementary imbedding with respect to the 2-ring functor K in the sense of Chapter II, §3. Let 2 be the universal hyperplane sheaf on P((f O+ Ox) (Chapter IV, §1) and let

q = [2]. Then fK(l)=A-I (q)

and

f*(2)_S.

Proof. By Proposition 2.7 of Chapter IV we know that X is the zeroscheme of a regular section of the locally free sheaf 2". The first formula giving fK(1) follows from Proposition 4.3(a), and the second giving f *(2) follows from Proposition 3.2(b) of Chapter IV. This concludes the proof.

Theorem 6.3. If f : X - Y is a regular imbedding, then Riemann-Roch holds for f with respect to (K, t/i', K), with multiplier B'(Wxlr). In other words, the diagram

K(X) IxI K(Y) commutes.

K(X) Jx

K(Y)

[V, §6]

ADAMS RIEMANN-ROCH FOR IMBEDDINGS

143

Proof. Since f admits a basic deformation to an elementary imbedding, Theorem 1.3 of Chapter II tells us that it suffices to prove Rie-

mann-Roch for the deformation f. But Lemma 6.2 shows that the abstract conditions of Riemann-Roch in Chapter II, Theorem 3.1 are satisfied here, and an application of this previous theorem concludes the proof.

Application to the Graded Degree

In Chapter III, we related the Adams Riemann-Roch theorem with the graded degree of fK. Using the results of Chapter III, we can now prove that if f : X -+ Y is a morphism in E, then fK : QK(X) - QK(Y)

has a graded degree in the sense of Chapter III, §2, thus completing the last preparations for the Riemann-Roch theorems of the next section. Proposition 6.4. (a)

If f : X -> Y is a regular imbedding of codimension d, then for all n,

.fx(QF"K(X)) c QF"+dK(Y) (b)

If 9 is a locally free sheaf of rank r + 1 on a scheme Y, X = P(s), and f : X -> Y is the projection, then for all n, .fK(F"K(X)) c

F"-rK(Y)

Proof. (a) follows from the preceding Theorem 6.3, and the implication (1) of Chapter III, Theorem 4.1; (b) follows from Corollary 2.4 and Chapter III, Corollary 1.3. (2)

Warning. Although part (b) shows that fK has a graded degree on the

filtration for K, part (a) gives this result only after tensoring with Q. This is apparently essential, cf. [SGA 6], XIV. This implies that the Riemann-Roch theorem in K-theory will have denominators. Proposition 6.5. If f : X -+ Y is a regular morphism of codimension d, then

fK(QF"K(X)) c QF'

K(Y)

for all n e Z.

Proof. The proposition follows from Proposition 6.4 by factoring f into a closed imbedding followed by a projection.

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THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

From Proposition 6.5 we conclude that fK induces homomorphisms fG:

QGr" K(X) -+ QGr"+a K(Y).

It is convenient here to put G = QGr K. Theorem 6.6. The association X H G(X) = QGr K(X) is a covariant functor from our category E to graded groups. Furthermore,

(K, c, QGr K) is a Chern class functor, and (K, ch, QGr K) is a Riemann-Roch functor in the sense of Chapter II, §1. Proof. Proposition 6.5 shows that all morphisms in our category have

a graded degree in the sense of Chapter III, §2, and that our present situation fits the axiomatized considerations therein, including the statement of the present theorem. Of course, the nilpotency is guaranteed by the much stronger condition of Corollary 3.10, that for each X there is an integer d such that Fe+'K(X) = 0. We are now in a position to repeat Lemma 6.1 for the graded functor.

Lemma 6.7. Let f : X -+ Y be a regular imbedding. Let Y' = P(`BxJr O+ (9x), and let f' be the zero section of X in Y'. Then the deformation to the normal bundle constructed in Chapter IV, §5 makes f a basic deformation of f with respect to the Riemann-Roch functor (K, ch, QGr K) in the sense of Chapter II, §1.

Proof. Same as for Lemma 6.1, using Proposition 4.4(b) instead of 4.4(a).

V §7. The Riemann-Roch Theorems We continue with the same category described before §6.

Let f : X - Y be a morphism in E and let

be a factoring of f into a regular imbedding i followed by a smooth morphism p. Define the tangent element T f = [i*( lJY)V )] - [('XJp)"] =

where .pJy is the relative tangent sheaf and

41

],

,xlp the normal sheaf. Often Tf is called the virtual tangent bundle of f. But it is not a bundle, it is an element of the K-group K(X). Also see Remark 1 below.

CV, §7]

145

THE RIEMANN-ROCH THEOREMS

Proposition 7.1.

(i) (ii)

The element Tf in K(X) is independent of the factorization of f If g, f are regular morphisms such that g -f is defined, then

T9,f=f'T+Tf. Proof. Given another factorization

form a diagonal diagram as in the proof of Theorem 5.1:

Y

where Q = P x y P'. By Chapter IV, Proposition 3.9 there is an exact sequence 0

-,Wx/p-'''X/Q-3J*fQIP-4 0.

Since 0Q/p = q'*S1P.1Y, this yields CWX/QJ = L W X/PJ - 1i'*f1P'7Y1-

By symmetry, L X/QJ = L X/P'J -1i*QP/YJ-

Comparing these two equations and applying the involution v gives the required equality in K(X). This proves the first part of the proposition. The second assertion of the proposition is an immediate consequence

of the first assertion, together with the relations in the K-groups obtained from the short exact sequences of Chapter IV, Propositions 3.4, 3.7, 3.9. Each one of these exact sequences gives an additive relation of the desired type for the tangent element in special cases of composites, which when put together give the general relation as stated here.

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THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

[V, §7]

Remark 1. If f is an imbedding then T f is the negative of the class of the normal sheaf. If f is a smooth morphism, then Tf is the class of the relative tangent sheaf in the ordinary sense. Thus in general, Tf unifies these two notions in the K-group.

Remark 2. Since the Todd map is a homomorphism, the additivity of Proposition 7.1(ii) implies the multiplicativity of -r f = td(Tf) in towers, namely T s. f = f Gig T p

where G = QGr K.

We now give for the Grothendieck Riemann-Roch theorem the statement corresponding to Lemma 6.2 for Adams Riemann-Roch.

Lemma 7.2. Let 9 be a locally free sheaf on X and let .f : X - P(9 (D(9x)

be the zero section imbedding. Let G = QGr K. Then f is an elementary imbedding with respect to the Chern class functor (K, c, G) in the sense of Chapter II, §2. If 2 is the universal hyperplane sheaf on P(off p (9x) and q = [2], then .fK(l) = A-1(q)

and

.fG(1) = c` '(q v )

Proof. This is an immediate consequence of Proposition 4.3 and Lemma 6.2.

Theorem 7.3 (Grothendieck Riemann-Roch). For any f : X --+ Y in E, Riemann-Roch holds for f with respect to (K, ch, QGr K), with multiplier T f = td(Tf).

In other words, the following diagram is commutative: K(X)

td(T') c h,

fsI K(Y)

IfQGrK

ch

QGr K(Y)

Proof. Factor f into p o i, with i a regular imbedding and p a projective bundle projection. In Chapter II, Theorem 1.1 we showed that the Riemann-Roch theorem for two morphisms implies Riemann-Roch for

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THE RIEMANN-ROCH THEOREMS

147

their composite with a multiplier which is obtained precisely satisfying the formalism of the tangent element of Proposition 7.1(ii). Therefore it suffices to prove the Riemann-Roch theorem in the present context for a regular imbedding and a projection separately. For an imbedding, we can use Lemma 7.2 and Lemma 6.7. They allow us to apply Theorem 2.1 of Chapter II, which says that RiemannRoch is valid for elementary imbedding and Theorem 1.3 of Chapter II which says that if a morphism admits a basic deformation to an elementary imbedding, then Riemann-Roch holds for this morphism. Note that

if f and f are as in Lemma 6.7 or 7.2, then Tf = Tf, = td('X/Y)-1 = td(f'K(q"))-',

where q = [2] and 2 is the universal hyperplane sheaf on P(S O (9x) This concludes the proof for regular imbeddings.

For the case of a projection f : P(s) - Y, it follows from §2 that f is an elementary projection in the sense of Chapter II, §2, so RiemannRoch holds with multiplier td(te"). By Chapter IV, Proposition 3.13, there is an exact sequence

so that td(ee") = td((0P'(9)/Y)"),

as required. This concludes the proof of the Grothendieck RiemannRoch theorem.

We make no attempt to list applications of Riemann-Roch here, but include the following famous special case. Suppose that X is a local complete intersection of dimension n over a field k. Let Y= Spec(k). Then

fK: K(X) -> K(Y) = Z

can be identified with the Euler characteristic Xx = X(X, -), where n

X(X, (') = E (- 1)` dimk H`(X, g). i=O

On the graded side,

fc: QGrK(X)-QGrK(Y) = Q is called the top graded degree, and then fG is often denoted by J. For a further description of fc in this case, see Chapter VI, the example

148

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[V, §7]

following Corollary 5.4. Finally the tangent element denoted by [.Tx]

is the class in the K-group of the tangent sheaf if X is smooth, otherwise is defined as we did previously using a regular embedding of X into a smooth variety, or into a projective space over k. Therefore the Grothendieck Riemann-Roch theorem implies:

Corollary 7.4 (Hirzebruch Riemann-Roch). Let X be a local complete intersection of dimension n over a field k. Then for any locally free sheaf 8 on X, we have

AX, .0 = f

x X

As an application, if [.ix] = n where n = dim X (for instance if X is an abelian variety so 9x is trivial), then for any invertible sheaf 2' we get 1

X(X, 2') = -n!deg c l(2')',

which is the usual formulation of Riemann-Roch on abelian varieties. We refer to Hartshorne [H], Appendix A4, to see how the Hirzebruch Riemann-Roch theorem implies the more classical Riemann-Roch theorem on curves and surfaces, except that Hartshorne's references to the Chow ring should be replaced by references to QGr K. Theorem 7.5. Let f : X -* Y be a regular imbedding of codimension d, .9

a locally free sheaf of rank r on X. Let e = [4'] and q = [%X/Y] in K(X). Then c(fK(e)) = 1 + ff(P..d(e, q))

in QGr K(Y).

Here P,,d is the universal polynomial defined in Chapter II, §4.

Proof. This follows from Theorem 4.3 of Chapter II and the deformation to the normal bundle, as in Lemma 6.1, 6.2, and 7.2.

Remark. Since the covariant map fG is defined only for G = QGr K, after tensoring with Q, the preceding theorem is not a Riemann-Roch theorem "without denominators". In other theories, when Gr K is re-

placed by the Chow ring, then the same type of proof does give a Riemann-Roch without denominator. For relations among K, Gr K, and rational equivalence, we refer to [SGA 6], [BFM 1], or [F 2]. The Adams Riemann-Roch theorem for imbeddings in §6 required no denominators. The next theorem gives the general version for 4i', valid

[Appendix]

NON-CONNECTED SCHEMES

149

after inverting j. By precisely the same reasoning as for Grothendieck Riemann-Roch (Theorem 7.3), we have:

Theorem 7.6 (Adams Riemann-Roch). For any f : X - Y in E, Riemann-Roch holds for f with respect to (K, ili', Z[1/j] ® K), with multiplier

9'(T;)

1.

Appendix. Non-connected Schemes For a Noetherian scheme X which may not be connected, to specify a locally free sheaf d on X is the same as giving a locally free sheaf 8a on each connected component Xa of X ; each ea has constant rank, but these ranks may differ from component to component. When one defines K(X) as in §1, one has a canonical isomorphism of rings giving a product decomposition

K(X) = 1J K(X,,).

Each K(Xa) is a A-ring, but K(X) is not a A-ring as we have defined it in Chapter I. The augmentation (i.e. rank homomorphism) is a sum of the augmentation on each K(Xa): e: K(X) --> Znacx>,

where iro(X) is the set of connected components of X. Rather than develop a theory of .%-rings with such augmentations, we

have preferred to concentrate on the connected case. At any rate, any assertions for general X follow readily from the product decomposition. For example, the operations R`, y', and 0` operate on K(X) via their action on each K(Xa). For the y-filtration, F"K(X) = fl F"K(Xc). a

Hence, if dim X < d, then Fa+'K(X) = 0, and ch : QK(X) - QGr K(X)

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THE K-FUNCTOR IN ALGEBRAIC GEOMETRY

[Appendix]

is an isomorphism. For a morphism f : X -> Y, f maps each connected component X. of X to some component Ye(a) of Y The pull-back fK and push-forward fK are defined by

fK(n Yp ) = Hf%#(,%)), 9

a

a

xa) = II ( d

Mic.) f.

a)=B

The Grothendieck and Adams Riemann-Roch theorems of the preceding two sections are valid without change for schemes which may not be connected; indeed, they follow immediately from the connected cases and the product decomposition.

CHAPTER VI

An Intersection Formula. Variations and Generalizations

The first point of this chapter is to develop a commutative diagram similar to that of the Riemann-Roch theorems, and called the Intersection Formula for the K-functor. In particular, this will show how the product in the ring K(X) relates to the geometric intersection of subschemes of X. From this intersection formula for K we deduce a corresponding formula for Gr K, which is analogous to the "excess intersection formula" of [FM], cf. [F 2], Theorem 6.3. Special cases of the intersection formula are contained in [SGA 6] and [Man], but the general version given here for K-theory seems to be new. Our proof elimin-

ates the use of Tor, and gives another striking illustration of the deformation formalism of Chapter II. We then introduce the Grothendieck group of coherent sheaves on a scheme, and show how this group relates with the K-groups studied in Chapter V. In particular, this involves looking at two separate functors, K' and K. which are contravariant and covariant respectively. The functor K' is the Grothendieck group of locally free sheaves as before, but K.

is the Grothendieck group of coherent sheaves. Our discussion sheds further light on the Grothendieck filtration by relating it to more geometric properties. We shall apply special cases of the Intersection Formula (known pre-

viously) to determine the structure of K of a blow up. This is both a complement to the K-theory of blow ups, and also illustrates geometric techniques. We follow [SGA 6] and [Man], §15, with some simplifica-

tions. We thought it would be useful for the reader to see how this material follows directly from what we have already done. Note that in both [SGA 6] and [Man] the calculation of K of a blow up played an important role in the proof of Riemann-Roch theorem, while our proofs required no such calculation. Next we discuss a filtration for K. and relate it to the filtration for K' when comparable. This gives more geometric insight into the Grothendieck filtration and topological filtration. The groups K' and K. and their graded groups are also basic for an extension of Riemann-Roch to schemes with arbitrary singularities. We state this singular Riemann-Roch without proof. Similarly, in the rest of

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the chapter, we indicate other related results of a "Riemann-Roch" nature, especially in the context of schemes, where one can use some of the formalism or results of Chapters I-V. We make no attempt to survey the extensive literature in this active area, however. In particular we ignore recent Riemann-Roch theorems for analytic spaces, for arithmetic surfaces, or involving higher K-theory, as well as relations with rational equivalence and intersection theory going beyond what we did in §3. We refer to the literature for most of the proofs. The reader may also find a more general and powerful formalism in [FM].

VI §1. The Intersection Formula Throughout this section, we work with the same objects as in the category l of Chapter VI, §6, §7 namely connected schemes quasi-projective

over an affine Noetherian base. Not all morphisms are subject to the same restrictions, however, and the context will make the restrictions precise.

We shall be concerned with a fibre square

X1 A

FS 1.

01

I'i 19

XfY Unless otherwise specified, the vertical morphisms >li, qp are morphisms of

schemes, but the horizontal morphisms f, f1 are assumed to be regular morphisms. We let d, d1 be their respective codimensions.

Remarks. Since a regular morphism is one which can be factored into a local complete intersection imbedding, and a projective bundle projection, it follows that a regular morphism is proper. Even though we make no restrictive assumptions on gyp, cli we note that the contravariant maps (pK and 0K are defined on the K-groups. We needed restrictions only to define the covariant maps. If we factor f into a regular imbedding is X -> P followed by a projective bundle projection p: P -> X, we obtain a fibre diagram

Xl FS2.

qJ, I

' PI

1

11 1'P

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153

with p1 o it = fl and it a regular imbedding. Since the ideal sheaf of X in P generates the ideal sheaf of X 1 in P1, the left square yields a surjection 0*lex/p - wxi/Pi -4 0.

We let iff be the kernel, which is a locally free sheaf on X1, so we have the exact sequence

0- 9-

(1.1)

Arguing as in the proof of Proposition 7.1, Chapter V, one verifies easily that (ff is independent of the factorization off We may call 8 the excess

conormal sheaf for the diagram FS 1. We let e = [8'] be its class in K(X 1), so we have

e = [ *`Bx/P] - [`ex,/P,7.

The rank m of 9 is called the excess dimension

m=d - dl. If f, tp are regular imbeddings, then X 1 is the intersection of Y1 and X in Y Classically, this intersection is called proper if the excess dimension is equal to 0. Proposition 1.1. If the excess dimension is 0, that is, f, f1 have the same codimension, then the following diagram commutes:

K(X) -A-, K(Y) OKI

K(X 1)

1(pK

f-

K(Y1)

Proof. Factoring f as above, it suffices to prove the proposition when f is a closed imbedding or a projective bundle projection. The imbedding case was proved in Chapter V, Proposition 4.5. For the projection case, suppose X = P(5) with locally free on Y, and f: P(90) -+ Y is the projection. Then X 1 = P((p*I)

and

fl : X 1 -> Y1

is the projection.

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To prove the assertion, it will suffice to prove:

Lemma 1.2. Let X = P(g) and let f : X -* Y be the projection. Let F be a regular locally free sheaf on X. Then *,F is regular on X1, and f,-V/*.

(0*f*.F

Proof. By Chapter V, (2.5) there is a canonical resolution of .F:

0-'(f*

(-r)_...-

0->

-,0.

on Y, .To = f*.F Since these sheaves

with locally free sheaves .%i =

are locally free, the pull-back of this sequence by ,* is exact on X1. Since

*f* =f*(P*

and

f*arW(1) =

we get an exact sequence on X1: 0

r)

...

d'

d'

,f*(P*Jo

d

0-0 i//*.

30.

The lemma follows readily from this resolution, namely let

9i = Ker di, so there are short exact sequences for i > 0: (A)

0 - Yi - (f *I

i) - ffi-1 - 0

and for i = 0, (B)

0 -> ffo - f i

>//*.F -> 0.

Starting with °LP, = 0 one uses the long exact cohomology sequence of (A) to show by descending induction that (1) is regular, and fl*.Ti = 0 for

all i > 0. From (B) one deduces that >li*.F is regular, and that f1*(f *10-0) _ A*41*.F

is an isomorphism. Since 90 = f*.F it follows that

(P*f*- =fi*fi((P*f*F) =fi*(fi(P*5o), which proves that Qp*f*.

This proves the lemma.

[VI, §1]

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155

Theorem 1.3 (Intersection Formula). Given a fiber square FS 1 with

excess conormal sheaf 9, let e be the class of 9 in K(X 1). Then the following diagram is commutative:

K(X)

K(Y)

&)OK1

j4PK

.1

K(X 1)

AK

K(Y1)

Before presenting the proof, we record some special cases.

1.3.1. Excess Dimension 0 (Proper Intersection). In this case, 9 = 0, L-1(e) = 1, and the formula reduces to that of Proposition I.I. 1.3.2. Self Intersection Formula. This is the other extreme, when

Y1=X, (f = f and f is a regular imbedding. Then X 1 = X, E = TxlY is the conormal sheaf, and the formula reads f KfK(x) = a _ 1(C)x,

where c = [WxiY].

1.3.3. Blow Up or Key Formula. In this case, f is a regular imbedding,

and p : Yi - Y is the blow up of a regular imbedding f : X -> Y, so Y1 = Blx(Y). Then X 1 = P(` XIY),

'x1/Y, N opwx/Y(l),

and the exact sequence (1.1) is the universal exact sequence

0-+9-+O*cXIY->cpwx,Y(1)-0. We usually let e = [Vpwx,Y(l)] _ [WxiY,], and then e = 0"(c) - t', where c = [WxiY]

We may now pass to the graded case.

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[VI, §1]

Corollary 1.4. Let G be the functor G = QGr K. With assumptions as in Theorem 1.3, we have a commutative diagram G(X)

fG

, G(Y)

C,(e")bG1

(p

G

1

G(X 1)

fic

G(Y1)

Proof. Given x e GkX choose a representative x" for x in QFkKX. By Proposition 2.1(i) of Chapter III, A-1(e) is a representative for cm(e") in FmKX 1. By Theorem 1.3, f1K(.-1(e)'YKx) = 4,KfK(x)

in QFk+dK(Y1), and this represents the required equation in Gk+d Y1, thus proving the corollary. As in the non-graded result, we have the three special cases:

1.4.1. Excess Dimension 0. The formula reads f1G OG = cpGfG

1.4.2. Self Intersection Formula. If qp= f is a regular imbedding, then fGf (x) = cm(e")x, with m = d. In this case, e' is the normal sheaf.

1.4.3. Blow Up or Key Formula. Here f is a regular imbedding, Y1 = Blx(Y),

and l is the universal subsheaf on'' X// 1 = P('x11//.). Then f1G(cd-1(e")Y'G(x)) = 9GfG(x)

Remark. The proof we shall give for the theorem can be modified slightly to prove the corollary directly: one replaces K by G, .1_1(e) by cm(e"), and Proposition 4.4(a) by Proposition 4.4(b). This proof has the advantage that it works in other contexts, such as rational equivalence theory or other cohomology theories where it is not necessary to tensor with Q.

[VI, §2]

PROOF OF THE INTERSECTION FORMULA

157

VI §2. Proof of the Intersection Formula Factoring f into a regular imbedding followed by a projection, it suffices

as usual to prove the formula in each case. The projection case is covered by Proposition 1.1, so we may assume that f is an imbedding. As in Chapter II, we meet a situation which splits in two parts, one formal the other not. This involves deforming an imbedding to an "elementary imbedding" (suitably defined for the present application), proving the formula formally for "elementary imbeddings", and showing that if the formula is true for a morphism, then it is true for a "deformation", suitably axiomatized.

So we start with the axiomatization. Let be a category. We have already observed the need for two kinds of morphisms, so we have to build this into the axioms. Hence we suppose given for each two objects a subset of their morphisms, called restricted morphisms, such that the restricted morphisms form a subcategory. By a k-ring functor K we now mean that the association X H K(X) is contravariant for all morphisms, also covariant for restricted morphisms,

and satisfies the projection formula for restricted morphisms. Let K be such a functor. In Chapter II, §3 we defined an elementary imbedding with respect to K. Given a morphism f : X -' Y the surjectivity of fK: K(Y) - K(X) will here come from the fact that f is a section of a morphism p: Y-+ X. The other condition was that fK(l) = A_ 1(q) for some element q e K(Y). Both these conditions are going to play a role. In addition, let X1

X

Yi

A

f `Y

be a commutative square in CE with f, f1 restricted. We shall say that the intersection formula holds for this square with multiplier A_1(e) for some element e e K(X1) if the following diagram commutes: K(X) -A_ ,(e)tKi

fK

K(Y)

i

9)

K

K(X 1)-- AK `` K(Y1)

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[VI, §2]

AN INTERSECTION FORMULA

We shall say that a commutative square in S as above is elementary if the following properties are satisfied: ES 1.

The morphisms f, f1 are sections of morphisms and

p: Y-> X

p1: Y1 - X1

such that pO(p = 41-pl. ES 2.

There exists /elements q c- K(Y) and q1 E K(Y1) such that

and

fK(l) = A-1(q) ES 3.

f1K\1) = 2-1(q1)

There exists an element e E K(X 1) /such that (pK2-1(q)

= pK2-1(e)''1-1(q1)

Proposition 2.1. Assume that the commutative square is elementary.

Then the intersection formula hods with multiplier A-1(e).

Proof. For x e K(X) we have: reasons (p KfK(x) = (pKf ( KpKx)

ES 1,pof=id

= cpK(pKx fK(1))

projection formula

= cpK(pKx A-1(q))

ES 2

= PK(pKx). pK,1- 1(q)

ES 1,p-(p=` -p1

= pK/(WI'Kx)'p1 _ (e)'A-1/(q1)

ES 3

=

ES 2

=f1K( K1

pK(qI I

=f1

Kx/)'2-1(e))

2 -1(e))

projection formula

ES 1, p1 -f, = id

This proves the proposition. Geometric Construction of an Elementary Square

We shall now construct a situation in the geometric category which satisfies the axioms of an elementary square. We suppose given:

a morphism 0: X 1 -> X ;

locally free sheaves F on X and F1 on X 1 and a surection

a:0*.F->.F1.

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PROOF OF THE INTERSECTION FORMULA

159

With such data, we let Y=

and

G (9x)

Y1 = P(3111-1 Q+ Ox,)

together with their projections p : Y - - ). X

and

pi: Y1- > X 1.

f:X ->Y

and

f1:X1->Y1

Finally, we let

be the zero section imbeddings. The homomorphism a induces a morphism (P: P(ffl;1 O+ Oxl) -* P(F Q+ Ox)

giving a fibre square as in FS 1. In this case, F and A1 are the conormal sheaves to f and f1 respectively, so the excess sheaf d is the kernel

of a. We let e = [9]. Moreover, if 2 and 21 are the universal hyperplane sheaves on Y and Yl as in Chapter IV, §1, and q, q1 are their respective classes in K(Y) and K(Y1), then ES 1 is trivially satisfied, and Proposition 4.3(a) of Chapter V shows that ES 2 is satisfied.

We shall now prove ES 3. We have a commutative diagram with exact rows and columns on Yl: 0

0

1

0

pi 9

Q,

1

2

*2

- (P -

1

1

0

' p * t°

*(p

1+

(D O x)

id

1

, P i-F, (1 O

From the top row, we deduce ES 3 as desired.

0

1

OPT (1)

0

x, -

0

0

->0

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[VI, §21

We return to the proof of Theorem 1.3 when f is a regular imbedding.

The idea is to use a deformation diagram as in Chapter II, §1. Again, since there is only one ring K here, we restate all hypothesis ab ovo, and we first describe the axiomatization. In part this amounts to repeating the BD conditions in the special case when A = K and p = id. The additional feature amounts to saying that the deformation diagram is functorial. On the other hand, we do not need all the BD axioms except for certain maps, so we list just the properties that we need. So again we let E be an arbitrary category with restricted morphisms,

and we let K be a A-ring functor on & Suppose given a commutative square in C with f, f1 restricted: X1

X

L f

Yi

Y

We shall say that this square admits a basic deformation to a square

X1 fi,Yi X ------ f_7___+ Y1

if there exist morphisms as shown on the following diagram called the deformation cube:

such that all the horizontal morphisms are restricted, and there exists a finite number of restricted morphisms h1i: C,, -+ M1

[VI, §21

PROOF OF THE INTERSECTION FORMULA

161

with integers m, e Z, satisfying the following conditions: SBD 1.

For each x c- K(X) there exists z E K(M) such that and

fx(x) = gK(z) SBD 2.

fic(x) = g'K(z).

91K(l) = 91K(1) + Y_

SBD 3. For each z c- K(M) as in SBD 1, and all v, we have hK, cbK(z) = 0.

SBD 4. The four vertical faces going around the cube are commutative ;

g, is a section of x 1 and a,-g'1-f, = f1. Proposition 2.2. Suppose given a commutative square with restricted

horizontal morphisms, and that this square admits a basic deformation as above. If the intersection formula holds for the square of a basic deformation X1

X

J,

Yi

Y,

then the intersection formula holds for the given square with the same multiplier.

Proof. The proof consists in following the same pattern as the analogous statement of the Riemann-Roch formula, Theorem 1.3 of Chapter II. We just go around the cube as follows. Given x c- K(X) choose z e K(M) as in SBD 1. Then : reasons

91K((PKfK(x)) = 91x((pK9KZ)

SBD 1

=

SBD 4

= 91K(1)IKZ

projection formula

= 91K(1)(DKZ + E m,,hl,,K(1)(Dxz

SBD 2

= 91K(9'K(DKZ) + Y

projection formula

= 91K(P'K9'KZ + 0

SBD 4 and SBD 3

= 91 K 9'KfK(x)

SBD 1

= 91f 1 K(A -1(e)giK(x))

intersection formula

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AN INTERSECTION FORMULA

We now apply nlK. Since g1 is a section of n1, we have nIK91K = id,

so we find : (PKIK(x) = n1Kg1K '1KQ-1(e)Y'K(x))

= AK(A-I (e)0'(x))

by SBD 4.

This concludes the proof.

All that remains to be done to finish the proof of Theorem 1.3 (the Intersection Formula in the geometric context) is to prove: Proposition 2.3. Given a fibre square FS 1, with f, f1 assumed to be regular imbeddings. There exists a basic deformation of this square to an elementary square.

Proof. We already know that regular imbeddings f : X -> Y and f1 : X1 -> Y1 can be deformed to their normal bundles. We now note that the construction of this deformation is functorial. In Chapter IV, §5 we constructed from f a diagram

Y

Y,

/Y

Y

Given the morphism gyp: Y1 -> Y and fibre square FS 1, we obtain a similar square for f1: X 1 - Y1 and induced vertical morphisms giving rise to the deformation cube:

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PROOF OF THE INTERSECTION FORMULA

163

The morphism

(D:M,-*M is the morphism of blow ups

Blx,x,.(Y, x P') -+ B1xx.(Y x P1) induced by

(pxid:Y1xPI->YxPI. Conditions SBD 4 (the commutativity properties) are then automatically satisfied, and of course we have some others not listed in SBD, like and

it o g = idy

7r-g'-f' =f,

which had not been necessary in the proof of Proposition 2.2. The left back vertical square is then an "elementary square" satisfying ES 1, ES 2, ES 3, as constructed previously:

X,

fi , Y, = P('x11 1 O+ Ox,)

X

Y' = P(Wx/r O+ (9x)

and the Intersection Formula holds by Proposition 2.1. From the deformation to the normal bundle of Chapter IV, §5 we also have the residual schemes Y and Y, with their imbeddings in M, and M respectively, and an induced morphism between them as shown on the following commutative diagram :

M,

01

M

1(p

h

We shall also need the imbedding

Y

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[VI, §3]

and the corresponding commutative square Y1' n Y1

h'

M1

I

M

Y' o Y h'

The top deformation together with Proposition 4.4(a) of Chapter V gives the equation 91K(1) = 9'IK(1) + h1K(1) - hiK(1)-

The construction of the basic deformation for f being the same as in the part of this book dealing with the Riemann-Roch theorem, we know from Lemma 6.1 of Chapter V that given x e K(X), there exists z e K(M) such that f'x(x) = 9,K(z), fK(x) = 9K(z), hK(z) = 0,

h'K(z) = 0.

Then hK(DK(z) = OOKhK(z) = 0,

and similarly h (DK(Z) = (pFKhIK(Z) = 0.

This proves SBD 2 and SBD 3, and concludes the proof of all the SBD conditions. It also concludes the proof of Proposition 2.3 and of Theorem 1.3.

VI §3. Upper and Lower K In this section, E denotes a category of Noetherian schemes, each of which has an ample invertible sheaf. For example, E may be the category of quasi-projective schemes over a fixed affine Noetherian base scheme. Morphisms are arbitrary scheme morphisms.

The purpose of this section is to introduce two different K-functors. Among other things, these two functors make it possible to deal with more general singularities than have been considered up to now. For X in (l; we let

K'(X) = Grothendieck group of locally free sheaves on X; K.(X) = Grothendieck group of coherent sheaves on X.

[VI, §3]

UPPER AND LOWER K

165

We denoted K'(X) by K(X) before, but now shall view K'(X) as a contravariant functor with respect to all scheme morphisms. We let 6: K'X -> K.X

be the homomorphism induced by the inclusion of JJ, in the category of coherent sheaves on X. This homomorphism is called the Poincarehomomorphism.

Proposition 3.1. If X is regular, then 6 is an isomorphism.

Proof. Over a regular local ring, every finitely generated module has finite homological dimension ([Mat], 18C, Theorem 45, Serre's Theorem); so if X is regular then every coherent sheaf on X has a finite locally free

resolution using the basic condition (*) of Chapter V, §4 and the introductory remarks of that chapter. Hence 6 is an isomorphism by Proposition 4.1 of Chapter V. For a regular scheme X, we may use S to identify K'X with K.X, and we write K(X) = K'X = K.X.

Next we consider a useful exact sequence which shows the advantage of dealing with K. in certain contexts. Let i:X -+ Y

be a closed imbedding. If F is a coherent sheaf on X, then i,, is the sheaf on Y obtained by extending .F to 0 outside X. Then i* is an exact functor, and therefore induces a homomorphism

iK:K.X->K.Y by

On the other hand, let j: U -' Y be the inclusion of an open subscheme U of a scheme Y. There is a restriction homomorphism jK.: K.(Y) - K.(U),

which takes the class [F ] of a coherent sheaf F on Y to the class [.F I U] of the restriction of .F to U.

166

[VI, §3]

AN INTERSECTION FORMULA

Proposition 3.2. Let i : X -+ Y be the inclusion of a closed subscheme, let U be the complement of X in Y, and let j: U -> Y be the inclusion. Then the sequence

K.(X)

K.(Y)

IK

- K.(U)

-0

is exact.

Proof. It is obvious from the definitions that the composite is zero, so there is a homomorphism K.(Y)/Im(iK) -> K.(U).

To prove that his homomorphism is an isomorphism, we use Appendix

3.5 and 3.6 that a coherent sheaf F on U is the restriction of some coherent sheaf 9 on Y and any short exact sequence of coherent sheaves on U is the restriction of an exact sequence on Y Assigning [.4] to [.F] then determines a homomorphism K.(U) -> K.(Y)/Im(iK )

which is inverse to the above homomorphism; all we have to prove is that [,] mod Im(iK) is well defined. By Lemma 3.7 it suffices to prove that if F2 are two extensions of .F to Y with a homomorphism .F1--> 2 on Y which is an isomorphism on U, then [F1] _ [ffl2] mod ImK. But the kernel and cokernel of .F1 -+ F2 have support in the complement of U, thus proving the assertion and concluding the proof of Proposition 3.2.

The definition of iK for a closed imbedding i was given ad hoc. We now study the covariant functoriality of K. more systematically. Let f : X -- Y be a proper morphism. We define the push-forward fK.: K.(X) - K.(Y)

by the formula

fK.[F] = E (-1)`[Rf*F] izo

The long exact cohomology sequence shows that fK is well defined on K.(X), and the spectral sequence for a composite shows that (f°9)K. =fK.°9K.,

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so K. is covariant for proper morphisms. For a proof, see for instance [L], Chapter IV, Theorem 9.8. We note that if f : X - Y is a closed imbedding, then for all i > 0,

R; f* F = 0

and consequently the above definition coincides with the ad hoc definition given for closed imbeddings in the preceding section. Indeed, f* is

an exact functor (extension by 0 outside X), and hence R`f* = 0 for i > 1.

The definition of fk above is compatible with the previous definition of fK whenever it is possible to compare them. More precisely: Proposition 3.3. The following diagram is commutative:

K'(X) ) K.(X) K'(Y) _ K.(Y) It suffices to prove this when f is a closed imbedding or when f is a projective bundle projection. We have just made the relevant remark for a closed imbedding. For a projection, fK. was defined on regular sheaves

S by

fK[F]=U*F], and here again this is the same as the new definition because for regular sheaves, R`f*.F = 0 for i > 0. Tensor product makes K.X a module over K'X K'X ©K.X -> K.X

by

[S] ' [-F] _ [.9 ®F]

For example, the Poincare homomorphism

6:K'X->K.X takes an element x in K'X to the element x [OX] in K.X. From R3 of Chapter V, §2 one deduces the Projection Formula: Proposition 3.4. For f : X -(-> Y/ proper, x c- K.X, y e K' Y, we have fK.(f K (y) x) = y 'J K (x)

We shall meet still another projection formula in Proposition 6.2.

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[VI, §3]

Basic Lemmas

Throughout this appendix, we let U be an open subscheme of a Noetherian scheme X.

Lemma 3.5. Let 9v be a coherent sheaf on U. Then there exists a coherent sheaf 9 on X such that

gIU=gu. If F is a quasi-coherent sheaf on X and <jlv is given as a subsheaf of

F I U, then

can be taken as a subsheaf of A

Proof. We give the proof in the case of the given . and 9v subsheaf of F. The proof in the absolute case without F is obtained by deleting all references to

.

Consider all pairs (9, W) consisting of an open subscheme W of X and a coherent subsheaf 9 of F I W extending (!Yv, U). Such pairs are partially ordered by inclusion of W's, and are in fact inductively ordered because the notion of a coherent sheaf is local, so the usual union over a totally ordered subfamily gives a pair dominating every element of the family. By Zorn's lemma, there exists a maximal element of the family, say (9, W). We reduce the proposition to the affine case as follows. If W 0 X then there is an affine open subscheme V = Spec(A) in X such that V ¢ W Then W n V is an open subscheme of V, and if we have the proposition in the affine case, then we extend 9 from W n V to V, thus extending 9 to a larger subscheme than W, contradicting the maximality.

We now prove the lemma when X is affine. In that case, note that the coherent subsheaves of 9v satisfy the ascending chain condition. We let Y 1 be a maximal coherent subsheaf of 9u which admits a coherent extension 9 which is a subsheaf of F. We want to prove that T1 = !Y1, .

If 91 # Lv then there exists an affine open Xf c U and a section s e 9(X f) such that s o g1(X f). By [H], II, Lemma 5.3, there exists n such that f"s extends to a section s' e .F(X), and the restriction of s' to U is in F(U). By the same reference, there exists a still higher power f'" such that ft(s' I U) = 0 in (.F/W)(U).

Then 71 + fms'(9x is a coherent subsheaf of F which is bigger than 91, contradiction. This concludes the proof of Lemma 3.5. Lemma 3.6. A short exact sequence of coherent sheaves on U is the restriction of an exact sequence of coherent sheaves on X.

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169

Proof Let

be an exact sequence of coherent sheaves on U. By Lemma 3.5 there is a coherent extension 9 of lU to X, and there is an extension W to #'U

to a coherent subsheaf of q on X. We let T" = 9/9' to conclude the proof.

Lemma 3.7. Let .F be coherent on U and let A1, F2 be coherent on X such that their restrictions to U are isomorphic to A Then there exists a coherent sheaf 9 on X and homomorphisms

which are isomorphism on U.

Proof. Let Vu be the graph of an isomorphism on U between A I U and F2 1 U. By Lemma 3.6 there exists a coherent subsheaf I of Al ®A2 whose restriction to U is LU. This subsheaf 9 has the required property, the homomorphisms to A and A2 being the projections.

VI §4. K of a Blow Up In the first part of this section, we let f: X -+ Y be a regular imbedding in the category (E of Chapter V, §6, §7. We let X1

JI

Yl

X-Y be the blow up diagram of X in Y. We let K = K unless otherwise specified. Note that cp,

are regular morphisms.

The next proposition gives one more geometric result about blow ups. Proposition 4.1. In the blow up diagram, the map (PK(PK: K(Y) -* K(Y)

is the identity map, so (PK(1) = 1.

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Proof. The special case (pK(1) = 1 is equivalent to the general formula by the projection formula. So we prove the special case. In fact, we shall prove it only under the assumption that X and Y are regular (so X 1, Y1 are also regular). The general case requires the Remark following Propo-

sition 5.1 of Chapter V, see [SGA 6], VII, Proposition 3.6. Under the regularity assumption, we have K = K. and we can use the definition 00

(PK (1) = E (-1)`[R`(P*OY,] i=o

Thus it suffices to prove: R '(P* (9y, =

(9, 0

if i=0, if i>0.

Let f be the ideal sheaf of X1 in 01.,. By Lemma 4.1 of Chapter IV, we know that . .: 01.,(1) and is invertible. Tensoring with Oy,(n) the exact sequence

0J, -4 Or,-'.f1*(9x,-40. yields the exact sequence 0 --)- Oy,(n + 1) -* Oy,(n) -> f1*(Ox,(n))

0.

We apply the functor qp*. We note that p*f1 = f*0 Furthermore f, f1 are closed imbeddings, so Rf* = 0 and Rf1* = 0 for i >_ 1. Then we get: R`(p*(.f1*Ox,(n)) = R`((o*.f1*)((9x,(n))

= R`(.f*+/*)(&x,(n))

if i = f*(R`i*)(Ox,(n)) = IO*(Sym" exit') if i

0,

by the fundamental properties R 5 and R 6 of the cohomology, Chapter V, §2. The long cohomology sequence then yields an isomorphism 0 -> R`cp*Oy,(n + 1)

R`cp*Oy,(n) -> 0

for i >_ 1 and all n > 0. By R 4 (Serre's theorem), R`(p*Oy,(n) = 0 for n sufficiently large, so = 0 for all n >_ 0. Thus

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Now let i = 0 and n >_ 0. We get an exact sequence 0 -' p*ty,(n + 1) -> (p*&y,(n) ---f* Sym"(W) -, 0. Since

Y1 = Proj

(c) n-0

there exist canonical

homomorphisms

of

sheaves

in -If* V y,(n)

giving rise to the commutative diagram 0

in+1

I

in -

-- jn/ jn+1

0 - Q*ar,(n + 1) --> T*ar,(n)

- f*Sym"(W)

0

, 0.

The left vertical arrow is an isomorphism for all sufficiently large n by Serre's theorem. The right vertical arrow is an isomorphism by Corollary 2.4 of Chapter IV. By descending induction on n it follows that the center arrow is an isomorphism for all n > 0. This concludes the proof of Proposition 4.1. Proposition 4.1 was the last geomeric fact needed to determine most of the structure of K of a blow up, and all of it in the case when the schemes are regular. We shall now enter into formal considerations, so we make a precise list of what we use. Let K be a 2-ring functor. Let f: X --> Y be a morphism. We say that f satisfies the self intersection formula with multiplier A _ 1(c) for some c e K(X) if fKfK(x) = A=1(c)x

for all x E K(X).

Consider a commutative diagram: X1

0i

f, ' 1'i IT

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[VI, §4]

We say that this diagram is a blow up diagram with respect to K if the following conditions are satisfied: 111 1.

' is an elementary projection with respect to K, in the sense of Chapter II, §2.

We recall what this means: K(X1) as K(X)-algebra via V/K is isomor-

phic with the extension K(X), of K(X) for some positive element c c- K(X) (cf. Chapter I, §2), and 4K corresponds to the associated func-

tional 0c. We let K(X)c = K(X)[8],

where 8 is the canonical generator. 111 2.

coK(1) = 1 and therefore cpKCpK: K(Y) - K(Y) is the identity (by the projection formula).

B13. f and f1 satisfy the self intersection formula with multipliers A-1(c) and A_1(e) respectively. 111 4.

Let e = c - e, or more precisely e = frK(c) - 8. Then the diagram satisfies the Intersection Formula cpKfK(x)

for all x a K(X).

We have proved that the blow up diagram arising from blowing up a regular imbedding in the category of schemes satisfies the BI properties:

BII comes from Chapter IV, Lemma 4.1; Bl2 is Proposition 4.1; BI 3 comes from Theorem 1.3, special case 1.3.2; and 1314 is once more the Intersection Formula of Theorem 1.3, special case 1.3.3. We now work only with these properties, unless otherwise specified.

Lemma 4.2. Let >ti be as in BI 1, and let f1 satisfy the self intersection formula with multiplier A_#) as in BI 3. Let e = o'(c) - 8. If x1 a KerflK, then x1 ='1-I(e)Y'KY'IC(xl)-

Proof. By the self intersection formula for f1 we have

0 =.fif1K(xl) ='1-1(8)x1 = (1 - 8)x1.

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Hence the lemma results from the next lemma on A-rings, cf. [SGA 6], VI, Proposition 5.10. Lemma 4.3. Let K be 2-ring. Let c be a positive element in K and let K, = K[ef] be the extension of Chapter I, §2 with associated functional tic: Kc -,. K. Let z e K, and assume that z(1 - f) = 0. Then z = qlc(z)1-1(c - 0.

Proof. Let e(c) = r + 1. Write z as a linear combination r

z = Y ai(&' - 1)i

with

ai E K.

i=o

From z(C - 1) = 0 we get r+1

0= Xai-1 i=1

On the other hand, by Proposition 1.1(a) of Chapter III, with t = 1 we know that the equation for e over K can also be written r+ 1

0=

(c

Y_(-1)iyr+1

i=0

-r

Multiplying this equation by ar and comparing coefficients show that ar(_ 1)r+1-iyr+I-i(C - r - 1) = ai_1

for i = 1,..., r + 1.

Hence r+1

r+1

z=

1)r+1-iyr+I-i(c

// ai-1(e - 1)i-1 = ar

i=1

- r - 1)(i° -

1)i

1

i=1 r

= ar(-1)r y

yr-i(C

i=0

- r - 1)y`(1 - t°)

= a,.(- 1)ryr(C - r - e)

because y is a A-operation

=art-1(c-e) by putting t = 0 in Proposition 1.1(a) of Chapter III. Applying 0, and using Corollary 2.3 of Chapter I with t = -1 yields Oc(z) = ar.

This concludes the proof of Lemma 4.3, and hence of Lemma 4.2.

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We come to the desired exact sequence for K of a blow up. The following theorem axiomatizes [SGA 6], VII, Theorem 3.7.

Theorem 4.4. Let

X f '' Y be a blow up diagram, and let e = ql'(c) - e. Then the following sequence is exact:

0 --K(X)

"

) K(X1) (@ K(Y)

"

)K(Y1)

where u, v are the homomorphisms defined by: u(x) = (- a,

fx(x))

v(x1, y) = flK(xl) + ggK(y)

The sequence splits with the left inverse u' for u given by u'(x1, y) = - WK(xl), that is u'u = idK(x) .

Proof. We proceed stepwise. u is injective, split by. u'. Indeed, by the./, projection formula /'

u'u(x) = WK(A-1(e),K(x)) = Y'K(2-1(e))x = x

by Corollary 2.3 of Chapter I, with t = -1. v o u = 0 is just the Intersection Formula BI 4, because v(u(x)) = WKfK(x) -fiK(1-1(e),Kx).

Ker v c Im u. Since u' splits u we have a direct sum decomposition

K(X1)G+ K(Y) = Imu$Keru',

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where directly from the definition, Ker u' = Ker OK Q+ K(Y).

Let (x1, y) a Ker u' so that Y'K(x1) = 0, `and suppose v(x1, y) = 0, that is flK(xl) + gg1(Y) = 0.

Applying TK and using Bl 2 yields 0 = coKflK(xl) + (PK(pK(Y) = fKIK(xl) + Y = Y

Thus y = 0. Then f1K(x1) = 0 and x1 = 0 by assumption and Lemma 4.2. This concludes the proof that the sequence is exact.

Remark. If, as in the next result, v is surjective, then v gives an additive isomorphism v: Ker OK O+ K(Y)

K(Y1).

The result depends on more than the formal Bl conditions. Theorem 4.5. In the blow up diagram as at the beginning of the section, suppose that X and Y are regular schemes, so X1, Y1 are also regular. Then v is surjective, and hence we have the exact sequence 0

K(X)-'K(X1) O+ K(Y).K(Yi)- 0.

Proof. Under the regularity assumption and Proposition 3.1 we can identify K = K = K. so we can use the exact sequence of Proposition 3.2, which yields in the present instance exactly K(X1) fix

, K(Y1) 3' 'K(Y1 - X1)-0,

where j1: Y1 - X1 -> Y1 is the inclusion, and similarly K(Y)

-K(Y-X)-0.

Furthermore, cp induces an isomorphism (pr-x: Y1 - X1 -> Y - X. Hence given y1 e K(Y1) there exists y e K(Y) such that jiY1 = (pr-xjK(Y) =ji4pKY

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Hence j'(y1 - coKy) = 0 so there exists x1 E K(X1) such that Yl - (PKY = .fiKxl

This proves that v is surjective, and concludes the proof of the theorem.

The rest of the section goes back to the formal BI conditions. Proposition 4.6. Let f : X -* Y be a morphism satisfying the self intersection formula with multiplier _ 1(c). Then .fK(A-1(c)xx') = .fK(x).fK(x').

Proof. The self intersection formula reads J KfK(x) = A-1(C)x.

Then fK(A - 1(C)xx') = fK(f KfK(x)x) = fK(x)fK(x')

by the projection formula. This proves the proposition.

The above proposition suggests redefining a product in K(X) in such a way that fK becomes a multiplicative homomorphism, namely we define x * x' = 2 _ 1(c)xx'.

This product is associative and commutative, and makes K(X) into an algebra (Z-algebra), even into a K(Y)-algebra via fK as one immediately verifies using the projection formula. Note however, that this star multiplication does not necessarily have a unit element. Similarly, we redefine the multiplication in K(X 1) by

X1 * x1 = A-1(t)x1x1 = (l - t)x1x1. Then f1K is a multiplicative homomorphism for this star multiplication.

Warning. Even though we are used to imbedding K(X) in K(X1) via 1'/K, the multiplication we have just defined in K(X1) does not induce the

star multiplication in K(X). Indeed if we identify K(X) in K(X 1) then we have

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and since A

K OF A BLOW UP

177

is a homomorphism, we get A-1(c) = A-1(e)2-1(e) = A- 1(e)(1 - 6').

The star multiplication in K(X1) could be denoted more accurately by

xl*lx1 but for simplicity of notation, we shall omit the index on this star.

The groups K(X) and K(X 1) with the star multiplication will be denoted by K(X)*

and

K(X1)*

respectively.

We introduce a star multiplication on the direct sum K(X1)* p+ K(Y) by defining (x1, Y)(xi, Y') = (z1, YY'),

where z1 = A_1(6)xlx'1 + x'1OKf"y + x1ifKfKy'.

This makes the direct sum into a commutative algebra. Note that the summands K(X1)* and K(Y) have the star and ordinary multiplications in their natural imbedding in the direct sum. Theorem 4.7. With the star multiplications in K(X)* and K(X 1)* and the above multiplication on the direct sum: (i) u and v are multiplicative homomorphisms, and u is a homomorphism

of K(Y)-algebras. (ii) Im u is an ideal in K(X1)* Q+ K(Y), and in fact (xl, Y)u(x) = u(fK(Y)x)

(iii) Im u and K(X 1)* are orthogonal with respect to the multiplication in K(X1)* p+ K(Y).

Proof That u is a homomorphism follows at once from the definitions and Proposition 4.6, using 2-1(c)

_-1(e)-1()

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There is also no difficulty in verifying that u is a homomorphism of K(Y)-algebras. Similarly one verifies that v is a homomorphism. We write out in full the proof of (ii). We have: (x1, y)u(x) = (x1, y)(-2_ 1(e)Y'Kx,fKx) = (Z1, .yfK(x)),

where

Z1 = -A_1(?)2_1(e)x10Kx + x1i"fKffx - A_1(e),Kx0KfKy. By the self intersection formula fKfKx = A -1(c)x of 1113, and the projection formula we get (x 1, y)u(x)

= u(f K(y)x).

This shows both that the image of u is an ideal, and also that the image of u is orthogonal to K(X 1)* (when y = 0), thereby concluding the proof of the theorem.

VI §5. Upper and Lower Filtrations In this section we work with the same category (E as in §3, that is a category of Noetherian schemes each of which has an ample invertible sheaf The morphisms are arbitrary scheme morphisms.

We discuss a filtration on K.X compatible with the filtration of K'X defined in Chapter V, §1. We define the lower filtration: FmK.X = set of elements x e K.X such that there exist coherent sheaves A,, A2 satisfying

x = [Al] - [.y2]

and

dim Supp(.yi) < m,

i = 1, 2.

Proposition 5.1. The subgroup FmK.X is generated by the classes [0y], where V runs through the integral closed subschemes of X of dimension at most m.

Proof. It suffices to prove that for a coherent sheaf .7 with dim Supp(.F) < m, we have

(5.1)

[fl _ E,'y(

)[&v] mod Fm-1K.K,

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179

where the sum is over the m-dimensional irreducible components V of Supp(F), and ((F) denotes the length of the stalk of .F at the generic point of V. For sheaves F with support contained in a given closed subset Z of dimension at most m, both sides of (5.1) are exact, so one may induct on tv(ffl;-). If 5 is the ideal sheaf of a component V of Z, the exact sequence 0

0

and the fact that 5".F = 0 for n large shows that we may assume V = Z

and F is a coherent sheaf of Ov-modules. If r = e ,(F), there is a non-empty open set U of V and an isomorphism of O®r with F I U. By Lemma 3.7 there is a coherent sheaf 9 on V and homomorphism Or v

and

I-*F

which are isomorphisms over U. Since the kernel and cokernels of these homorphisms define classes in Fin_1K.X, it follows that

[y] _

[d®r]

_ v()[Ov] mod Fm-1K.X,

as required. Proposition 5.2. Under the product K'X O K. X -> K. X, we have the inclusion '

c F,_,K.X. Proof. We show in fact that Fn,,PK'X FmK.X c F. _ "K. X, which is

stronger by Chapter V, Theorem 3.9. Given x e F;"PK'X,

y e FmK.X, we may assume y = [F] for a coherent sheaf F whose support Y has dimension at most m. Then x is represented by a complex e' of locally free sheaves which is exact off a closed subset Z of X with codim(Z n Y, Y) >_ n;

therefore dim(Z n Y) < m - n. Then x'y=Y(-1)`[8`©F]=I(-1)`[

oiv.OO F)],

and Supp(*`(t' O.y)) c Supp(OV) n Supp(F) c Z n Y, which proves that x y as in F. -"K. X, and concludes the proof of the proposition.

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We recall that S: K'X -+ K.X was the natural homomorphism induced by the inclusion of the category of locally free sheaves into the category of coherent sheaves.

Proposition 5.3. Let d be the dimension of X. Then b(F"K'X) c Fd_"K.X.

Proof This follows immediately from Proposition 5.2.

Let G"X = Gr" K'X = F"K'X/Fn+1K'X be the associated graded group studied in Chapters III and V, and set

G'X = n G"X = ( Gr" K'X n?0

n20

Define the lower graded component GmX = Grm K.X = FmK.X/Fm_1K.X,

and set

G. X = Q Gm X = Q Grm K. m20

m20

By Proposition 5.2, tensor product induces a "cap" product G"X (D GmX n -Gm-nX,

making G.X into a graded G'X-module. (The notation "n" is to suggest the cap product of topology.) By Proposition 5.3 we conclude that 8 induces a homomorphism on the graded groups SG: Gr K'(X) - Gr K.(X)

such that bG(x) = x n [(9X]. Actually we have an induced map on each graded component SG: Gr" K'(X) -> Grd_" K.(X).

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The commutativity relation of Proposition 3.3 for S now gives the corresponding relation in the graded context: Corollary 5.4. Let G = QGr K. For any regular morphism f : X -> Y, the following diagram commutes. G'(X)

.

G.(X)

fG1

G.(Y)

I fG.

G.(Y)

aG

Example. Let k be a field and Y = Spec(k). Let f: X - Y be a regular morphism and let d = dim X. By Proposition 5.1, QGro K.(X) = G0(X) is generated by the classes [(9P], where P ranges over the closed points. In this case, using the functoriality on the composite P -> X -+ Y, one sees

at once that fc [ar] = [k(P): k][ay] = [k(P): k], where we identify G.(Y) with Q via the basis element [Oy]. With this identification the commutative diagram of Corollary 5.4 on the component of top degree reads: QGro K.(X)

QGr`°P K'(X) fG

=fG If. Q

This gives the promised geometric interpretation of fG in top graded degree, relevant for the complete interpretation of the Hirzebruch Riemann-Roch theorem of Chapter V, Corollary 7.4. Indeed, fG is the ordinary "degree" of 0-cycles, in which case we have a preconceived geometric notion of "number of points". In the preceding chapter, we compared the y-filtration F"K(X) with a

topological filtration F OPK(X). There is another natural filtration of K(X) when X is regular. We let: FtOPK(X) = subgroup of K(X) generated by classes [.F] of coherent sheaves F whose supports have codimension

at least n in X.

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As in Proposition 5.1, 'P,,PK(X) is generated by the classes [(9v], where V runs through the'integral closed subschemes of X of codimension at least n. Note that in case X has dimension d, and dim(V) + codim(V, X) = d

for all such V (for example, if X is a variety over a field), then FtoPK(X) =

On a general scheme, however, one must distinguish these notions.

Proposition 5.5. If X is regular, then F"K(X) c F, PK(X) c'FIOpK(X), and

QF°K(X) = QFic,PK(X) = Q'Fi, K(X) in QK(X).

Proof. The first inclusion was proved in Chapter V, Theorem 3.9. The second follows from the fact that if 9" is any bounded complex of locally free (or coherent) sheaves, with homology sheaves A", then

Y- (-1)`[4` ] = Y- (-1)`[ °` ] in K(X).

To show that all three agree after tensoring with Q, we must show that if V is an integral closed subscheme of X, and n = codim(V, X), then

[av] E QF"K(X)

By Noetherian induction, we may assume this has been proved for all proper closed integral subschemes of V There is a proper closed subscheme S of V such that the inclusion

j:V-S-X-S is a regular imbedding of codimension n. One sees this by taking n equations which generate the ideal of V in the local ring of X at the generic point of V; such a sequence is regular on some open set U, and one may choose S so that its complement in V is V n U.

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183

Since j is a regular imbedding, we have seen that

jx(FkK(V - S)) c QFk+"K(X - S) (Chapter V, Proposition 6.4). In particular,

[0v-s] E QF"K(X - S). Consider the exact sequence of Proposition 3.2. QK(S) -> QK(X) --> QK(X - S) -> 0.

Since the restriction map K(X) --+ K(X - S) is, a surjection of %-rings, it

maps F"K(X) onto F"K(X - S). Therefore there is an x in QF"K(X) such that y = [Uv] - x e Im(QK(S) --> QK(X)).

Expressing y as a rational combination of classes [(9w], for W integral closed subschemes of S, and applying Noetherian induction to these W, gives

[(9v] = x + y e QF"K(X), as required.

Finally we deal with the functoriality of fK with respect to the filtration.

Proposition 5.6. If f : X -+ Y is proper then fK(FmK.X) c FmK.Y

Proof. It suffices to note that Supp(R`f*F) c f (Supp

),

and dim f(Z) < dim Z for any Z closed in X.

The lemma tells us that fK is compatible with the lower filtration. Therefore we have an induced functorial homomorphism fc : G.(X) - G.(Y)

for a proper morphism f

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By the projection formula of §3 for K' and K. and Proposition 5.6 we deduce the projection formula for the graded functors .fG.(.f G (Y) n x) = Y n fc (x)

for f: X - Y proper, x e G.X and y e G' Y.

VI §6. The Contravariant Maps f K- and f G.. From here on, we only give indications of proofs, if at all.

We have defined K. as a covariant functor for proper morphisms. We now wish to define K. as a contravariant functor. In order to take care of the open subschemes as in §3, we let:

E = category whose objects are the same as in §3 and whose morphisms are those which can be factored as p o i, where p is smooth and i is a regular imbedding. Note here that the only difference with our previous regular morphisms is that p is not assumed proper. We assume now that morphisms are in this category.

Suppose first that f : X -+ Y is flat. Then the obvious desideratum gives us the contravariant map, namely for 9 coherent on Y,

fK [T] = U*T]. This does give a homomorphism K.(Y) - K.(X) since f * is exact. If f is smooth, then f is flat, and this definition applies. If f is not flat, there is a technical complication, and we have to go

through the same rigamarole as before, which we summarize. Let us begin by a sheaf-theoretic remark. Let X be a closed subscheme of Y Let . be a coherent sheaf on Y, supported by X. Let I be the ideal sheaf defining X in Oy. Then there is some power j' such that .f"`?°=0.

Therefore there is a filtration

[VI, §61

THE CONTRAVARIANT MAPS J" AND J°

such that each factor sheaf r /fir+1

185

is a coherent sheaf over OX.

We define

where the subscript X indicates the class in K.(X), which is defined for each term on the right-hand side, and thus defines the left-hand side. Lemma 6.1. Let is X -. Y be a closed imbedding. Let K.(Y,X) be the Grothendieck group of coherent sheaves on Y supported by X. The homomorphism

iK : K.(X) - K.(Y, X) induced by i,, is an isomorphism, whose inverse is given by

[n Y H

[iX

as defined above.

Proof. This is an easy consequence of the Jordan-Holder theorem, which we leave to the reader.

Note. For clarity we indexed the class of a sheaf by Y and X respectively. In practice, we may also drop the indices by making the identification via the isomorphism of the lemma, or we may just write the X as an index to make the distinction clear. Suppose that is X -> Y is a regular imbedding. There is a finite resolution ,ff.->OX -+ 0

by locally free sheaves on Y (e.g. the Koszul complex). For any coherent

sheaf V on Y we then obtain a complex 9. ®9. We define IK-[I] = L (- 1)"[.fk(?. © W)AX, k

where W,(wO. 05) is the k-th homology of X. Q 9r, and is supported by X. By basic abstract nonsense of homological algebra (sheaf Tor), the sheaf homology is independent of the resolution. Since in addition 9r S. px 9 is exact, we obtain a well-defined map 1K.: K.(Y). K.(X).

186

AN INTERSECTION FORMULA

[VI, §6]

If f : X -+ Y is factored as f = p o i, then by using the same steps as in the definition of fK in Chapter V, §5 one can prove that the map iK

o pK

is independent of the factorization, and defines f K. functorially. It can also be verified that for flat f the map fK obtained from a factorization is the pull-back that we mentioned first. Having defined K. as contravariant functor, it is then natural to obtain the corresponding Projection Formula:

Proposition 6.2. Let f: X -' Y be a regular morphism. For x e K'(X) and y e K.(Y) we have .fK.(x . f K (Y)) = .fK (X). y

Proof. By factoring f into a regular imbedding and a projection, it suffices to prove the formula in each case. In the case of a projection, the formula follows from R 7 of Chapter V, §2 just as the first projection formula of Proposition 3.4 followed from R 3 after we use the fact that the classes of sheaves [&(n)] generate KP over the base for a projective bundle P. In the case of a regular imbedding, the formula involves two resolutions, and a proof can be given by constructing a double complex, in the style of general homological algebra.

The next results have to do with the graded properties off'-, and so involve dimension as well as codimension. This means that one has to be careful about the schemes involved. Therefore we assume in addition, for the rest of this section, that all schemes are over a field.

Schemes of finite type over a regular base would suffice, provided that an appropriate notion of dimension is used. See [F 2], Chapter 20. Next we pass to the grading properties of fl.

Proposition 6.3. Let f : X - Y be a morphism in Cr, of codimension d. Then

c Fm-dK.X so f'- induces a functorial homomorphism fG.:

Gm(Y) - Gm-d(X)

[VI, §6]

THE CONTRAVARIANT MAPS fX AND jG

187

Proof. It suffices to prove the proposition when f is smooth and when f is a regular imbedding. In the first case, one uses Proposition 5.1, and the assertion is immediate by applying fK. to [Ov] where V has dimen-

sion m. In the case of a regular imbedding, a proof can be given by deformation to the normal bundle. Note that the proof of Chapter V, Proposition 6.4 that fK has a graded degree also went through deformation to the normal bundle, via Adams Riemann-Roch. Then we have the projection formula for the graded map: Proposition 6.4. Let f : X -> Y be a regular morphism. Let G = QGr K. For x e G'(X) and y e G.(Y) we have

fc.(x n fc.(Y)) =

fc-(x)

n y.

Proof. This is an immediate consequence of the non-graded Proposition 6.2 together with the compatibility with filtrations and the induced graded maps which has been proved in all cases. In the next section, we shall state a Riemann-Roch theorem involving the contravariant maps introduced above. A particular case of Proposition 6.3 occurs for the restriction

fK:K.Y-+K.U to an open subscheme of Y, and the induced map f c : Gm(Y) -> Gm(U).

Proposition 6.5. If U is the complement of a closed subscheme X of Y, then we have an exact sequence Gm(X) -+ Gm(Y) -+ Gm(U) - 0

Although this proposition looks innocuous, and is the graded analogue of Proposition 3.2, we don't know any proof which does not involve using the Singular Riemann-Roch theorem with values in the Chow group, of [BFM 1], which we discuss in the next section.

188

AN INTERSECTION FORMULA

[VI, §7]

VI §7. Singular Riemann-Roch This section uses only §3, §5, and §6. We continue with the same category and with the same notation.

The Riemann-Roch theorem of Chapter V, §6 yields a formula for the

Euler characteristic X(X, 6) of a locally free sheaf of on a projective scheme X over a field k only if X is a local complete intersection, i.e. the morphism from X to Spec(k) is regular in the sense of Chapter V, §5.

We next give a statement of a Riemann-Roch theorem for more singular schemes. For simplicity we restrict our attention to schemes which are quasi-projective over a field k. Much of the theorem is valid without

this assumption. The main use of a ground field is to have dimensions and codimensions of closed and open subschemes behave nicely. For more general versions see [F 2], 20. We set S = Spec(k). The singular Riemann-Roch theorem constructs a homomorphism

T=r,:K.X-+QG.X satisfying the following properties: SRR 1.

(Covariance). If f: X -> Y is proper, then the following diagram commutes.

K.X fK,

SRR 2.

QG.X jIG.

(Module). For any X, the following diagram commutes.

K'X ®K.X

ch®t. QG'X ®QG.X In

1 K.X

' QG.X

For any X one then defines the Todd class Td(X) in QG.X by (7.1)

Td(X) = T([OX]).

[VI, §7]

SINGULAR RIEMANN-ROCH

189

One deduces from SRR 1 and SRR 2 a Hirzebruch Riemann-Roch formula (7.2)

X(X, e) = fxch(ol) n Td(X). J

Here Ix is the push-forward fG. for the morphism from X to S. If X is a local complete intersection over S, then (7.3)

Td(X) = td(Txis) n [0x]

This formula (7.3) is a special case of a Riemann-Roch theorem which is dual to the Grothendieck Riemann-Roch theorem.

SRR 3. (Verdier Riemann-Roch). If f: X -+ Y is a regular morphism, then the following diagram is commutative: *QG.Y

K.Y

fKI K. X

1fG

td(Tf)-1 n i

QG.X

Applying SRR 3 to Y = S, [(9s] e K.S, yields (7.3).

The construction of t may be sketched as follows. Given a coherent sheaf .F on X, one imbeds X in a scheme P which is smooth over S, and one resolves .F by a complex 9' of locally free sheaves on P. Since 9' is exact on P - X, the class Y, (- 1)` ch(gi) n [0p] e QG.P

restricts to zero on P - X. Proposition 6.5 motivates the existence of a class in QG.X whose image in QG.P is this class. An essential step is to construct a canonical such class chX((f') e QG.X.

Its construction is based on MacPherson's graph construction, which is a generalization of the deformation to the normal bundle. Then one defines

ix([tfl) = td(Tpls) n chx(g.).

190

AN INTERSECTION FORMULA

[VI, §8]

We refer to [BFM 1], [V], and [F] for details on MacPherson's graph construction, and the proof that T, is well defined and satisfies SRR 1,

SRR 2, SRR 3, as well as for applications in the case of algebraic schemes over a field.

There is an entirely parallel discussion for the Adams operators

0:K'->K' One constructs q1j: K.X --> Z[1/j] Q K.X

satisfying the analogues of SRR 1, SRR 2, SRR 3. As in Chapter V, §7, O (T f)-1 replaces the Todd classes td(Tf). The construction of ([.F])

is also analogous to that of i, by imbedding X in a smooth P and resolving F by a complex of locally free sheaves; it would be interesting

to find a more direct description of O;[.]. For an extension to higher K-theory, which follows the same pattern, see Soule [S].

Remark. It follows easily from Proposition 5.1 that there is a functorial surjective homomorphism Am(X) -+ Gm(X )

from the group of m-cycles module rational equivalence to the associated

graded group to K.(X). In fact, the Riemann-Roch theorem is proved with values in QAm(X), from which it follows that the above homomorphism becomes an isomorphism after tensoring with Q. For details and more on rational equivalence, see [F 2].

VI §8. The Complex Case For schemes over S = Spec(C), one has topological functors, the (singular, even) cohomology

H'(X) _

HZ`(X; Z)

and

Kt p(X),

[VI, §8]

THE COMPLEX CASE

191

the Grothendieck group of topological vector bundles on X. These are contravariant, ring-valued functors. Since vector bundles on X have Chern classes in H'X, there is a Chem character, which we denote ch' : Kt'0 -+ QH'

as in Chapter II, which is a natural transformation of contravariant functors; here QH' denotes Hf( ; Q). If f : X -> Y is a projective local complete intersection morphism, there are push-forward homomorphisms and

fH : H'X --+ H' Y

fx,01,: KtoPX -* K- 0P Y,

so that the diagram td(Tf) ch

K;0

QH'X IfR

i QH. Y ch

Kiop

commutes, i.e. Riemann-Roch holds for f with respect to (K;op, ch', QH'), with multiplier td(Tf). For the constructions, see [BFM 1] and [BFM 2]. There is a homomorphism

Y= K'X - Keo,X which takes chi to V( (ff V ), where V(9 V) is the vector bundle whose sheaf

of sections is a (Chapter IV, §1). This a' gives a natural transformation of contravariant functors. If f : X -> Y is a projective local complete intersection morphism then the diagram K'X

K' X

K* Y

) K'OPY

OP

commutes, i.e. Riemann-Roch holds for f with respect to (K', a', K00P) (with multiplier l!). It follows (Chapter II, Proposition 1.4) that Riemann-Rock holds for the composite functor, i.e., K'X

;JY

commutes.

td(Tf) ch' o «

ch' o a'

QH'X

192

AN INTERSECTION FORMULA

[VI, §9]

Once the push-forward homomorphisms are constructed for K'°P and if, these Riemann-Roch theorems may be proved by exactly the same procedure as in this treatise: by deformation to the normal bundle, the general case is reduced to the case of elementary imbeddings and projections.

In topology cohomology theories If and Kt°P are dual to homology theories H. and K°P, ch' corresponds to a natural transformation ch.: K`°P - H.

satisfying analogues of SRR 1, SRR 2. With K. as in §3, one can construct a.: K. ->KLOP

"dual" to a', satisfying analogues of SRR 1, SRR 2, SRR 3. The proof follows the same pattern (cf. [BFM 2]). A more intrinsic construction of a., valid for arbitrary complex analytic spaces, and extending to higher K-theory, has recently been given by R. Levy.

VI §9. Lefschetz Riemann-Roch The formalism developed here can also be used in another situation, to study equivariant sheaves.

Let k be an algebraically closed field, S = Spec(k), and let n be a positive integer not divisible by the characteristic of k. Let l be the category whose objects are pairs (X, hx), where X is a smooth projective scheme over S, and

hx:X -+X is an endomorphism such that hX = idx. A morphism

f:(X,hx)-'(Yhr) is a morphism f: X -> Y such that h1 ° f = f ° hx. The hypotheses imply that the fixed point scheme of hx on X, denoted X', is also smooth over S. A morphism f as above induces a morphism fh: Xh

of the fixed point schemes.

Yh

[VI, §9]

LEFSCHETZ RIEMANN-ROCH

193

An equivariant (locally free) sheaf on (X, hx) is a (locally free) sheaf 9 on X together with a homomorphism CpE: hX8 -> S.

(Note that it is not required that PE have finite order.) Homomorphisms and exact sequences of equivariant sheaves are defined in an evident way, so that one has a Grothendieck group K(X, hx) of equivariant locally free sheaves.

If hx acts trivially on X, then K(X, hx) = K(X) ®z Z[k],

where Z[k] is the free abelian group on the elements of k, a ring with multiplication induced by multiplication in k. This is because any equivariant 9 is a finite sum of sheaves da, for eigenvalues a e k, on which T. - a is nilpotent. Fix a Z[k]-algebra A such that for every n-th root of unity a in k, a 1, the image of [1] - [a] in A is invertible. (Note that such A can have characteristic zero, even if k has positive characteristic; e.g. A may be a Witt ring.) For any (X, hx) in C, the conormal sheaf W = WxhIX to X" in X is an equivariant sheaf on X, all of whose eigenvalues are non-trivial roots of unity. It follows that the element Ax = E (-1)'[A'W] e K(X') ®z A is invertible. The functor (X, hx) H- K(X, hx) is both contravariant and covariant on

C, just as in the absolute case. For this one needs to know that any equivariant coherent sheaf is the image of an equivariant locally free sheaf; this follows from the fact that any (X, hx) admits a closed imbedding into (P, hp) where P is a projective space over k, and hp is a linear endomorphism. An equivariant locally, free sheaf on (X, hx) restricts to an equivariant

locally free sheaf on (X', id), giving rise to a homomorphism p : K(X, hx) - K(X ") ® A

Thus if one defines L(X, hx) = K(Xh) ® A, then (K, p, L) is a RiemannRoch functor in the sense of Chapter II, §1.

194

AN INTERSECTION FORMULA

[VI, §9]

The Lefschetz Riemann-Roch theorem asserts that for a morphism f: (X, hX) -+ (Y, hi), the diagram K(X, hx)

Af P K(X h) ® A

K(Y,,hY) P 'K(Y'')Qx A commutes, where

Af=AXIfL(AY)EK(X")®A.

The reader should find the proof a pleasant exercise: One factors f into a closed imbedding followed by a projection. The case of an imbed-

ding is handled by deforming to the normal bundle, and calculating directly for an elementary imbedding. For a projection one proves equivariant analogues of the results of Chapter V, §2; the calculations for a projection are easiest for one of the form (Y, hY) x s (P, hp) - (Y, hY),

with (P, hp) as above. For details, as well as generalizations to the singular case, see [BFQ]. As a special case, one has a fixed-point formula. If the fixed point set X' is finite, then

(- 1) tr(H`(X, 4')) = E tr(4(P))/det(I - ti). PEXF

{

Here for an equivariant vector space V over k, tr(V) is its image in A under the canonical homomorphism K(S, id) = Z[k] -> A,

4'(P) is the fibre (restriction) of 9 at P, and tP: TP*X -+ TpX

is the map on the cotangent space TpX = WP,X induced by hX.

[VI, §9]

LEFSCHETZ RIEMANN-ROCH

195

By Proposition 1.4 of Chapter II, this Lefschetz Riemann-Roch theorem can be composed with Grothendieck Riemann-Roch, yielding a commutative diagram K(X, h.)

if ch , .) QGr K(X") O A

fK

I fh QG,K

QGr K(Y") Q A

K(Y, h,.)

where T f = td(Tf,.) ch(2, f).

Or one may compose with Adams Riemann-Roch... .

As a final exercise, the reader may work out the analogous theorem when & is replaced by the category of smooth projective schemes X over a finite field k = F9, and X" is the set of F9 valued points of X (the fixed points of the Frobenius on X). Let K'(X) be the Grothendieck group of locally free sheaves .9 on X together with q-linear endomorphism, cpg:B-+9 (i.e. qp, is additive, and rp,(as) = a9p.(s) for a and s sections of (9X and 9

over an open set of X). When X = Spec(F9), such space with a linear map, and K'(Spec(F9))

is just a vector

' F.

tI

Therefore for any X, K'(X") is a vector space over F. with basis the points in X". Restriction gives a Riemann-Roch functor p: K'(X) -> K'(X").

For f: X -+ Y, one has a Frobenius Riemann-Roch theorem: the diagram K'X

n

,) K'(X")

All

I

K'Y

K'(Y")

K'

196

AN INTERSECTION FORMULA

[VI, §9]

commutes. In particular, given .9 on X, we have the formula

(- 1)` tr(H'(X, i')) _ Y tr(e(P)). PeXh

For example, if H'(X, Ox) = 0 for i > 0, and X is geometrically connected, then

# X(F9) - 1

mod p,

where q is a power of the prime p, a Chevalley-Warning formula. For details and a generalization to the singular case, see [F 1].

References

[EGA]

A. GROTHENDIECK, With J. DIEUDONNE, Elements de geometrie alge-

brique, Publ. Math. I.H.E.S. 4, 8, 11, 17, 20, 24, 28, 32 (1961-1967) [SGA 6]

P. BERTHELOT, A. GROTHENDIECK, L. ILLUSIE, et al., Theorie des in-

tersections et theoreme de Riemann-Roch, Springer Lecture Notes 225, 1971

[AK]

A. ALTMAN, S. KLEIMAN, Introduction to Grothendieck duality theory,

[At]

M. ATIYAH, K-Theory, Benjamin, 1967

[At-Hi]

M. ATIYAH and F. HIRZEBRUCH, Cohomologie-Operationen and charak-

Springer Lecture Notes 146, 1970

teristische Klassen, Math. Z. 77 (1961) pp. 149-187 [AT]

M. F. ATIYAH, D. O. TALL, Group representations, 1-rings and the Jhomomorphism, Topology 8 (1969) pp. 253-297

[BFM 1] P. BAUM, W. FULTON, R. MACPHERSON, Riemann-Roch for singular varieties, Publ. Math. I.H.E.S. 45 (1975) pp. 101-145 [BFM 2]

Riemann-Roch and topological K-theory for singular varieties, Acta. Math. 143 (1979) pp. 155-192

[BFQ]

P. BAUM, W. FULTON, G. QUART, Lefschetz-Riemann-Roch for singular varieties, Acta. Math. 143 (1979) pp. 193-211

[BS]

A. BOREL, J: P. SERRE, Le theoreme de Riemann-Roch (dapres Grbthendieck), Bull. Soc. Math. France 86 (1958) pp. 97-136

[B]

M. BORELLI, Some results on ampleness and divisorial schemes, Pacific

J. Math. 23 (1967) pp. 217-227 [Bo]

R. BoTT, Lectures on K(X), Benjamin, 1969

[Ev]

L. EVENS, On the Chern classes of representations of finite groups, Trans. Amer. Math. Soc. 115 (1965) pp, 180-193

[Ev-K]

L. EVENS and D. S. KAHN, An integral Riemann-Roch formula for induced representations for finite groups, Trans. Am. Math. Soc. 245 (1978) pp. 809-330

[F 1]

W. FULTON, A fixed point formula for varieties over finite fields, Math. Scand. 42 (1978) pp. 189-196

[F 2]

W. FULTON, Intersection Theory, Springer-Verlag, 1984

REFERENCES

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[FM]

W. FULTON, R. MACPHERSON, Categorical framework for the study of singular spaces, Mem. Amer. Math. Soc. 243, 1981

[Gr]

A. GROTHENDIECK, Classes de Chern et representations lineaires des groupes discrets, Dix exposes sur la cohomologie etale des schemas, North-Holland, Amsterdam, 1968

[H]

R. HARTSHORNE, Algebraic geometry, Springer-Verlag, 1977

[Hi]

F. HIRZEBRUCH, Neue topologische Methoden in der algebraischen Geo-

metrie, Ergebnisse der Mathematik, Springer-Verlag, 1956; Translated and expanded to the English edition, Topological Methods in Algebraic Geometry, Grundlehren der Mathematik, Springer-Verlag, 1966

[J]

J. P. JOUANOLOU, Riemann-Roch sans denominateurs, Inv. Math. 11, (1970) pp. 15-26

[Ke]

M. KERVAIRE, Operations d'Adams en theorie des representations lineares des groupes finis, l'Ens. Math. 22 (1976) pp. 1-28

[Kn]

1. KNOPFMACHER, Chern classes of representations of finite groups, J.

London Math. Soc. 41 (1965) pp. 535-541 [Kn]

D. KNUTSON, d-rings and the representation theory of the symmetric group, Springer Lecture Notes 308, 1973

tKr]

Ch. KRATZER, Operations d'Adams et representations de groupes, 1'Ens.

Math. 26 (1980) pp. 141-154 [L]

S. LANG, Algebra, second edition, Addison-Wesley, 1984

[Man]

Y. I. MANIN, Lectures on the K-functor in algebraic geometry, Russ. Math. Surveys 24, No. 5 (1969) pp. 1-89

[Mat]

H. MATSUMURA, Commutative algebra, second edition, Benjamin/ Cummings, 1980

[Mi]

A. MICALI, Sur les algebres universelles, Ann. Inst. Fourier, Grenoble 14 (1964) pp. 33-88

[Q]

D. QUILLEN, Higher algebraic K-theory: I, Springer Lecture Notes 341, 1973 pp. 85-147

[S]

C. SOULS, Operations en K-theorie algebrique, CNRS preprint, 1983

[Th 1]

C. B. THOMAS, Riemann-Roch formulae for group representations,

Mathematika 20 (1973) pp. 253-262

[Th 2]

C. B. THOMAS, An integral Riemann-Roch formula for flat line bundles,

Proc. London Math. Soc. XXXIV (1977) pp. 87-101 [V]

J.-L. VERDIER, Le theoreme de Riemann-Roch pour les intersections completes, Asterisque 36-37 (1976) pp. 189-228

Index of Notations

A Ac

A(X) Blx(Y) c, ct

ch chq, W

7 , yr

e" E Ee e

fA, fA

f', fK f', fG

P

F. F,nop

fe G 9e

Gr`

Gr(K) K Ke

K(X) K(a)

K', K. e,

L A

A' A,

the other ring in Riemann-Roch, 11, 28 extension of A determined by c, 15 receives values of Chern character, 17, 28 blow up of X in Y 91 Chern class and power series, 12, 54 Chern character, 17, 125 Chem character associated with power series gyp, 17 conormal sheaf, 77 Grothendieck operations and power series, 47 involution of e, 20 positive elements in K, 3 positive elements in extension Ke, 9 augmentation, 3 homomorphisms induced by f in A, 28 homomorphisms induced by f in K, 28 homomorphisms induced by f in G, 28, 144 Grothendieck y-filtration, 48 lower filtration, 178 topological filtration, 120 canonical functional from Ke to K, 10 in practice, Gr K or QGr K, 61 canonical functional from A, to A, 15 Grothendieck graded component, 54 Grothendieck associated graded ring, 54 A-ring and K-functor, 3 extension of K determined by e, 7 Grothendieck group of X, 102 Koszul complex, 70, 106 upper and lower K-groups, 164 canonical generator of Ke, 8, 15 line elements in K, 4, 53 as in A-ring, 3 lambda operations, 3 lambda power series, 3

200

INDEX OF NOTATIONS

polynomial equation defining A,, 15 polynomial equation defining Ke, 7 P(d) Projective bundle Proj Sym(1ff), 67 Pk, Pk,; certain universal polynomials, 9 Pic(X) isomorphism classes of invertible sheaves, 103 universal hyperplane bundle, 67 .9 higher direct images, 105 R !f, 91p Regular sheaves, 107 0, 0, Adams operations, 23 ax coherent sheaves on X having finite locally free resolutions, 126 a, at related to the classes of Sym`, 7, 117 canonical sheaves in canonical resolution of .F, 113 Tf tangent element, 144 Tf Riemann-Roch multiplier, 28 td Todd homomorphism, 19 tdgp, t Todd power series, 20 B' Adams multiplier, 24 V(m) m-th eigenspace for Adams operations, 60 V(s) vector bundle of d, 68 93X category of locally free sheaves on X, 102 xv involution of x, 20 PC

Pe

(.F)

Index

A

D

Adams character 23, 60 Adams multiplier 24 Adams operations 23, 58 Adams Riemann-Roch 37, 63, 119, 142, 146, 149, 190

Deformation cube 160 Deformation diagram 30, 99 Deformation to normal bundle 96, 142, 144, 160 Dimension 6 Direct images 105 Double complex 140 Doubly variant functor 37

Ample 52, 118 Associated functional

10, 15, 117 Associated Hirzebruch polynomial

17,

19

Augmentation 3 Augmented Koszul complex

71

B

Basic deformation 30, 142, 144, 160 Blow up 91, 97, 169, 172,-177 Blow up diagram 91, 172 Blow up formula 155, 156 Bott's cannibalistic classes 24 C

Canonical generator 8, 15, 115 Canonical positive structure- 9 Canonical resolution 113 Canonical section 77 Cap product 180 Chern character 17, 125 Chern class 12, 54 Chern class functor 31, 142, 144, 146 Chern class homomorphism 12 Chern polynomial 13 Chern root 14 Chevalley-Warning formula 196 Codimension 86, 89, 120 Complex

119

Conormal sheaf 77, 153

Contravariance for lower K 184 Cotangent sheaf

81

Covariance 28, 37, 116, 127, 134, 144, 166

E

Eigenspace decomposition for Adams character 60 Elementary imbedding 32, 37, 57, 68, 142, 146

Elementary projection 32, 38, 57, 115, 117, 147

Elementary square 158 Exceptional divisor 91 Excess conormal sheaf 153 Excess dimension 153

Extension of 2-ring K 4, 7, 115 F Filtration 48, 61, 117, 120, 124, 178, 182, 186 Finite-dimensional A-ring 6

Fixed point formula 194 Formal group 40 Frobenius Riemann-Roch 195 Functional of extension 10, 15, 117 G

y-filtration

48, 122, 124, 179, 182 Graded degree 55, 65, 143, 183 Graded filtration 48

Graded K 54, 61 Graded splitting 49

INDEX

202

Grothendieck filtration 48 Grothendieck group 102 Grothendieck operations 47 Grothendieck Riemann-Roch

N 146

Newton polynomial 23 Nilpotence 52, 125

P H

Pic

Hirzebruch-Newton polynomials Hirzebruch polynomials 17 Hirzebruch Riemann-Roch 148 Homology isomorphism 140 Howe's proof 34 Hyperplane at infinity 68

23

I

Imbedding 68 Integral Riemann-Roch 43, 46, 148 Intersection formula 131, 155, 157 Involution 20

167, 184, 186 Projective bundle 67, 104, 115 Projective completion 69 Proper intersection 153, 155 Proper transform 94 Push forward 116, 127, 134, 144, 166 Q

Quasi-equal 111 Quasi-finitely generated Quillen's proof 114

K K-functor

53, 103

Poincare homomorphism 165, 181 Positive element 3 Positive structure 3, 9 Principal element 32 Projection formula 28, 118, 128, 139,

111

134

K of blow up 169 K of projective bundle

115

K(X) 103 Key formula 155, 156 Koszul complex 70, 76, 106 Koszul resolution 76, 107 L 1-dimension 6 A-operations 3 2-ring 3, 103 2-ring functor 37, 139, 157 Lefschetz Riemann-Roch 194 Line elements 4, 53, 103 Local complete intersection 86 Locally free sheaf 67, 100 Locally free resolution 100, 126 Lower filtration 178 Lower grading 180

Lower K 164

R

Regular complex 123 Regular imbedding 77, 126 Regular intersection 131 Regular morphism 86, 134 Regular section 76 Regular sequence 71 Regular sheaf 107 Relative dimension 89 Represented by a complex 119 Residual scheme 93 Resolution 76, 100, 113, 126 Restricted morphism 157 Riemann-Roch, Adams and Grothendieck 63, 142, 146, 149 Riemann-Roch character 28 Riemann-Roch element 45 Riemann-Roch functor 28 Riemann-Roch for imbeddings 32 Riemann-Roch multiplier 28 Riemann-Roch for projections 33

M

S

Meet regularly 80, 128, 131 Micali's theorem 73 Multiplier 24, 28, 157, 171

Self intersection 155, 171 Singular Riemann-Roch 188 Smooth 81

203

INDEX

Special 2-ring 6 Splitting principle 13 Splitting property 4, 49, 118 Staircase decomposition 87 Star multiplication 177 S upport

119

Symmetric functions 4 Symmetric powers 7, 117

Total Chern class

13

Total complex 140

U Universal exact sequence 67 Universal hyperplane sheaf 67, 68 Universal polynomials 5

Upper K 164 Upper and lower filtration

T Tangent bundle 144 Tangent element 144 Tautological exact sequence 67 Todd class 20, 188 Todd homomorphism 19, 20, 24 Top Chern class 14 Top graded degree 147, 181 Topological filtration 120, 122, 125, 179, 181

V

Vector bundle 68 Verdier Riemann-Roch 189 Virtual tangent bundle 144

Z Zero scheme 76, 128 Zero section 68

178

Grundlehren der mathematischen Wissenschaften Continued from page ii

235. 236

Dynkin/Yushkevich. Markov Control Processes and Their Applications Grauert/Remmert: Theory of Stein Spaces

237. 238. 239.

Kothe. Topological Vector-Spaces II Graham/McGehee Essays in Commutative Harmonic Analysis Elliott. Probabilistic Number Theory 1

240 241. 242. 243 244 245

Elliott: Probabilistic Number Theory II Rudin: Function Theory in the Unit Ball of C" Huppert/Blackbum. Finite Groups I Huppert/Blackbum. Finite Groups 11 Kubert/Lang Modular Units Cornfeld/Fomin/Sinai Ergodic Theory

246. 247

Naimark/Stem. Theory of Group Representations Suzuki: Group Theory I

249.

Chung Lectures from Markov Processes to Brownian Motion

250 252 253

Arnold Geometrical Methods in the Theory of Ordinary Differential Equations Chow/Hale: Methods of Bifurcation Theory Aubin: Nonlinear Analysis on Manifolds, Monge-Ampere Equations Dwork: Lectures on p-adic Differential Equations

254 255

Freitag: Siegelsche Modulfunktionen Lang: Complex Multiplication

256 258

Hormander The Analysis of Linear Partial Differential Operators 1 Hormander The Analysis of Linear Partial Differential Operators 11 Smoller Shock Waves and Reaction-Diffusion Equations

259

Duren Univalent Functions

260 261.

Freidlin/Wentzell: Random Perturbations of Dynamical Systems Remmert/Bosch/Giintzer: Non Archimedian Analysis-A Systematic Approach to Rigid Analytic Geometry Doob: Classical Potential Theory & Its Probabilistic Counterpart Krasnosel'skii/Zabreiko. Geometrical Methods of Nonlinear Analysis Aubin/Cellina Differential Inclusions Grauert/Remmert Coherent Analytic Sheaves de Rham. Differentiable Manifolds Arbarello/Comalba/Griffiths/Hams. Geometry of Algebraic Curves, Vol 1 Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol II Schapira. Microdifferential Systems in the Complex Domain Scharlau Quadratic and Hermitian Forms Ellis. Entropy, Large Deviations, and Statistical Mechanics Elliott: Arithmetic Functions and Integer Products Nikolskij: Treatise on Shift Operators Hormander: The Analysis of Linear Partial Differential Operators III Hormander: The Analysis of Lineal Partial Differential Operators IV Liggett: Interacting Particle Systems Fulton/Lang Riemann-Roch Algebra Barr/Wells: Toposes, Triples, and Theories Bishop/Budges: Constructive Analysis

251

257.

262 263 264 265. 266 267. 268 269 270 271. 272

273. 274. 275 276 277 278 279