Jantzen - Representations of Algebraic Groups

Representations of Algebraic Groups

This is volume 131 in PURE AND APPLIED MATHEMATICS H. Bass, A. Borel, J. Moser, and S.-T. Yau, editors Paul A. Smith and Samuel Eilenberg, founding editorsA list of titles in this series appears at the end of this volume.

Representations of Algebraic GroupsJens Carsten JantzenMathematisches Seminar der Universitat Hamburg Federal Republic of Germany

ACADEMIC PRESS, INC.Harcourt Brace Jooanovich, Publishers

Boston Orlando San Diego New York Austin London Sydney Tokyo Toronto

Copyright 01987 by Academic Press, Inc. All rights reserved. N o part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc.Orlando, Florida 32887United Kingdom Edition published b y ACADEMIC PRESS INC. (LONDON) LTD. 24-28 Oval Road, London N W l 7DX

Library of Congress Cataloging-in-Publication Data Jantzen, Jens Carsten. Representations of algebraic groups.

(Pure and applied mathematics; v. 131) Bibliography: p. Includes index. 1. Representations of groups. 2. Linear algebraic groups. I. Title. 11. Series: Pure and applied mathematics (Academic Press); 131. QA3.P8 510 s C512.21 86-26619 LQA1711 ISBN 0-12-380245-8 (alk. paper)

87 88 89 90 9 8 7 6 5 4 3 2 1 Printed in the United States of America

ContentsIntroduction Part I: General Theory 1: Schemes 2: Group Schemes and Representations 3: Induction and Injective Modules 4: Cohomology 5 : Quotients and Associated Sheaves 6: Factor Groups 7: Algebras of Distributions 8: Representations of Finite Algebraic Groups 9: Representations of Frobenius Kernels 10: Reduction mod p Part 11: Representationsof Reductive Groups 1: Reductive Groups 2: Simple G-Modules 3: Irreducible Representations of the Frobenius Kernels 4: Kempfs Vanishing Theorem 5 : The Borel-Bott-Weil Theorem and Weyls Character Formula 6: The Linkage Principle 7: The Translation Functors 8: Filtrations of Weyl Modules 9: Representations of G, T and G, B 10: Geometric Reductivity and Other Applications of the Steinberg Modules 11: Injective G,-Modulesvii1 3 21 43 55 73 95 109 129 145 161 171 173 197 213 227 243 259 28 1 297 319 337 351V

vi

Contents

12: Cohomology of the Frobenius Ke'rnels 13: Schubert Schemes 14: Line Bundles on Schubert Schemes References Index

369 38 1 395 417 435

Introduction

I This book is meant to give its reader an introduction to the representation theory of such groups as the general linear groups GL,(k), the special linear groups SL,(k), the special orthogonal groups SO,(k), and the symplectic groups Sp,,(k) over an algebraically closed field k. These groups are algebraic groups, and we shall look only at representations G + G L ( V ) that are homomorphisms of algebraic groups. So any G-module (vector space with a representation of G) will be a space over the same ground field k. Many different techniques have been introduced into the theory, especially during the last fifteen years. Therefore, it is necessary (in my opinion) to start with a general introduction to the representation theory of algebraic group schemes. This is the aim of Part I of this book, whereas Part I1 then deals with the representations of reductive groups.I1 The book begins with an introduction to schemes (Chapter 11) and to 2. (affine) group schemes and their representations (Chapter 1 ) We adopt the functorial point of view for schemes. For example, the group scheme SL, over Z is the functor mapping each commutative ring A to the group SL,(A). Almost everything about these matters can also be found in the first two chapters of [DG]. I have tried to enable the reader to understand the basic definitions and constructions independently of [DG]. However, I refer to

vii

viii

Introduction

[DG] for some results that I feel the reader might be inclined to accept without going through the proof. Let me add that the reader (Df part I) is supposed to have a reasonably good knowledge of varieties and algebraic groups. For example, he should know [Bo] up to Chapter 111, or the first seventeen chapters of [HuZ], or the first six ones of [Sp2]. (There are additional prerequisites for Part I1 mentioned below.) In Chapter 13, induction functors are defined in the context of group schemes, their elementary properties are proved, and they are used in order to construct injective modules and injective resolutions. These in turn are applied in Chapter 14 to the construction of derived functors, especially to that of the Hochschild cohomology groups and of the derived functors of induction. In contrast to the situation for finite groups, the induction from a subgroup scheme H to the whole group scheme G is not exact, only left exact. The values of the derived functors of induction can also be interpreted (and are so in Chapter 15) as cohomology groups of certain associated bundles on the quotient G/H (at least for algebraic schemes over a field). Before doing that, we have to understand the construction of the quotient G/H. The situation gets simpler and has some additional features if His normal in G. This is discussed in Chapter 16. One can associate to any group scheme G an (associative) algebra Dist(G) of distributions on G (called the hyperalgebra of G by some authors). When working over a field of characteristic 0, it is just the universal enveloping algebra of the Lie algebra Lie(G) of G. In general, it reflects the properties of G much better than Lie(G) does. This is described in Chapter 17. A group scheme G (say over a field) is called finite if the algebra of regular functions on G is finite dimensional. For such G the representation theory is equivalent to that of a certain finite dimensional algebra and has additional features (Chapter 18). For us, the most important cases of finite group schemes arise as Frobenius kernels (Chapter 19) of algebraic groups over an algebraically closed field k of characteristic p # 0. For example, for G = GL,(k) the map F : G -,G sending any matrix (a,) to (a;) is a Frobenius endomorphism. The kernel of F' (in the sense of group schemes) is the rth Frobenius kernel G, of G. The representation theory of G , (for any G) is equivalent to that of Lie(G) regarded as a p-Lie algebra. In order to apply our rather extensive knowledge of the representation theory of groups like SL,(C) to that of SL,(k), where k is a field of prime characteristic, one uses the group scheme SL, over Z. One chooses SL,stable lattices in SL,(C)-modules and tensors with k in order to get SL,(k)-modules. Some general properties of this procedure are proved in Chapter 110.

Introduction

ix

From Part I, the contents of Chapters 1 (until 1.6), 2, 3, 4 (until 4.18), 5 (mainly 5.8-5.13), and 6 (until 6.9) are fundamental for everything to follow. The other sections are used less often. In Part 11, the reader is supposed to know the structure theory of reductive algebraic groups (over an algebraically closed field) as to be found in [Bo], [ H u ~ ] ,[Sp2]. The reader is invited (in Chapter 111) to believe that there is for all possible root data a (unique) group scheme over Z that yields for each field k (by extension of the base ring) a split reductive algebraic group defined over k having the prescribed root data. Furthermore, he has to accept that all standard constructions (like root subgroups, parabolic subgroups, etc.) can be carried out over Z. (The sceptical reader should turn to [SGA 31 for proofs.) I have included a proof (following Takeuchi) of the uniqueness of an algebraic group with a given root datum (over an algebraically closed field) that does not use case-by-case considerations.111 Let me describe a selection of the contents of the remaining chapters in more detail. Assume from now on (in this introduction) that k is an algebraically closed field and that G is a (connected) reductive algebraic group over k with a Bore1 subgroup B c G and a maximal torus T c B. Let X ( T )be the group of characters of T. In case char(k) = 0 the representation theory of G is well understood. Each G-module is semi-simple. The simple G-modules are classified (as in the case of compact Lie groups or of complex semi-simple Lie algebras) by their highest weights. Furthermore, one has a character formula for these simple modules. In fact, Weyls formula for the compact groups holds when interpreted in the right way. (For us, the character of a finite dimensional G-module will always be the family of the dimensions of its weight spaces with respect to T. As the semi-simple elements in G are dense in G and as each semi-simple element is conjugate to one in T, the character determines the trace of any g E G on the Gmodule.) The situation in prime characteristic is much worse. Except for the case of a torus, there are non-semi-simple G-modules. Except for a few low rank cases, we do not know a character formula for the simple modules, and Weyls formula will certainly not carry over. Only one property survives: The simple modules are still classified by their highest weights, and the possible highest weights are the dominant weights in X ( T ) . (The notion of dominant depends on the choice of an ordering of X ( T ) .We shall always work with an ordering for which the weights of T in Lie(B) are negative.) This classification is due to Chevalley, cf. [SC]. Let L(A) denote the simple module with the highest weight A.

X

Introduction

The difference of the situations in zero and in prime characteristic can be illustrated already in the case G = SL,(k). Let H ( n ) be the nth symmetric power of the natural representation of G on k 2 . If char&) = 0, then H(n) = L(n) for all n E N. (For SL, we can identify X ( T ) N Z in such a way that the dominant weights correspond to N.) char(k) = p # 0, obviously not all H ( n ) If can be simple: For all r, n E N, > 0 the map f H f p r maps H ( n ) to a proper submodule of H(p'n), so H(p'n) is not simple. It is not too difficult to show for any n that H ( n ) contains L(n) as its unique simple submodule, and that H ( n ) = L(n) if and only if n = up' - 1 for some a, r E N with 0 < a I p . So for all other n the module H ( n ) is not semi-simple. For arbitrary G one gets L(A) as the unique simple submodule of an in, duced module Ho(L): One extends 1 E X ( T ) to a one dimensional representation of B such that the unipotent radical of B acts trivially. Then Ho(A) is the G-module induced by this B-module. It is nonzero if and only if A is dominant. (In the case G = SL,(k) the H o ( L ) are just the H ( n ) from above.) This is the main content of Chapter 112. The case G = SL,(k) with char&) = p # 0 can serve to illustrate other general results also. For any vector space I/ over k let be the vector space that is equal to I/ as an additive group and where any a E k acts as ap-' does on I/. Then the map f H f p r is linear when regarded as a map H(n)(')--t H ( p r n ) , hence a homomorphism of G-modules. It is not difficult to show: If n= aipi with 0 I ai < p for all i, thenf, 0 f i 0 . . 0, H $ f is an isomorphism

XI=,

nr=,fpi

H(a,) 0 H(a1)(') . . . 0 H(a,)(') 2 L(n). 0This result was generalized in [Steinberg 21 to all G: A suitable p-adic expansion of the highest weight 1 leads to a decomposition of L(L) into a This tensor tensor product of the form L(1,) 0 L(A1)(')0 . . . 8 L(Ar)(r). product theorem reduces the problem of calculating the characters of all simple G-modules to a finite problem (for each G). Steinberg's proof relied on a theorem from [Curtis 13 on the representations of Lie(G).In the special case of G = SL,(k), it can be proved in a quite elementary way. It says then: Each L(n) with n < p remains simple for Lie(G), and each simple module of Lie(G) regarded as a p-Lie algebra is isomorphic to exactly one L(n)with n < p . More generally, each L(n) with n < p' is simple for the rth Frobenius kernel of SL,(k), and we get thus each simple module for this infinitesimal group scheme. This result again has an extension to all G and then leads to a rather simple proof of Steinberg's tensor product theorem, discovered by Cline, Parshall, and Scott. (All this is done in Chapter 113.)

Introduction

xi

The choice of the notation H o ( i ) for the induced module has been influenced by the fact that Ho(%)is the zeroth cohomology group of a line bundle on G/B associated to %.Let Hi(%)denote the ithcohomology group (for not any 1 E X(T), only for dominant ones). We could have constructed Hi(%) also by applying the i derived functor of induction from B to G to the onedimensional B-module defined by A. Another result from characteristic zero that does not carry over to prime characteristic is the Borel-Bott-Weil theorem. It describes explicitly all H i ( p ) with i E N and p E X ( T ) :For each p there is at most one i with H i ( p ) # 0, and this H i ( p )can then be identified with a specific L ( i ) .We know already that we cannot expect the simplicity of the H i ( p )in prime characteristic. But, even worse, there can be more than one i for a given p with H i @ ) # 0, and the character of H i ( p ) will depend on the field. (This was first discovered by Mumford.) It is crucial for the representation theory that one special case of the Borel-Bott-Weil theorem holds for any k: If % is dominant, then Hi(%)0 for all i > 0. This is Kempfs vanishing = theorem from [Kempf 11. The proof here in Chapter 114 is due to Haboush and Andersen (independently). In Chapter 115, we give Demazures proof of the Borel-Bott-Weil theorem in case char&) = 0. Furthermore, we prove (following Donkin) that Weyls character formula yields the alternating sum (over i) of the characters of all Hi(%). Assume from now on char(k) = p # 0. Kempfs vanishing theorem implies with A dominant by starting with that one can construct for any k the Ho(%) the similar object over C,taking a suitable lattice stable under a Z-form of G, and then tensoring with k. To construct representations in this way has the advantage that one can carry out specific computations more easily. Several examples computed especially by Braden then led Verma in the late sixties to several conjectures (cf. [Verma]) that had a great influence on the further development of the theory. One conjecture is the linkage principle (Chapter 116): If L ( p ) is a composition factor of Ho(A) (or, more generally, if L ( p ) and L(A) are both composition factors of a given indecomposable Gmodule), then p E W,.%. Here W, is the group generated by the Weyl group W and by all translations by pa with a a root. The dot is to indicate a shift in the operation by p, the half sum of the positive roots (i.e., w.A = w(A + p ) - p). For large p this principle was proved in [Humphreys 13. The result was then extended by several people to almost all cases, but a general proof appeared only in 1980 (in [Andersen 41). It relies on an analysis of the failure of Demazures proof (of the Borel-Bott-Weil theorem) in prime characteristic. Another conjecture of Verma was a special case of the translation , principle (Chapter 117): If two dominant weights i p belong to the same

xii

Introduction

"facet" with respect to the affine reflection group W,, then the multiplicity of any L(w.l) with w E W, as a composition factor of Ho(A) should be equal to that of L ( w . p ) in H o ( p ) . This was proved (modulo the linkage principle) in [Jantzen 21. The approach to the Ho(A) via representations over Z also has the advantage that it allows the construction of a certain filtration (Chapter 118)of Ho(A).One can compute the sum of the characters of the terms in the filtration ([Jantzen 31 for large p , [Andersen 121 in general) and use this information to get information about composition factors. For example, for G = SL,(k) or for G of type G2 this leads to a computation of the characters of all simple modules. are If I and 1+ pv for some v E X(T) weights that are "small" with respect to p 2 and are "sufficiently dominant" (confer 11.9.14-9.15 for a more precise condition), then one gets the composition factors of H o ( I + pv) from those of Ho(l)by adding p v to the highest weights. This was proved first in [Jantzen 4) using involved computations. Later on it was realized that it follows rather easily if one develops the representation theory of the group scheme G,T. For 1 as above experimental evidence (cf. [Humphreys lo]) indicated that the H'(w.1) with w E W satisfy a weak version of the Borel-Bott-Weil theorem ( W ( w . 2 ) # 0 for at most one i). This was then proved in [Cline, Parshall, and Scott lo] using the representation theory of the group scheme G,B. All this is described in Chapter 119. Let us assume that G is semi-simple and simply connected. There is for each r 2 1 a unique simple G-module that is simple and injective for G,. It is called the rth Steinberg module and was first discovered by Steinberg within the representation theory of finite Chevalley groups. We do not look at its great importance there but discuss some applications to the representation theory of G (Chapter 1110). It plays a crucial role in Haboush's proof that G is geometrically reductive. One may wonder whether any injective G,-module can be extended to a G-module. For large p this was proved by Ballard. We discuss this (with some applications to the representation theory of G) in Chapter 1111. One can write down the character of a simple G-module L(A) if one knows all extension groups Ext",(L(A), Ho(p)),cf. 11.6.21. Unfortunately, rather little is known about these groups. There has been a considerable amount of work (especially by Cline, Parshall, and Scott) to understand better especially Hochschild cohomology groups H"(G, M ) 1 Ext",k, M ) . : One has H " ( G , M )N l&H"(G,,M) if dimM c 00, so one may hope to get information on G-cohomology from information on G,-cohomology. Here the most remarkable theorem known is due to Friedlander and

Introduction

xiii

Parshall: For large p the cohomology ring H(G,, k ) is isomorphic to the ring of regular functions on the nilpotent cone in Lie(G). This result can be found in Chapter 1112. The orbits of B on G / B are isomorphic to affine spaces. They are called Bruhat cells, their closures are called Schubert varieties. For example, G / B itself is a Schubert variety. One can extend Kempfs vanishing theorem to any Schubert variety Y c G/B:If one restricts to Y the line bundle on G / B corresponding to a dominant weight A, then all higher cohomology groups vanish. As an application one can prove the normality of Y and a character formula for the space of global sections. These results have been proved only recently (1984) using old ideas of Kempf and Demazure by Mehta, Ramanathan, Seshadri, Ramanan, and Andersen. One can find this in Chapter 1114, whereas I1 13 provides the necessary background on the Schubert varieties. The last seven chapters of Part I1 can be divided into three groups (118, 119-12, 1113-14), which are independent of each other. Also, the logical interdependence of chapters 1110-12 is rather weak.

IV Suppose that Fq is a finite field and that k is an algebraically closed extension of Fq.The representation theory of groups like GL,(k) or Sp,,(k) has always been developed in close interaction with that of groups like GL,(Fq) or Sp,,(F,). It would therefore have been desirable to have a third part of the book dealing with representations of finite Chevalley groups (say over fields of the same characteristic as that over which they are defined). In fact, I promised to write such a part in a preliminary foreword to a preprint version of Part I. However, I hope to be forgiven if I break this promise, as otherwise the book would have grown to an unreasonable size. Furthermore, I feel that people most interested in these finite groups would prefer another book where they would not have to devour at first all of Parts I and 11.V In the summer of 1984, I gave a series of lectures on some topics discussed in this book at the East China Normal University in Shanghai. I had been asked in advance to provide the audience with some notes. When doing so I was still undecided about the precise contents of my lectures. I therefore included more material than I could possibly cover in my lectures. This book has grown out of those notes. I should like to use this opportunity to thank the mathematicians I met in Shanghai, especially Professor Cao Xihua, for their hospitality during my stay and for the patience with which they listened to my lectures. Thanks are also due to Henning Haahr Andersen, Burkhard Haastert, and Jim Humphreys for useful comments on my manuscript and for providing lists of misprints.

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Part 1

General Theory

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1

Schemes

It is the purpose of this first chapter to give the necessary introduction to schemes following the functorial approach of [DG]. This approach appears to be most suitable when dealing with group schemes later on. After trying to motivate the definitions in 1.1, we discuss affine schemes in 1.2-1.6. What is done there is fundamental for the understanding of everything to follow. As far as arbitrary schemes are concerned, it is most of the time enough to know that they are functors with some properties so that all affine schemes are functors and so that over an algebraically closed field any variety gives rise to a scheme in a canonical way. Sometimes, e.g., when dealing with quotients, it is useful to know more. So we give the appropriate definitions in 1.7-1.9 and mention the comparison with other approaches to schemes and with varieties in 1.11. The elementary discussion of a base change in 1.10 is again necessary for many parts later on. There is also a discussion of closed subfunctors and of closures (1.12-1.14). Finally, we describe the functor of morphisms between two functors (1.15)and prove some of its properties. Again, these results are used only in few places. A ring or an associative algebra will always be assumed to have a 1, and homomorphisms are assumed to respect this 1. Let k be a fixed commutative ring. Notations of linear algebra (like Hom, 0 )without special reference to a ground ring always refer to structures as k-modules. A k-algebra is always

3

4


assumed to be commutative and associative. (For noncommutative algebras we shall use the terminology: algebras over k.)1.1 Before giving the definitions, I want to point out how functors arise naturally in algebraic geometry. Assume for the moment that k is an algebraically closed field. Consider a Zariski closed subset X of some k" and denote by I the ideal of all polynomials f E k[T,, T,, . . . , T,,] with f(X) 0. Instead of looking = at the zeroes of I only over k, we can look also at the zeroes over any k-algebra A, i.e., at %(A) = {x E A" ( f ( x ) = 0 for all f E I } . The map A H % ( A ) from {k-algebras} to {sets} is a functor: Any homomorphism cp: A -+ A' of k-algebras induces a map cp": A" -+ (A')", (a,, a , , . . * a,) H (cp(a,), cp(a,), ' . * with f(cp"X)) = cp(f(x)) for all x E A" and f E k[T,,. . . , T,,]. Therefore cp" maps %(A) to %(A'). Denote its restriction by %(cp): % ( A ) -+ %(A'). For another homomorphism cp': A' -+ A" of k-algebras, one has obviously %(cp')%(cp) = %(cp' o cp), proving that % is indeed a functor. A regular map from X to a Zariski closed subset Y of some k" is given by rn polynomials f 1 , f 2 , . . .,f E k[T,, T,, . . ., T,] as f : X , Y, x H ( f , ( x ) , f 2 ( x ) ,. . .,fm(x)). The fi define for each k-algebra A a map f(A): + A", A" x H ( f , ( x ) , . .. , f m ( x ) ) .The comorphism f ( k ) * :k [ T , , . .., T,] -+ k[Tl,. . . ,T,] maps the ideal defining Y into the ideal I defining X . This implies that any f(A) maps %(A) into Y(A). The family of all f(A) defines a morphism f : % - + % of functors, i.e., a natural transformation. The more general discussion in 1.3 (cf. 1.3(2)) shows that the map f H f is bijective (from {regular maps X --+ Y } to {natural transformations % -+ Y}). Taking this for granted, we have embedded the category of all affine algebraic varieties over k into the category of all functors from {k-algebras}to {sets}as a full subcategory. This embedding can be extended to the category of all algebraic varieties, see 1.11. One advantage of working with functors instead of varieties (i.e., of working with % instead of X ) will be that it gives a natural way to work with "varieties" over other fields and over rings. Furthermore, we get new objects over k (algebraically closed) in a natural way. Instead of working with I , we might have taken any ideal I' c k[T,, . . .,T,,] defining X , i.e., with X = {x E k" I f ( x )= O for all f~ 1') or, equivalently by Hilbert's Nullstellensatz, with f i = I. Replacing I by I' in the definition of X, get a functor, say X, we with %'(A) = %(A) for each field extension A 2 k (or even each integral domain), but with %'(A) # %(A) for some A if I # 1'. Such functors arise in a7

9

+

Schemes

5

natural way even when we deal with varieties, and they play an important role in representation theory. Before giving the proper definitions, let us describe the functor % without using the embedding of X into k. For each k-algebra A , we have a bijection ... Homk-,,,(k[Tl, T,,. . . ,T,], A ) --f A, sending any a to (a(T1),a(T2), ,a(T,)). The inverse image of %(A) consists of those a with 0 = f(a(T1),. . .,a(T,,)) = a ( f ) for all f E I, hence can be identified with Horn,.,,,(k[ Tl, T,, . . . , T,,]/Z, A ) . As k[Tl, T,, . . .,T,,]/Z is the algebra k[X] of regular functions on X, we have thus a bijection .%-(A) N Homk.,,,(k[X], A ) . If cp: A + A is a homomorphism of k-algebras, then %(q) corresponds to the map Homk~,l,(k[X],A ) + Hom,,,,(k[X], A ) with u H cp a. A morphism f: X + Y is given by its cok morphism f*: [ Y ] + k[X]. Then the morphism p: 3+ g is given by / ( A ) : Hom,_,,,(k[X], A ) Hom,_,,,(k[ Y ] ,A ) , ct H ct f* for any k-algebra A .0--f

0

1.2 (&-functors)Let us assume k to be arbitrary again. In the definitions to

follow, we shall be rather careless about the foundations of mathematics. Instead of working with all k-algebras, we should (as in [DG]) take only those in some universe. We leave the appropriate modifications to the interested reader. A k-functor is a functor from the category of k-algebras to the category of sets. Let X be a k-functor. A subfunctor of X is a k-functor Y with Y ( A )c X(A) (~ and Y(cp) = X ( V ) / ~ for )all k-algebras A, A and all cp E Homk-,&i, A). Obviously, a map Y that associates to each k-algebra A a subset Y ( A )c X(A) is a subfunctor if and only if X(cp)Y(A)c Y(A) for each homomorphism cp: A + A of k-algebras. For any family ( Q oof subfunctors of X, we define their intersection r 5 through (flier) ( A )= y q ( A )for each k-algebra A. This is again a subfunctor. The obvious definition of a union is not the useful one, so we Y;:. shall not denote it by UiEl For any two k-functors X,X, we denote by Mor(X,X) the set of all morphisms (i.e., natural transformations) from X to X . For any f E Mor(X,X) and any subfunctor Y of X, we define the inverse image f - ( Y ) of Y under f through f - ( Y ) ( A ) = f ( A ) - ( Y ( A ) )for each kalgebra A . Clearly f - ( Y ) is a subfunctor of X. (The obvious definition of an image of a subfunctor is not the useful one.) Obviously, f- commutes with intersections. For two k-functors X,, X,, the direct product XI x X, is defined through (X, x X,)(A) = X,(A) x X,(A) for all A. The projections pi:X, x X, 4 Xi

niEr

niEI

6


are morphisms and ( X , x X,,p,,p,) has the usual universal property of a direct product. For three k-functors X , , X,, S and two morphisms f,:X , + S, f, : X 2 + S , the jibre product X , x s X , (relative to f,, f2) is defined through

The projections from X , x s X , to X , and X , are morphisms, and X , x s X,, together with these projections, has the usual universal property of a fibre product. Of course, we may also regard X , x s X 2 as the inverse image of the diagonal subfunctor D, c S x S (with Ds(A) = {(s,s) I s E S ( A ) }for all A ) under the (obvious) morphism ( f 1 , f 2 ) : X , X , -+ S x S. (On the other hand, x inverse images and intersections can also be regarded as special cases of fibre products.)1.3 (Affine Schemes) For any n E N,the functor A" with A"(A) = A" for all A and A"(q)= q " : ( a , , . . .,an)H ( q ( a l ).,. . ,q(a,)) for all cp: A + A' is called the afJine n-space over k. (We also sometimes use the notation A; when it may be doubtful which k we consider.) Note that A' is the functor with Ao(A) = (0) for all A . Hence there is for each k-functor X exactly one morphism from X to Ao (i-e., Ao is a final object in the category of k-functors), and we can regard any direct product as a fibre product over A'. For any k-algebra R, we can define a k-functor Sp,R through (Sp,R)(A) = HOmk-alg(R, ) for all A and (Sp,R)(q): HOmk.alg(R, ) + HOmk-alg(R, A A A'), a H q 0 a for all homomorphisms q : A + A'. We call Sp,R the spectrum of R. Any k-functor isomorphic to some Sp,R is called an afJine scheme over k. (Note that the Sp,R generalize the functors 9" considered in 1.1.) For example, the affine n-space A" is isomorphic to Sp,k[T,, . . . , T,] (and will usually be identified with it), where k[T,, ..., T,] is the polynomial ring over k in n variables T,, ...,T,. We can recover R from Sp,R. This follows from:

f H f(R)(id,) is a bijection

Yoneda's Lemma: For any k-algebra R and any k-functor X , the map

Indeed, take any k-algebra A and any a E Homk-alg(R, ) = (Sp,R)(A). As f is A a natural transformation, we have X ( a ) 0 f(R) = f(A)0 (Sp,R)(a). Let us ab-

Schemes

7= a, we

breviate xs = f(R)(id,). As (SpkR)(a)(idR)= a 0 id,(1)

get

f(A)(a) = X ( a ) ( x s ) .

This shows that f is uniquely determined by x f and indicates how to construct an inverse map. For each x E X ( R ) and any k-algebra A , let f,(A): ( S p k R ) + X ( A ) be the map Z H X(r)(x). Then one may check that f , ~ Mor(SpkR,X ) and that x H f, is inverse to f H x f . An immediate consequence of Yonedas lemma is

(2)

MOr(SpkR,SpkR) 1 HOmk-a,g(R, ) R

for any k-algebras R , R . We denote this bijection by f ~ f and call * f * the comorphism corresponding to f. As Homk-,,,(k[T1], R ) r R under a H a( T,) we get especially

(3)

Mor(SpkR,A ) 3 R .

For any k-functor X , we denote M o r ( X , A ) by k [ X ] . This set has a natural structure as a k-algebra and (3) is an isomorphism k [ S p k R ] 7 R of k-algebras. (For f i , f 2 E k [ X ] , define f, f2 through (f, f , ) ( A ) ( x ) = f , ( A ) ( x ) f , ( A ) ( x ) for all A and all x E X ( A ) . Similarly, f1f2 and af, for a E k are defined.) We shall usually write f ( x ) = f ( A ) ( x ) for x E X ( A ) and f E k [ X ] . Note that for X = SpkR and f E R N k [ X ] we have f ( x ) = x ( f ) for X E (SPkR)(A)= HOmk-alg(R, ) . A The universal property of the tensor product implies immediately that a direct product X, x X , of affine schemes over k is again an affine scheme over k with k [ X , x X , ] N k [ X , ] 0 k [ X , ] . More generally, a fibre product X , x s X , with X , , X , , S affine schemes is an affine scheme with

+

+

+

1.4 (Closed Subfunctors of AffineSchemes) Let X be an affine scheme over k.

For any subset I c k [ X ] , we define a subfunctor V ( I )of X through(1)

V ( I ) ( A )= {x E X ( A )I f ( x ) = 0N

for all f

E

I}

{ a E Homk-,,,(k[X],A)Ia(I) =

o}

for all A . (One can check immediately that this is indeed a subfunctor, i.e., that X ( ( p ) V ( I ) ( Ac V ( I ) ( A for any homomorphism q : A + A.) ) ) Of course, V(1)depends only on the ideal generated by I in k [ X ] . We claim: (2) The map I H V ( I )from {ideals in k [ X ] } to {subfunctors of X } is injective.

8


More precisely, we claim for two ideals I , I of k [ X ] : I c I 0 V ( I ) 3 V(Z). (3) Of course, the direction +- is trivial. O n the other hand, consider the canonical map a : k [ X ] + k [ X ] / I , which we regard as an element of X ( k [ X ] / I ) .As a(Z) = 0, it belongs to V(Z)(k[X]/Z). V(Z) c V ( I ) , then If a E V ( Z ) ( k [ X ] / I and a ( I ) = 0, hence I c I. ) We call a subfunctor Y of X closed if it is of the form Y = V ( I )for some ideal I c k [ X ] . Obviously, any closed subfunctor is again an affine scheme over k as

(4) V ( I ) = SPk(kCXI/I). For any family ( I j ) j E of ideals in k [ X ] , one checks easily

Thus the intersection of closed subfunctors is closed again. For each subfunctor Y of X , there is a smallest closed subfunctor F of X with Y ( A ) c P ( A ) for all A . (Take the intersection of all closed subfunctors with the last property.) This subfunctor 7 is called the closure of Y. We really do not have to assume here that Y is a subfunctor: Any map Y will do that associates to each A a subset Y ( A ) c X ( A ) .We can, for example, fix an A and consider a subset M c X ( A ) . Then the closure M of M is the smallest closed subfunctor of X with M c M ( A ) . Let I , , I , be ideals in k [ X ] . Because of (3), the closure of the subfunctor A H V ( I , ) ( A )u V ( I , ) ( A ) equal to V ( I , n I,). If A is an integral domain, then is one checks easily that V ( I , ) ( A )u V(Z,)(A)= V(I1 n I,)(A). For arbitrary A, this equality can be false. Still, we dejne the union as V ( I , )u V ( I , ) = VUl n 121. Let f:X --t X be a morphism of affine schemes over k. One easily checks for any ideal I of k [ X ] that

(6)

f-V(I) = V(k[X]f*(I)).

Thus the inverse image of a closed subfunctor is again a closed subfunctor. For ) any ideal I c k [ X ] , the closure of the subfunctor A H f ( A ) ( V ( I ) ( A )is V ( ( f * ) - I ) . This functor is also denoted as f(V(Z)), but we do not want to define f (V(I)) here. For two affine schemes X , , X , over k and ideals I , c k [ X , ] , I , c k [ X , ] , one checks easily(7)V(I1) x V(I2) N V(I1 0 k [ X , l

+k[Xil@

Schemes

9- S, ,

If S is another affine scheme and if morphisms X, one gets

X,

--+

S are fixed, then

(8)

v(zl) x S v ( z 2 )

2 V(zl 1

@k[S]k[x21

+ k[xll @k[S]I 2 ) *

(Use, e.g., that V ( I , ) xs V(Z2)= p i V(Il) n p i V(I,) together with (5), (6), where pi:XI xs X, - Xi for i = 1,2 are the canonical projections.) ,

1.5 (Open Subfunctors of Affine Schemes) Let X be an affine scheme over k. A subfunctor Y of X is called open if there is a subset I c k[X] with Y = D ( I ) where we set for all k-algebras A :(1)

D ( I ) ( A )= {x E X(A) I

ier

1Af(x)= A )

= { a E HOmk-,,,(k[X],A)I A a ( l ) = A }

Note that (1) defines for each ideal I a subfunctor: For each cp E Homk.alg(A, A ) and each x E D(I)(A), one has Af(X(cp)x) = Acp(f(x)) = A q ( x , , , Af(x)) = Acp(A) = A. Obviously:

xfEr

ErBr

(2) If A is a j e l d , then D ( I ) ( A )= U f B r ( x X(A) (f(x) # 0). EOf course, the right hand side in (2) would be the obvious choice for something open. But it does not define a subfunctor, as homomorphisms between kalgebras are not injective in general. Therefore, we have to take (1) as the appropriate generalization to rings. For I of the form I = {f} for some f E k[X], one writes X, = D ( f ) = D({f}) and gets(3)

Xf(A) = { .

Homk-alg(k[X1, A ) 1 @ ( f ) E A x 1,

hence

(4)

xf

Spk(k[XIf)

where k[X], = k[X][f-] is the localization of k[X] at f. So the open subfunctors of the form X, are again affine schemes. For arbitrary I, however, D ( I ) may be no longer an affine scheme. Obviously, D ( I ) depends only on the ideal of k[X] generated by I . As any proper ideal in any ring is contained in a maximal ideal, we have for any AD ( I ) ( A )= { a E Homk-,,,(k[X], A ) I a ( ] ) $ m

for any M E Max(A)}

= { a E Homk-,,,(k[X], A ) a, E D ( I ) ( A / ~ ) any for

I

rn E Max(A)}

where Max(A) is the set of all maximal ideals of A and a, is the composed map k[X] 4 A % A/rn. This shows that D ( I ) is uniquely determined by its

10


values over fields and especially that D ( I ) = D ( J I ) for any ideal I c k [ X ] . Denote for each prime ideal P c k [ X ] the quotient field of k [ X ] / P by QPand the canonical homomorphism k [ X ] + k [ X ] / P + Q Pby a p . Thenup 4 D ( I ) ( Q p )0 a p ( I )= 0 0 P3

I.

As &is the intersection of all prime ideals P ideals I , I of k [ X ](5)

I> I

of k [ X ] , we get for any two

D ( I ) c D(I) o 8 c f i+ {open

Thus I H D ( I ) is a bijection (ideals I of k[X] with I = J I } subfunctors of X } . For two ideals I , I in k [ X ] , one checks easily(6)D ( I ) n D(I) = D(I n 1 ) = D ( I 1 )

.

and gets especially for any f,f E k [ X ](7)

X, n X,. = X,,,.

For any ideal I in k [ X ] one has (8) If A is a field, then X ( A ) is the disjoint union of D ( I ) ( A )and V ( I ) ( A ) , For arbitrary A , the union may be smaller. Also, the next statement may be false for arbitrary A : Consider a family ( I j ) j e J of ideals in k [ X ] . Then obviously (9) If A is a field, then(10)

ujEJI j ) ( A )= D ( c j E JIj)(A). D(f-D(I) = D(k[X]f*(I))

For any morphism f:X + X of affine schemes over k, one has for any ideal I c k [ X ] , as one may check easily. We get especially for any

f E 1(11) f - ( x , . ) = X&. For any fibre product X , xs X , of affine schemes over k (with respect to suitable morphisms) and any ideals I, c k[X,], c k [ X , ] , one has I,

(12)(Argue as for 1.4(8).)

DV,) xs W

Z )

= D(I1 Ok(S1IZ).

1.6 (Affine Varieties and Affine Schemes) An affine scheme X is called algebraic if k [ X ] is isomorphic to a k-algebra of the form k[T,, . . . , % ] / I for some n E N

Schemes

11

and a finitely generated ideal I in the polynomial ring k[Tl,. . . , T I .It is called reduced if k [ X ] does not contain any nilpotent element other than 0. Assume until the end of Section 1.6 that k is an algebraically closed field. Any affine variety X over k defines as in 1.1 a k-functor 3 which we may identify with S p , [ X ] . One gets in this way exactly all reduced algebraic affine schemes over k. For two affine varieties X , X , one has M o r ( X , X )= Hom,-,,,(k[X], k [ X ] ) v Mor(X, 3). So we have indeed embedded the category of affine varieties as a full subcategory into the category of affine schemes. When doing this, one has to be aware of several points. Any closed subset Y of an affine variety X is itself an affine variety. The functor CiY is the closed subfunctor V ( I )c 3 where I = { f E k [ X ] I f(Y ) = O}. In this way one gets an , embedding {closed subsets of X } + {closed subfunctors of X}. On the level of ideals (cf. 1.4(2)), it corresponds to the inclusion {ideals I of k [ X ] with I = f i }c {ideals of k [ X ] } . The embedding is certainly compatible with inclusions (i.e., Y c Y c> Y c CiY ), but in general not with intersections: It may happen that Y n Y is strictly larger than the functor corresponding to Y n Y. Take for example in X = k 2 (where k [ X ] = k[Tl, T,]) the line Y = {(a,O) 1 a E k } and the parabola Y = {(a,a) I a E k } . Then Y n Y = {(O,O)]. The ideals I,I of Y , Y are I = (T,) and I = ( T t - T,), hence I + I = (Tf, # (TI, T2) and Y n CiY = V ( I )n V(I)= V(I + 1)differs from the T,) subfunctor V(Tl, T2)corresponding to Y n Y. So, when regarding affine varieties as (special) affine schemes, we have to be careful, whether intersections are taken as varieties or as schemes. The same is true for inverse images and (more generally) for fibre products. Similar problems do not arise with open subsets. To any open Y c X we can [ associate the ideal I = { f k~ X ] I f ( X - Y ) = 0) and then the open subfunctor D ( I ) , which we denote by Y. Because of 1.5(5), the map Y H Y is a bijection from {open subsets of X > to {open subfunctors of z } that is compatible with intersections. It follows from 1.5(10), (12) that this bijection is also compatible with inverse images and fibre products. (In case Y is affine, the notation C?l is compatible with the earlier one.)

1.7 (Open Subfunctors) (Let k again be arbitrary.) Let X be a k-functor. A subfunctor Y c X is called open if for any affine scheme X over k and any morphism f:X + X there is an ideal I c k[X ] with f -(Y) = D ( I ) . Note that this definition is compatible with the one at the beginning of 1.5 because of 1.5(10). From 1.5(6) one gets

(1) I f Y, Y are open subfunctors of X , then so is Y n Y.

12


Let f:X

+X

be a morphism of k-functors. Then one has, obviously,-I(

(2) If Y is an open subfunctor of X , then f

Y ) is an open subfunctor of X.

Let X , , X,, S be k-functors and suppose X , x s X , is defined with respect to some morphisms. Then one gets (using Y, xs Y = p;(Y,) n p i 1 (Y,)) ,(3) If Y, c X , and Y, c X, are open subfunctors, then Y, xs Y2 is an open subfunctor of X , xs X , .

Let Y, Y be open subfunctors of X . Then

Y o Y ( A ) = Y ( A )for each k-algebra A that is a jield. (Of course 3 trivial. In order to show e suppose Y # Y . Then is , there is some k-algebra A with Y ( A )# Y(A).Assume that there is x E Y ( A ) with x # Y(A). Via Y ( A ) N Mor(SpkA,Y ) c Mor(SpkA,X), regard x as a morphism SpkA + X . Then idA E x - ( A ) ( A ) , # x - ( Y ) ( A ) , hence x - ( Y ) # x-(Y). Now the result follows from the discussion preceding 1.5(5).) A family ( y j ) j E of open subfunctors of X is called an open covering of X , if X(A)= ? ( A ) for each k-algebra A which is a jield. If X is affine and if 5 = D ( I j )for some ideal Ij c k [ X ] , then formula 1.5(9) implies that the D ( I j )form an open covering of X if and only if k [ X ] = Ij. A comparison with the case of an affine variety shows that this is the appropriate generalization of the notion of an open covering. Note that especially=

(4) Y

uje

cj,

( 5 ) Let X be afine and f i , f 2 ,...,f, k [ X ] . Then the X f i form an open E covering of X if and only if k [ X ] = k[Xlfi.

ci=,

Let Y c X be an open subfunctor, and let cp: A + A be a homomorphism of k-algebras. Then

( 6 ) If A is a faithfully flat A-module via cp, thenY ( A )= X ( q ) - Y ( A ) . We have to prove only 3. Suppose at first that X is affine. Then Y = D(Z) for some ideal Z c k [ X ] . Consider some a E X ( A ) = Hom,,,,,(k[X],A) with cp 0 a = X(cp)(.) E Y(A), i.e., with A = Acp(a(I)). The isomorphism A OA % A, a 0 a H cp(a)a induces an isomorphism Aa(I)0, A r A Acp(a(1)). Therefore A = Acp(a(1))together with the flatness of A implies ( A / A a ( I ) ) A = 0, hence Aa(I) = A by the faithful flatness. 0, For arbitrary X , we regard x E X ( A ) as a morphism x: SpkA + X with x(A)(id,) = x , hence with X(cp)x = x(A)Sp,(cp)(id,). So, if x E X(cp)- Y(A), then id, E Spk(cp)-(x-(Y)(A)),hence id, E x - ( Y ) ( A ) ,as x - ( Y ) is an open subfunctor of the affine scheme Spk(A).Now x = x(A)(id,) E Y ( A )as desired.

Schemes

13

Of course, ( 6 )implies that we can restrict to algebraically closed fields in (4). Also, a family ( I.;)je of open subfunctors of X is an open covering of X if and only if X ( A ) = U j , ? ( A ) for all k-algebras A that are algebraically closed fields.1.8 (Local Functors) As the notion of an affine scheme generalizes the notion of an affine variety, we want to define the notion of a scheme generalizing the notion of a variety. Certainly a scheme should (by analogy) be a k-functor admitting an open covering by affine schemes. This is, however, not enough. Consider two k-functors X , Y and an open covering ( I.;)j, of Y. If X , Y correspond to geometric objects, then a morphism f: Y + X ought to be determined by its restrictions jirJ all I.;. Furthermore, given for each j a to morphism fj: I.; -+ X such that filrJ r,, = I y, y J , for all j, j E J , then there ought to be a (unique) morphism f: Y -+ X with fir, = fi for all j. In other words, the sequence

ought to be exact where a( f ) = (fir,)jcJ and p((fj)j,J) resp. ~ ( ( f j ) ~ , ~ ) has (j,j)-component fil YJnY,. resp. f j , Y , n Y , . . For arbitrary X , Y,(I.;),the sequence (1) will not be exact. So we define a kfunctor X to be local if the sequence (1) is exact for all k-functors Y and all open coverings ( I.;)jc (One can express this as saying that the functor Mor(?,X ) is a sheaf in a suitable sense.) For any k-algebra R and any fl,. . . ,f, E R with RJ;.= R, the Sp,(Rf,) form an open covering of the affine scheme SpkR.In this case the sequence (1) takes (because of Yonedas lemma) the formJ .

where the maps have components of the form X ( a )with a one of the canonical maps R + Rji or Rj, + R f i f j Now one can prove (cf. [DG], I, 51, 4.13). .f l ,. . . ,f , E R with

Proposition: A k-functor X is local if and only if for any k-algebra R and any RJ;.= R the sequence (2) is exact.

(Note that in [DG] the second property is taken as the definition of local.) For R and f l,. . . ,f, as in (2) the sequence (3)

14


(induced by the natural maps R --+ R f i and R f i + R f i f j is exact. (This is really ) the description of the structural sheaf on Spec R, e.g., in [Ha], II,2.2.) For an affine scheme X over k the exactness property of Hom,,,,(k[X],?) = X ( ? ) shows that the exactness of (3) implies the exactness of (2). Thus we get(4) Any ajine scheme over k is a local k-functor.

Consider k-algebras A , , A , , . . , ,A , and the projections p j : Ai + A j . If we apply (2) to R = A iand the f i = (0,. . . ,0,1,0,. . . ,O),then we get

( 5 ) If X is a local functor, then X ( n l = , algebras A , , A , , . . . , A , .(The bijection maps any x to ( X ( p i ) x ) l s i s , , . )

nl=,

nl=, Ai) n;=, r

X(Ai) for all k-

1.9 (Schemes) A k-functor is called a scheme (over k) if it is local and if it admits an open covering by affine schemes. Obviously, 1.8(4) implies(1) Any ajine scheme over k is a scheme over k.

The category of schemes over k (a full subcategory of {k-functors})is closed under important operations:

( 2 ) If X is a local k-functor (resp. a scheme over k) and i f X is an open subfunctor o X, then X is local (resp. a scheme). fIn the situation of 1.8(1), the injectivity of tl for X implies its injectivity for X . In order to show the exactness for X , one has to show then for any f~ M o r ( Y , X ) such that each f i Y j factors through X , that also f factors through X . The assumption implies r j c f - ( X ) for each j , hence by the definition of an open covering that f - ( X ) ( A )= Y ( A )for each k-algebra A that is a field. Then 1.7(4) implies Y c f - ( X ) and f factors through X . In order to get the affine covering of X in case X is a scheme, one can restrict to the case where X is affine, hence X = D ( I )for some ideal. Then the ( X f ) f E I form an open affine covering. Let X , , X , , S be k-functors and form X , x s X , with respect to suitable morphisms. Then:

(3) If X , , X,,S are local (resp. schemes), then so is X , xs X,,The proof may be left to the reader. The most important non-affine schemes are the projective spaces and, more generally, the Grassmann schemes gr,,for each r, n E N. For any k-algebra A , one sets gren(A) equal to the set of direct summands of the A-module A + ,

Schemes

15

having rank r. (Then P = 3,,is the projective space of dimension n.) In [DG], are 1.1.3.9/13, there is a proof that all gr, schemes.1.10 (Base Change) Let k be a k-algebra. Any k-algebra A is in a natural way also a k-algebra, just by combining the structural homomorphisms k - k and , k A . We can therefore associate to each k-functor X a k-functor Xk! by Xk.(A)= X(A) for any k-algebra A. For any morphism f:X - X of k, functors, we get a morphism fk: Xk, -+ Xi, of k-functors simply byfk.(A)= f(A) for any k-algebra A . In this way we get a functor X H &,, f H fk, from {k-functors} to {k-functors}, which we call base change from k to k. For any subfunctor Y of a k-functor X, the k-functor G. is a subfunctor of xkf.Furthermore, the base change commutes with taking inverse images under morphisms, with taking intersections of subfunctors, and with forming fibre products. The universal property of the tensor product implies that (SpkR),. = Spk.(R0 k) for any k-algebra R. In other words, if X is an affine scheme over k, then XT is an affine scheme over k with k[Xk.] N k[X] 0 k. For any k ideal I of k[X], one gets then V(& = V ( I 0 k) and D(I)kr = D(I 0 k). ( w e really ought to replace 1 0 k in these formulas by its canonical image in k[X] 0 k, but for once we shall indulge in some abuse of notation.) For any k-algebras A, R one has

(Spk,R)(A)= Homk,-,,g(R,A) C HOmk-alg(R,A ) = (SPkR)k(A). Thus we have embedded Spkf as a subfunctor into (SpkR),,. For any ideal I of R R, denote the corresponding subfunctors as in 1.4/5 by V ( I ) ,D ( I ) c SpkRand l$(I), Dkf(I) Spk,R.Then one sees immediately Dkf(I)= (Sp,.R) n D(I)k,and c h,(I) (Spk.R) V(I)k,. = Using the last results, it is easy to show for any open subfunctor Y of a kfunctor X that G. is open in Xk,.If X is a local k-functor, then obviously Xk. is a local k-functor. Now it is easy to show that Xk! is a scheme over k if X is one over k. Let k, be a subring of k. We say that a k-functor X is dejinedouer k, if there is a fixed k,-functor X, with X = (X1)k. 1.11 (Schemes) In text books on algebraic geometry (like that by Hartshorne to which I shall usually refer in such matters) another notion of scheme is introduced that I shall denote by schemes in case a distinction is useful. A scheme is a topological space together with a sheaf of k-algebras and an open covering by affine schemes. The affine schemes are the prime spectra Spec(R) of the k-algebras R endowed with the Zariski topology and a sheaf having sections R, on each Spec(R,) c Spec(R) for all f E R. To each

16


such scheme X one can associate a k-functor 9 via %(A) = Mor(Spec A , X ) 5 for all A . O n the other hand, one can associate in a functorial way to each k-functor X a topological space (XI together with a sheaf such that ISp,R( = Spec(R) for x each k-algebra R. It turns out that 1 1is a scheme if and only if X is a scheme and that X H 1 1and X H % are quasi-inverse equivalences of categories. x (This is the content of the comparison theorem [DG], I, 1,4.4.) One property of this construction is that the open subfunctors of any kfunctor X correspond bijectively to the open subsets of 1x1, cf. [DG], I, 41, 4.12. More precisely, if Y is an open subfunctor of X, then I Y I can be identified with an open subset of and the k-algebra of sections in I Y I of the structural x sheaf of 11is isomorphic to Mor(Y,A), ibid. 4. 14/15. Suppose that k is an algebraically closed field. Consider a scheme X over k that has an open covering by algebraic affine schemes. We can define on X(k) a topology such that the open subsets are the Y(k) for open subfunctors Y c X. The map Y H Y(k) turns out to be injective ([DG], I, 3,6.8). We can define a sheaf ox(,) on X(k) through ox(,)( Y(k)) = Mor( Y A), Then X H (X(k), Ox,,,) , is a faithful functor and its image contains all varieties over kin the usual sense. There are some fundamental notions of algebraic geometry (like smoothness and dimension) that we shall have to consider only in a few places. The necessary definitions and the main properties from the point of view of kfunctors are contained in [DG]. I do not want to repeat what is done there in order to keep the length of this book down. Any reader who is familiar with these notions in the context of schemes (e.g., from [Ha]) can use the correspondence of X and 1 1as above to translate. For example, a scheme Xis x x smooth if and only if 1 1is so.

11 x

1.12 (Closed Subfunctors) A subfunctor Y of a k-functor X is called closed if and only if for each affine functor X and any morphism f:X X of kfunctors the subfunctor f -(Y) of X is closed in the old sense (as in 1.4). Because of 1.4(6), this is compatible with the old definition in case X is affine. The following statements are clear from 1.4(5) or by definition:--f

(1) If ( y i ) i e , is a family o closed subfunctors of a k-functor X, then f yi is closed in X. (2) Let fi X X be a morphism o k-functors. If Y c X is a closed f subfunctor, then f -(Y) c X is closed. (3) Let X , , X,, S be k-functors with fixed morphisms X, -+ S and X, -+ S . If Y, c X, and Y, c X2 are closed subfunctors, then Y, xs Y2 c X, xs X, is closed.

nie,

Schemes

17

Because of (l), we can define the closure of any subfunctor Y of X as the intersection of all closed subfunctors containing Y. In order to get some deeper results we need (4) Let X be an afJine scheme and ( X j ) j , J an open covering o X . If Y, Y' f are local subfunctors of X with Y n Xj= Y' n X j for all j E J , then Y = Y'. If X j = D(lj) for some ideal 'j c k [ X ] , then cjsJlj [ X ] , cf. 1.7. We can =k choose a finite subset J, c J and & E lj for all j E Jo such that k [ X ] = J o k [ X ] f , . Then the D(4.) 5.with j E J, form also an open covering c of X (refining the original one). We have also Y n D(&) = Y ' n D(&)for all j E J,, so we may as well assume that J = {I, . . ,r> and X j = X f j for some 2,. f j E k [ X ] with k [ X ] = k[X]fj. Consider now x E X ( A ) = Hom,-,,,(k[X], A ) for some k-algebra A ; set f;= x(h) E A and x i E X ( A f ; )corresponding to the composed homomork [ X ] L = k [ X ] implies A = phism k [ X ] A A 2 A J i . Now A f i , so the local property of Y and Y' yields

cis

Xi=,

x E Y ( A )0 xi E Y ( A f ; )o x i E ( Y n Xi)(AfJ o xi E ox E

for all

i

for all i for all i

(Y' n X i ) ( A f i )Y'(A).

In the affine case, any closed subfunctor is again an affine scheme, cf. 1.4(4), hence local, so we can apply (4) to it.

(5) Let be an open covering of some k-functor X . If Y c X is a closed subfunctor with Y 3 Xjfor all j , then Y = X .Indeed, consider x E X ( R ) for some k-algebra R and let f:Sp,R + X be the morphism with f (R)(idR) x, cf. 1.3. We can apply (4) to the closed, hence = local, subfunctors f - I ( Y ) and f - ' ( X ) = Sp,(R) of Sp,(R) and the open hence idR E f - ' ( Y ) ( R ) and covering (f-(xj))j,J. We get f-'(Y) = f -(X), x E Y(R).

( 6 ) Any closed subfunctor Y of a local functor X (resp. a scheme X ) is again local (resp. a scheme).Indeed, consider a morphism f:X ' + X and an open covering ( X j ) j , , of X ' such that each fix; factors through Y, i.e., with X j c f - '( Y ) . As f - I ( Y ) is closed, ( 5 ) yields f - ' ( Y ) = X ' , hence f factors through Y. This together with the local property of X implies easily that Y is local. If X is a scheme and if

18


(Xj)js is an open covering by affine schemes, then ( Y n Xj)js is an open covering by closed subschemes of affine schemes, hence by affine schemes. The proof of the following statement is left as an exercise: (7) For any closed subfunctor Y of a k-functor X and any k-algebra k' the subfunctor &, o X,. is closed. f1.13 Lemma: Let X be a local functor and Y c X a local subfunctor of X. Let (Xj)je be an open covering of X. Then Y is closed in X if and only if each Y n Xj is closed in Xj.

Proof: One direction being obvious, let us suppose that each Y n Xj is closed in Xj. For any morphism J X ' -+ X with X' affine also, f -'(Y) z X'x,Y is local, the f - ' ( X j )are an open covering of X ' , and each f - '( Y) n f -'(Xj) = f - '( Y n Xj) is closed in f - (4.). as well assume that X is affine. So we may As in the proof of 1.12(4),we can assume J = { 1,2,. , . ,r } and 4. X, for = , somefi E k[X]. Let I resp. Ij be the kernel of the restriction map k[X] --* k[ Y] resp. k[X] -+ k [ X j n Y]. Then F = V ( I ) . As Y n X j is closed, we have Y n Xj = V(Zj),j. The Y n Xj form an open covering of Y. So the restriction induces an injective map k[Y] -+ k[Y n X,], hence I = 4. We have for all i,j

ns=

ns=l

hence (Ij),8f, = ( I J J i f j . So for any a E Ii,there is some n with (fih)"a Ijfor all j , E hence with f :a E 4 for all j and thus f :a E I = Zj. This implies ZSi = (Zi)si for all i, hence r n X i = Y n X , . Now apply 1.12(4) to the local subfunctors Y and F of and get Y = F.

r

1.14 (Closures and Direct Products) Let us assume in this section that k is noetherian (in order to simplify the following definition). A scheme X over k is called algebraic if it admits a finite open covering by affine subschemes which are algebraic in the sense of 1.6. (One can check that this yields the old definition in the affine case.) Let Y,Z be schemes and X a subscheme of Y such that X and Y are algebraic and such that Z is flat. (This means that Z admits open and affine covering ( Z i ) isuch that each k[Zi] is a flat k-module.) Then we have in Y x Z(1)

x x z = 5? x 2.

This follows, e.g., by applying [DG], I, $2, 4.14 to Y' = Y x Z and the projection Y x Z + Y.

Schemes

19

1.15 (Functors of Morphisms) For any k-functors X , Y , we can define a kfunctor A o + ( X ,Y ) through(1)

&oz(X, Y ) ( A )= M o r ( X , , Y,)

for any k-algebra A . For any homomorphism cp: A -+A' of k-algebras AG~(X, Y)(cp)maps any morphism f : X A + Y, to the morphism f A , : X A'v , ( X A ) A-+ ( YA)As Y,. using the structure of A' as an A-algebra via cp. f = The construction of A o 2 ( X , Y ) is clearly functorial: To each morphism X ' -+ X resp. Y' -+ Y of k-functors there corresponds an obvious morphism A o . c ( X , Y ) -+ A u t ( X ' , Y ) resp. A G 2 ( X , Y ' ) -+ A o h ( X , Y ) . If Y' is a subfunctor of Y , then we shall always regard &02(X, Y') as a subfunctor of &Oh(X, Y ) . Consider an open covering ( X j ) j , of X and a closed subfunctor Y' of Y. Let p j : A G ~ ( X , ) -+ &&(Xj, Y ) be the obvious restriction map. We claim Y

f E A o Q ( X ,Y ) ( A )= M o r ( X , , YA) for some k-algebra A with p j ( A ) f E M o r ( X j A ,Y a ) for all j E J, i.e., with X j A c f-'(YA) for all j. Now the ( X j A ) j E J are an open covering of X, and f-( Y a )is a closed subfunctor of X,, so 1.12(5) yields f - ' ( Y > ) = X,, hence f E A G ~ ( X , Y)(A).If X is an affine scheme, i.e., if there is a k-algebra R with SpkR = X , then A u h ( X , Y ) can also be described as follows: One has for any k-algebra A Auh(SpkR, Y ) ( A ) Mor((SpkR)A, YA)2 :

Of course, one inclusion (c") trivial. Consider on the other hand is

Mor(Sp,(R 0 A ) , Y ) N Y ( R 0 A ) .

Any Y E Y ( R 0 A ) defines by Yoneda's Lemma (1.3) a morphism f,: Spk(R 0 A ) Y mapping any y E Homk-AI,(R 0 A , B ) = Spk(R 0 A ) ( B ) to Y(y)(y).Using the identification as above, we get also a morphism fk: SpkA -+ A&h(SpkR, Y ) . For any k-algebra B, we can regard f ; ( B ) as the map Homk-AI,(A,B) -+ Y ( R 0 B ) with p H Y(idR0 p)(y). So f b ( B ) is the COmpOSitiOn Of p H id^ 0 p, Homk-A],(A, B ) + Homk-~],(R A , R 0 B ) 0 with fJR 0 B). We claim: If R is free as a k-module and if Y' c Y is a closed subfunctor, then fF'Ao@pkR, Y ' )is closed in SpkA. To start with, we know that f ; ' ( Y ' ) is closed in Spk(R0 A ) , so there is an ideal I' c R 0 A with f Y ' ) = V(1'). ;( The description of f; as above implies (for any k-algebra B )+

f;'&oh(SpkR,

Y ' ) ( B )= { p E HOmk.Alg(A, B ) I I' c ker(idR 0 p)}=

( p E Homk-~],(A, ) 11' c R 0 ker(p)) B

20

Representatiolrs of Algebraic Groups

using the freeness of R for the last equality. This freeness implies also R0 li = R 0 li for any family (li)i ideals in A . If we take as the of li all ideals with R 0 li 3 I, then I = li is the smallest idea of A with I' c R Q I. Then

n

n

n

I' c R Q ker(p) o ISO

t

ker(p) o p E V(I)(B),

f;'.,doh(Sp,R, Y ' ) = V ( I )is closed. As we can apply this to all A and all y , this implies that Jtio.t(Sp,R, Y ' ) is

closed in .,dob(SpkR,Y ) .Using ( 2 ) we get now( 3 ) Let X , Y be k-functors and Y' c Y a closed subfunctor. If X admits an open covering ( X j ) j E with afine schemes such that each k [ X j ] is free as a kJ module, then .,doh(X, Y ' ) is closed in J t i u ~ ( XY ) . ,

If X is a scheme, then X is called locally free if and only if there is an open covering as above.

L

Group Schemes and Representations

In this section we define group schemes and modules over these objects and discuss their fundamental properties. As in Chapter 1 we more or less follow CDGI. After making the definitions of k-group functors and k-group schemes in 2.1, we describe some examples in 2.2. The relationship between algebraic groups and Hopf algebras generalizes to group schemes. This is done in 2.32.4. (We always assume our group schemes to be affine.) We then discuss the class of diagonalizable group schemes in 2.5 and group operations in 2.6. We then go on to define representations (2.7) and to discuss the relationship between G-modules and k[G]-comodules (2.8).We generalize standard notions of representation theory to G-modules: submodules (2.9),fixed points (2.10), centralizers and stabilizers (2.12), and simple modules (2.14). The definition of a submodule has an unpleasant aspect that disappears only when G is a flat group scheme (ie., a group scheme such that k[G] is a flat k-module). This is the reason why we shall restrict ourselves to such groups later on. We show that representations of group schemes are locally finite (2.13). Furthermore, we describe representations of diagonalizable group schemes (2.1 1) and mention results about modules for trigonalizable and unipotent groups over fields (in 2.14).One can twist a given representation of an abstract group over k by composing with a group endomorphism. One can also

21

22


construct a new representation by changing the operation of k on the module by a ring endomorphism. This is generalized to group schemes in 2.15-2.16.2.1 (Definitions) A k-group functor is a functor from the category of all k-

algebras to the category of groups. We can regard any k-group functor also as a k-functor by composing it with the forgetful functor from {groups} to {sets}. In this way we can and shall apply all ideas and notions from section 1 also to k-group functors. For two group functors G, H, we shall denote by Mor(G, H) the set of all morphisms ( = natural transformations) from G to H considered as k-functors and by Hom(G, H) the set of all morphisms from G to H considered as k-group functors. So Hom(G,H) consists of those f E Mor(G, H) with f(A)a group homomorphism for each k-algebra A . These elements are called homomorphisms from G to H. Let Aut(G) be the group of automorphisms of the k-group functor G. A k-group scheme is a k-group functor that is an afJine scheme over k when considered as a k-functor. (Of course, we really ought to call such an object an affine k-group scheme and drop the word affine in the definition of a k-group scheme. But we shall consider only affine group schemes, and then it is more economical to call them group schemes.) An algebraic k-group is a k-group scheme that is algebraic as an affine scheme. A k-group scheme is called reduced if it is so as an affine scheme. Over an algebraically closed field, the category of algebraic groups as in [Hu2] or [Sp2] can be identified with the subcategory of all reduced algebraic k-groups. Let G be a k-group functor. A subgroup functor of G is a subfunctor H of G such that each H(A) is a subgroup of G ( A ) . The intersection of subgroup functors is again a subgroup functor. The inverse image of a subgroup functor under a homomorphism is again one. A direct product of k-group functors is again a k-group functor; so is a fibre product if the morphisms used in its definition are homomorphisms of k-group functors. A subgroup functor H of G is called normal (resp. central) if each H(A) is a normal (resp. central) subgroup of G ( A ) .Again, normality is preserved under taking intersections and inverse images under homomorphisms. The kernel ker cp of a homomorphism cp: G -+ G is always a normal subgroup scheme. A closed subgroup scheme of a k-group scheme G is a subgroup functor H which is closed if considered as a subfunctor of the affine scheme G over k. If G and H are algebraic k-groups, we simply call H a closed subgroup of G. A k-group functor G is called commutative if all G ( A )are commutative.2.2 (Examples) The notations introduced here for special group functors G and their algebras k[G] will be used always, The additive group over k is

Group Schemes and Represeotations

23

the k-group functor G, with G,(A) = ( A , +) for all k-algebras A . It is an algebraic k-group with k[G,] isomorphic to (and usually identified with) the polynomial ring k [ T ] in one variable. Any k-module M defines a k-group functor Ma with Ma(A)= ( M 0A, + ) for all A . (So we have G, N ka). If M is finitely generated and projective as a kmodule, then Ma is an algebraic k-group with k[M,] = S ( M * ) , the symmetric algebra over the dual k-module M *. In case M = k" for some n E N, we may identify Ma with G, x G, x . . . x G, ( n factors) and k[M,] with the polynomial ring k[T,, T',. . . ,T,]. The multiplicatioe group over k is the k-group functor G, with G,(A) = A" = {units of A ) for all A. It is an algebraic k-group with k[G,] = k[T, 7-11. Any k-module M defines a k-group functor G L ( M ) with G L ( M ) ( A )= (End,(M 0A))" called the general linear group o M. In case M = k", we f may identify G L ( M ) with GL, where G L , ( A ) is the group of all invertible (n x n)-matrices over A . Obviously, GL, is an algebraic k-group with k[GL,] isomorphic to the localization of the polynomial ring k [ q j , 1 Ii, j In] with respect to {(det)"I n E N}. More generally, if M is a finitely generated and projective k-module, then the k-functor A H End,(M 0A ) can be identified with the affine scheme ( M * @ M ) , from above and G L ( M )with the open subfunctor D(det). For such M (projective of finite rank), the determinant defines a homomorphism of algebraic k-groups G L ( M )+ G,. Its kernel is denoted by S L ( M )and is called the special linear group of M. It is an algebraic k-group. Similarly, we define SL, c GL,. Note that GL1 = G, and S L , = 1 = the group functor associating to each A the trivial group (1). For each n E N, let T, be the algebraic k-group such that T,(A)is the group of all invertible upper-triangular (n x +matrices of A , i.e., of all uppertriangular matrices such that all diagonal entries belong to A " . One may identify k [ T , ] 2: k [ q j l 1 I i I j I n, T i ' 11 I i I n]. Furthermore, let U, be the algebraic k-group such that each U,(A) consists of all g E T,(A) having all diagonal entries equal to 1. We may identify k[U,] = k [ T j I 1 I i < j I n]. For any n E N we denote by pc, the group functor with p(,)(A)= { a E A la" = l } for all A . It is an algebraic k-functor with k[p(,,] = k[T]/(T" - 1) and a closed subgroup of G,. Let p be a prime number and assume pl = 0 in k. Then we can define for each r E N a closed subgroup G,,, of G, through G,,,(A) = { a E A I apr = O}.2.3 (Group Schemes and Hopf Algebras) Let G be a k-group functor. The group structures on the G ( A ) define morphisms of k-functors m G : G x G + G (such that each mG(A):G ( A ) x G ( A )+ G ( A ) is the multiplication), and

24


l,:Sp,k + G (such that l,(A) maps the unique element of (Spkk)(A)to the 1 of G(A)),and i G : G + G (inducing on each G(A)the map g - g - ' ) . Now assume G to be a k-group scheme. Then these morphisms correspond uniquely to their comorphisms AG = m;: k[G] -+ k[G] Q k[G] (called comultiplication), E~ = 12: k[G] + k (called counit or augmentation), and aG= i a : k [ G ] -P k[G] (called coinverse or antipode). So, if AG(f) = fi Q f; for Some f E kCG1, then f ( g 1 s J = I;=, 91, g 2 E G(A) f i ( s l ) f : ( g J for each and any A . Furthermore, we have ~ ~ (= f(1) and a G ( f ) ( g )= f ( g - ' ) for any f ) g E G ( A )and any A. We shall drop in our notations the index G whenever no confusion is possible. As in the case of algebraic groups (cf. [Bo], 1.5 or [ H u ~ ] ,7.6 or [Sp2], 2.1.2), the group axioms imply that A, E, a satisfy(1)

(id Q A) A = (A Q id) A,0

0

(2)(3)

(E

@ i d ) A = id = (id @ E ) A,0

0

( a @ i d ) o A= B = ( i d @ o ) o A .

(Here we denote by cp@ $ the map ~ Q u ' Hcp(a)$(a') in contrast to cp Q $ : a Q a' H cp(a)Q $(a') and by Bthe endomorphism a H ~ ( a )of kCG].) l A morphism cp: G -P G' between two k-group schemes is a homomorphism if and only if its comorphism cp*: k[G'] -+ k[G] satisfies(4)

AG q* = ( ~ p * Q ~ p * ) AGg.0 0

If so, then one has automatically(5)&G

cp* = &G'

and(6)bG

q* = q*

QG'.

A Hopf algebra over k is an associative (not necessarily commutative) algebra R over k together with homomorphisms of algebras A: R --i R Q R, E : R -P k, and a: R -P R satisfying (1)-(3). A homomorphism between two Hopf algebras is a homomorphism of algebras satisfying additionally (4)-(6) (with the appropriate changes in the notation.) We call R commutative if it is so as an algebra, and cocommutative if s A = A, where s: R Q R + R Q R is the map a Q b H b @ a. Let R be a commutative Hopf algebra over k. Then we can define on each (Sp,R)(A)= Hom,(R, A ) a multiplication via ab = (a @ p) A. In this way we get on SpkR a structure as a k-group scheme. It is elementary to see that we0 0


25

get in this way a functor {commutative Hopf algebras over k } - {k-group , schemes} that is quasi-inverse to G H k [ G ] . Thus these categories are antiequivalent. Note that G is commutative if and only if k [ G ] is cocommutative. 2.4 (Properties of the Hopf Algebra Structure) Let us look at the Hopf algebra structures on k [ G ] in our examples in 2.2. In the case of G,, one has A(T) = 1 0 T + T 0 1, E ( T ) 0 and o ( T )= - T. Similar formulas hold = = for the Ga,r.In the case of G,, one has A(T) = T @ T, E ( T ) 1, and a(T) = T-'. In G L , , one has A(Tj) = q,,,0 Tmjand c ( q j ) = 6 , (the Kronecker delta). The formula for o ( T j ) is more complicated. Furthermore, one has A(det) = det 0 det, e(det) = 1, and o(det) = det-'. Let G be a k-group scheme and set I , = k e r ~ , augmentation ideal the in k [ G ] . One has k [ G ] = k l @ I,, and a H al, k - k l is bijective. This im, 0 plies k [ G ] 0 k [ G ] = k(1 0 1) @ (k 0 I , ) @ ( I , 6 k) 0 ( I , 0 1'). The formula 2.3(2) implies

,

(1)

A ( f ) ~ f O+ 10f+I,OZ, lo ( f )E -f

forall f e z ,

and then the formula 2.3(3) implies

(2)Set

+ 1:

for all f

E

I,.

(3)

X ( G ) = Hom(G, G,,,).

This is a commutative group in a natural way. The embedding of affine schemes G , c G, = A' yields an embedding

X ( G ) c Mor(G, G,) c M o r ( G , G,) N k [ G ]that is compatible with the multiplication. Take f E k [ G ] . One has f * ( T )= f. Therefore 2.3(4) implies easily

(4)

X(G)

{f kcGI I f(l)= 1, A G ( f )

=f 0f>*

Of course A G ( f ) = f 0 f implies f(1)' = f(1). If f(1) = 0, then f(g) = f ( g 1) = f(g)f(l) = 0 for all g E G ( A ) and all A , hence

-

(4') If k is an integral domain, then X ( G ) N (f E k [ G ] A&)

f ZO}.

I

=f

0 f,

Let me refer to [ D G ] , II,l, 2.9 for the proof of( 5 ) If k is a jeld, then X ( G ) is linearly independent.

26


(This is just another variation on the theme h e a r independence of characters.) Usually we shall write the group law in X ( G ) additiuely. Let I be an ideal in k[C]. Using 1.4(6), (7) one checks easily that V ( I ) is a subgroup functor if and only if(61

A(I) c I 0 k[G]

+ k[G]

@ I , &(I) = 0,a(I)c I.

If so, it will be a normal subgroup if and only if (7)

c * ( I ) c k[G] 0 I ,

where c* is the comorphism of the conjugation map c: G x G - G with , c(A)(g1,g2) g1g2gi1for all A and gl, g2 E G ( A ) .One may check that =

2.5 (Diagonalizable Groups) Let A be a commutative group (written multiplicatively) and let us identify A with the canonical basis of the group algebra k[A]. We make k[A] into a commutative and cocommutative Hopf algebra via A(i) = i0 iand &(A) = 1 and ~(i) all iE A. In this way we as= A- for sociate to A a k-group scheme which we denote by Diag(A). If A is finitely generated, then Diag(A) is an algebraic k-group. So Diag(A)(A) consists (for any k-algebra A ) of all group homomorphisms cp: A - A . The multiplication is the obvious one: (cpcp)(A) = cp(A)cp(A) for all , I. E A. We call a k-group scheme diagonalizable if it is isomorphic to Diag(A) for some commutative group A. For example G, Y Diag(Z) and p(, N Diag(Z/(n)) are diagonalizable. We get also direct products of these groups as Diag(Al x A2) Y Diag(A,) x Diag(A,) for all commutative groups Al, A 2 . Any group homomorphism a: A1 - A2 induces a homomorphism of group , algebras a*: k[AI] + k[A2] which is a homomorphism of Hopf algebras, hence we get a homomorphism Diag(a): Diag(A,) + Diag(A2) of k-group schemes. Thus A H Diag(A) is a functor from {commutative groups} to {kgroup schemes} that maps {finitely generated commutative groups) into {algebraic k-groups}. If LX is surjective, then Diag(cc)(A)is injective for each A, and we can regard Diag(A2) as a subfunctor of Diag(A,). If a is not surjective, then there a nontrivial homomorphism A 1 / a ( A 2 + A x for some k-algebra A , hence ) Diag(a)(A) is not injective.


27

Suppose that k is an integral domain. Then an easy computation shows (cf. [DG], 11, $1, 2.1 1) for all A, A(1)

X(Diag(A)) N A

( k integral)

and (2) Hom,,(A, A) r Hom(Diag(A),Diag(A))

(k integral)

Thus in this case Diag(?) is an anti-equivalence of categories from {commutative groups} to {diagonalizable k-group schemes}. Furthermore, A is finitely generated if and only if Diag(A) is an algebraic k-group. We get from (1) that a k-group scheme G is diagonalizable if and only if X(G) is a basis of k [ G ] (for k integral).2.6 (Operations) Let G be a k-group functor. A left operation of G on a kfunctor X is a morphism a: G x X + X such that for each k-algebra A the map a ( A ) :G ( A ) x X(A) + X(A) is a (left) operation of the group G ( A ) on the set X(A). We usually write gx instead of a(A)(g,x ) for g E G ( A )and x E X(A). We can similarly define right operations. For example, the conjugation map c in 2.4 is an operation of G on itself. Other operations of G on itself are by left ( a ( A ) ( g , g ) gg) and right = ( a ( A ) ( g , g )= g g - ) multiplication. Let k be a k-algebra. Then any operation of G on a k-functor X defines in a natural way an operation of Gkeon Xkn. For any operation a as above we set

(1)

XG(k)= { x E X(k) Igx = x

for all g E G ( A ) and all A } .

(This is done by some abuse of notation. The x in gx = x is really the image of x under the map X ( k ) + X ( A ) corresponding to the structural morphism k + A. We shall stick to this abuse.) We can define a subfunctor X G of X, the Jixed point functor via XG(A) = (X,)GqA)

(2)

= {x E X(A)(gx= x

for all g E G ( A ) and all A-algebras A } .

If Y is a subfunctor of X, then its stabilizer in G is the subgroup functor StabG( with Y)

(3)

Stab,(Y)(A) = { g E G ( A )1 gY(A) c Y(A)

for all A-algebras A }

28

Representatiolls of Algebraic Groups

for all A, and its centralizer is the subgroup functor CentG(Y) with

(4)

Cent,(Y)(A) = { g E G ( A )I gy

=y

for all y E Y(A) and all A-algebras A }

for all A . These constructions can also be described using the k-functors Aur(X1,X,) as in 1.15. The operation of G defines a morphism y:G-+Au+(X,X) as, , sociating to each g E G ( A ) the morphism X + X defined by the action of 9. For any subfunctor Y of X we get by restriction a morphism y y : G + Aur( Y, X). Now obviously(5)

StabG( = Y)

J ; )

A u h (Y,

Y).

Let cp: G -+ A u r ( Y, X x X) be the morphism associating to each g the morphism y H (gy,y). Using the notation D, for the diagonal subfunctor of X as in 1.2 we see(6)

Cent,(Y) = cp-Aur(Y, D,).

Let I): X -,Aur(G,X x X) be the morphism associating to each x the morphism g H (gx,x). Then(7)

xG I ) - ~ A ~ C ( G , = D,).

Now 1.15(3) implies: ( 8 ) I Y is a closed subfunctor of X and a locally free k-scheme, then StabG( ) f Y is closed in G. (9) If Y is a locally free k-scheme and if D, is closed in X x X, then CentG(Y) is closed in G. (10) If G is locally free and if D is closed in X x X, then X Gis closed in X. , (One calls X separate if D, is closed in X x X. Any affine scheme is separate.) Y) In case X = G with G acting via conjugation one usually calls StabG( the normalizer of Y and denotes it by NG(Y). Furthermore, we then usually write CG(Y) instead of Cent,(Y), and Z ( G )instead of CG(G).Of course, Z ( G )is just the centre of G . Consider a k-algebra k which is an algebraically closed field. Suppose that Xk, is algebraic and separate. The map Z H Z(k) is a bijection from {closed and reduced subfunctors of Xkr> {closed subsets of X(k)),cf. 1.6. We claim to for any closed subfunctor Y of X such that yk. is reduced:(11) (12)

StabG( Y)(k) = StabGog,( Y(k)), CentG( Y)(k) = Cent,(,,,( Y(k)).

Group Schemes and Representatiom

29

Indeed, if g E G(k),then g & and 5.are two closed and reduced subfunctors of xk,, hence g & = yk. if and only if (gY,.)(k)= gY(k)is equal to Y(k),i.e., if This yields (11). In order to get (12), we embed 5. via g E StabG(kt)(Y(k)). y H ( y ,y ) and via y H ( y ,gy) into & x Xk,. The images Y, and Y2 are closed subfunctors of Xkpx &, both isomorphic to &, hence reduced. Therefore Y, = Y2 if and only if Y,(k) = Y,(k). On the other hand, g E Cent,(Y)(k) resp. g E CentG(kr)(Y(k))if and only if Y, = Y2 resp. Yl(k) = Y,(k). This implies (12). Suppose G acts on another k-group functor H such that each G ( A )acts on H ( A ) through group automorphisms. Then we can define the semi-direct product G DC H where each (G PC H ) ( A ) is the usual semi-direct product G ( A ) DC H ( A ) . As a k-functor G DC H is of course the direct product of G and H . Let H , N be subgroup functors of G such that H normalizes N , i.e., that each H ( A ) normalizes N ( A ) . We can then construct H cx N as above and get a homomorphism cp: H DC N + G via (h, n) H hn for all h E H ( A ) , n E N ( A ) and all A . Its kernel is isomorphic to H n N under h H ( h , h - ) for all h E H ( A ) n N ( A ) and all A . If cp is an isomorphism, then we say that G is the semi-direct product of H and N and write G = H LX N . (If G is a k-group scheme and G = H DC N , then necessarily H and N are closed subgroup schemes.)2.7 (Representations) Let G be a k-group factor and M a k-module. A representation of G on M (or: a G-module structure on M ) is an operation of G on the k-functor Ma (as in 2.2) such that each G ( A ) operates on Ma@) = M 6 A through A-linear maps. Such a representation gives for each A a 3 I group homomorphism G ( A )+ End,(M @ A ) ,leading to a homomorphism G + GL(M) of group functors. Vice versa, any such homomorphism defines a representation of G on M. There is an obvious notion of a G-module homomorphism (or G-equivariant map) between two G-modules M and M . The k-module of all such homomorphisms is denoted by HomG(M,M). The representations of G on the k-module k, for example, correspond , bijectively to the group homomorphisms from G to G L , = G, ie., to the elements of X(G). For each 3, E X(G) we denote k considered as a G-module via 3, by k , . In case 3, = 1 we simply write k. Given one or several G-modules we can construct in a natural way other G-modules. For example

(1) Any direct sum of G-modules is a G-module in a natural way. (2) The tensor product of two G-modules is a G-module in a natural way.

30


f ( 3 ) Any symmetric and exterior power o a G-module is a G-module in a natural way. In (3), for example, we consider for each commutative ring R the functor F R from R-modules to itself with & ( M ) = SM. We have for each R-algebra R canonical isomorphisms FR(M) OR N FR,(M OR for all R-modules M , R R) i.e., the functors M H FR(M) ORR and M H FR,(M ORR) are isomorphic. If M is a G-module, then G operates on the functor A H F,(M 0 A ) , each g E G ( A ) via FA(g). By our assumption this functor is isomorphic to Fk(M),, hence we get a G-module structure on F,(M). The functor M H AM has the same property, hence we can argue as above. Our reasoning can easily be extended to functors in several variables and then yields (l), (2). If we deal with contravariant functors (FR)Rin our situation above, we ought to let g E G(A) act via F,(g-), This applies to the functor M H M * which will, however, commute with ring extensions only when restricted to finitely generated and projective modules. Thus we get(4) Let M be a G-module which is finitely generated and projective over k. Then

M * is a G-module in a natural way.

For M as in (4)one has canonically M * 0 M 4, k-module M . Combining (2) and ( ) we get

N

Hom(M, M ) for any

( 5 ) Let M , M be G-modules with M finitely generated and projective over k. Then Hom(M, M ) is a G-module in a natural way.The following result is obvious from the definitions:

( 6 ) Let k be a k-algebra and M a G-module. Then M @ k is a Gk.-mOdUk in a natural way.Another way, how representations arise, is from an operation of G on an affine scheme X. Then we get a G-module structure on k[X]: If g E G(A) and f E k[X] 0 A = A[X,] for some k-algebra A, then gf E A[X,] is defined through ( g f ) ( x )= f ( g - x ) resp. = f ( x g )(for a left resp. right operation) for all x E X ( A ) = X,(A) and all A-algebras A. (Again, the g in g - x or xg is really the image of g under G ( A ) -+ G(A). . .). In case G is a k-group scheme, we get thus the left and right regular representations of G on k[G] derived from the action of G on itself by left and right multiplications. We shall always denote the corresponding homomorphisms G -+ GL(k[G]) by pI and pr. The coinverse 0, is an isomorphism of G-modules from k[G] with pr to k[G] with p l . Furthermore, the


31

conjugation action of G on itself gives rise to the conjugation representation of G on k[G].

id,[,] E G(k[G]) = End,.,,,(k[G]) acts on M 0 k[G], so we get a k-linear map A M : M + M 0 k[G] with AM(m)= idktGl(m 1) for all m E M . We call A M the 0 comodule map of the G-module M . It determines the representation of G on M completely: For any k-algebra A and any g E G ( A ) = Homk_,,,(k[G],A) we have a commutative diagram

2.8 (The Comodule Map) Let G be a k-group scheme. If M is a G-module, then

G(kCG1) x ( M O kCG1)G(g)

G ( A ) x ( M 0A )

1

MOkCGl8 9)idM89> M O A0

1

by the functorial property of an operation. As G(g)cp = g cp for any cp E G ( k [ G ] ) ,we have g = G(g)id,[,,, hence g(m 6 1) = (id, 0 g) 0 AM(m) for all m E M . More explicitly, if AM(m)= m i0 fi, then

The fact that each G ( A ) operates on M O A (ie., g(gm) = (gg)m and l m = m)yields easily the following formulas: (2) and(AM

8 id,[,,)

O

AM

= (idM 8 A G )

O

AM

(3)

(idM Q E G )

0

AM

= id,.

If M is another G-module and if cp: M + M is a linear map, then cp is a homomorphism of G-modules if and only if (4)AM, Cp

= (P 0id,,,,)

AM.

A comodule over the Hopf algebra k[G] is a k-module M together with a linear map A M : M + M 0 k[G] such that (2) and (3) are satisfied. A homomorphism between two comodules is a linear map satisfying (4).So we have defined a faithful functor from {G-modules} to {k[G]-comodules). On the other hand, any k[G]-comodule gives rise to a G-module: Just take (1) as a definition. In this way we can see that the two categories of G-modules and of k [G] -comodules are equivalent. Let a: X x G + X be an action of G on an affine scheme X over k . Then k[X] is a G-module in a natural way (see 2.7) and the comodule map

32

Representatiom of Algebraic Croup

Akfx1: k[X] + k[X] @ k[G] is easily checked to be the comorphism a*. If we take X = G and the action by right multiplication, we get(5)

Apr = A G .

(We write APr and also Ap, below instead of AkIG] in order to indicate which representation is considered.) For the left regular action we get(6) Ap, = s O(JG

@ id,[,])

O

AG

with s(f @ f) = f @ f for all f,f. For the conjugation representation on k[G] the comodule map is equal to(7)tO

(&[GI

@ AG) O AG

where

t(fi@ f 2 @ f 3 ) = f 2 @ aG(fi)f3.

Remark: Suppose for the moment that k is an algebraically closed field and that G is a reduced algebraic k-group. There is a natural notion of representations of G(k) as an algebraic group (or of a rational G(k)-module),cf. [ H u ~ ] ,p. 60. One can show, as above, that the category of G(k)-modules is equivalent to the category of comodules over k[G(k)] = k [ G ] , hence to that of G-modules. (To a G-module M we associate the operation of G(k)on M given by the definition of a G-module.)Similarly one can show that the notions of Gsubmodules (to be defined in 2.9) and of G(k)-submodules coincide, using ) 2.9(1),and that M G ( k = M G (to be defined in 2.10), using 2.10(2).Furthermore, one has HomG(M,M ) = Hom,(,,(M, M ) for any two G-modules M , M (using (4) above).2.9 (Submodules) Let G be a k-group functor. If

submodule of a G-module M as a subspace N c M such that N @ A is a G(A)stable submodule of M @ A for each k-algebra A. Then N itself is a G-module in a natural way. For arbitrary k this works out well as long as the natural map N @ A + M @ A is injective for each A, e.g., if N is a direct summand of M . Taking only such pure submodules (as in [DG], 11, 1.3-4) would be too restrictive and not allow kernels and images of all homomorphisms. So let us define a submodule of a G-module M to be a k-submodule N of M that has itself a G-module structure such that the inclusion of N into M is a homomorphism of G-modules. If so, then M I N has a natural structure as G-module: We have for each A an exact sequence of G(A)-modules N @I A -+ M @I A + ( M / N )@ A + 0. We call M I N the factor module of M by N . It has the usual property of a factor module.

k is a field, we can define a


33

Still, our definition of a submodule has one disadvantage. A given ksubmodule N of M may conceivably carry more than one structure as a Gmodule. In order to prevent this we shall prefer to make special assumptions on our group and not on the modules. An affine scheme X over k is called flat if k[X] is a flat k-module. A k-group scheme is called flat, if it is so as an affine scheme. This property is obviously preserved under base change. Assume now that G is a flat k-group scheme. If N is a submodule of a Gmodule M, then N O k [ G ] is a G(k[G])-stable submodule of M O k[G] (by our assumption of flatness). Then we obviously get

(1)and (2)

AM(N) c N 0 k[G]

AN

= (AM)IN.

Together with 2.8 the second equality implies that the G-module structure on N is unique. O n the other hand, if N is a k-submodule of M satisfying (l),then ( 2 )defines a G-module structure on N and N is a G-submodule of M . So the Gsubmodules of M are exactly the k-submodules N satisfying (1). Using 2.8(4) one easily checks:

( 3 ) Let G be a flat k-group scheme. For each homomorphism cp: M + M' of G-modules its kernel ker(cp) and its image im(cp) are G-submodules o M resp. f M'.We get from this that the G-modules form an abelian category (for G flat). Under the same assumption intersections and sums of submodules are again submodules. Note that inductive limits exist in the category of G-modules (for G flat): Just take the inductive limit as k-modules. This is a factor module of the direct sum (which is O.K. by 2.7(1)) where we divide by a sum of images of homomorphisms.2.10 (Fixed Points) Let G be a k-group scheme and M a G-module. Set

(1) M G = { m ~ M I g ( m O 1 ) = m O 1 forall g E G ( A )

Jantzen - Representations of Algebraic Groups

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