Algebraic K-Theory 0821841033, 9780821841037

Citation preview

Advances in

SOVIET

MATHEMATICS Volume 4

Algebraic K-Theory

A. A. Suslin Editor

American Mathematical Society

Advances in

10.1090/advsov/004

Volume 4

Algebraic K-Theoiy A. A. Suslin Editor From The Seminar on Algebraic K-Theoiy held at Leningrad State University by A. A. Suslin

American Mathematical SocietyProvidence, Rhode Island

Advances

in

S o v ie t M a t h e m a t ic s

Editorial Committee

V. I. ARNOLD S. G. GINDIKIN V. P. MASLOV Translation edited by A. B. SOSSINSKY

1991 Mathematics Subject Classification. Primary 11R34, 11R52, 11R70, 11S31, 13D15, 14C15, 14C35, 16A54, 18F25, 18G30, 19D45; Secondary 11S99, 14L10, 14M15, 14M17, 19B20, 19D55, 19F05, 19E15, 19L20, 55S25.

ISBN 0-8218-4103-3 ISSN 1051-8037

COPYING AND REPRINTING. Individual readers o f this publication, and nonprofit libraries acting for them, are permitted to make fair use o f the material, such as to copy an article for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment o f the source is given. Republication, systematic copying, or multiple reproduction o f any material in this pub­ lication (including abstracts) is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Manager o f Editorial Services, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940-6248. The appearance o f the code on the first page o f an article in this book indicates the copyright owner’s consent for copying beyond that permitted by Sections 107 or 108 o f the U.S. Copyright Law, provided that the fee o f $1.00 plus $.25 per page for each copy be paid directly to the Copyright Clearance Center, Inc., 27 Congress Street, Salem, Massachusetts 01970. This consent does not extend to other kinds o f copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copyright © 1 991 by the American Mathematical Society. All rights reserved. Printed in the United States o f America. The American Mathematical Society retains all rights except those granted to the United States Government. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. @ This publication was typeset using the American Mathematical Society’s TgX macro system. 10 9 8 7 6 5 4 3 2 1

96 95 94 93 92 91

Contents Preface

v

Part I. Computations in ^-T heory

l

Chow Groups of Quadrics and the Stabilization Conjecture N. A. KARPENKO

3

Simplicial Definition of ^-Operations in Higher AT-Theory A. NENASHEV

9

On Algebraic AVTheory of Generalized Flag Fiber Bundles and Some of Their Twisted Forms I. A. PhNIN

21

On Algebraic K-Theory of Some Principal Homogeneous Spaces I. A. PANIN

47

Af-Theory and ^-Cohomology of Certain Group Varieties A. A. SUSLIN

53

SATi of Division Algebras and Galois Cohomology A. A. SUSLIN

75

Part II. M ilnor ^-T heory

101

On Class Field Theory of Multidimensional Local Fields of Positive Characteristic I. FESENKO

103

On p-Torsion in for Fields of Characteristic p O. IZHBOLDIN

129

Triviality of the Higher Chem Classes in the AT-Theory of Global Fields A. MUSIKHIN and A.A. SUSLIN

145

Milnor’s K 3 of a Discrete Valuation Ring A. A. SUSLIN and V. A. YAROSH

155

iii

Preface The volume contains previously unpublished papers on algebraic AT-theory written by Leningrad authors during the last few years. The papers are di­ vided into two parts. The main topic of Part I is the computation of AT-theory and AT-cohomology for special varieties such as group varieties and their principal homogeneous spaces, flag fiber bundles and their twisted forms, A-operations in higher AT-theory and Chow groups of nonsingular quadrics. Part II deals with Milnor Af-theory: Gersten’s conjecture for K .f of a discrete valuation ring, the absence of p-torsion in K™ for fields of charac­ teristic p , Milnor AMheory and class field theory for multidimensional local fields, the triviality of higher Chem classes for the A"-theory of global fields. A. A. Suslin

PARTI

Computations in K-theory

10.1090/advsov/004/01 ADVANCES IN SOVIET MATHEMATICS Volume 4, 1991

Chow Groups of Quadrics and the Stabilization Conjecture N. A. KARPENKO §0. Introduction Let F be an arbitrary field of characteristic different from 2 and (p be a nondegenerate quadratic form of dimension n over F . By X 9 we denote the projective quadric corresponding to

CKP(X) for each p , and the composition

CHP(X) -» Gp K(X) -» CHP(X) coincides with the multiplication by (-1 )p~ \p —1)!.

□

Corollary 1.2. I f TGPK(X) / 0 for some p , then TCHP(X)

0.

□

The aim of this section is to formulate the part of the results of [Kl] on Gp K(X ) which will be used in §2. Recall some notation: C0( CH!(Xp) — CH'(£7) — 0 is suijective. Considering the commutative square C H '-1^ )

G‘~l K(X¥)

G' K (X J

we see that the lower arrow is suijective too. Since s(y/) = 0, Theorem 1.6 says that TGl~x K(X^) = 0 for all i . In view of 1.7, the last statement implies that G'_1 K(X^) is generated by h‘~l for /—1 f (dim X if/)/2 = p - 1. Thus G' K(X?)) is generated by hl and therefore has no torsion for i f i p . On the other hand, y> is anisotropic and s( 1 , there exists a Ap-dimensional quadratic form (p (over a suitable field F) such that TCHP(X(/I) 0. Construction . Let k be an arbitrary field and t0, tx, . . . , t2p_x be in­ dependent variables, where p > 1. We put F = k(t0, tx, . . . , t2p_ l) , p = (t0, tx, . . . , t2p_ ,), and finally y>= p ® (1, ( - l ) p d e tp ). We claim that

rC H p( ^ ) ^ 0 .

CHOW GROUPS OF QUADRICS

7

P roof. A reduced notation dp for the term ( - l ) p det p will be used. It is sufficient to show that CHP(U) ^ 0, where U is the affine quadric defined over F by the equation p±{dp) ® {tx, . . . , t2p_ {) = ~{dp)t0 . Projecting on the coordinates which are terms of the second summand in the left part of the equation, we obtain a flat morphism U -* A^f- 1 . Since CHP([7) -» CHP({7), where U is the fiber over the generic point, is epimorphic, it suffices to prove that CHP(I7) ^ 0. The variety U is the affine quadric over E = F( x l , , x 2p_x) defined

by the equation pE = ~{ dp) a, where a = tfi+t {x l H----- 'r t2P- \ xiP- \ ■Denote by X the projective closure of U and set Y = X \ U . The projective quadric Y is determined by the form pE , and CHP_1(7) = Z ■hp~l in view of 2.2. We have a sequence consisting of isomorphisms and an equality: CHp(t/) ~ CHP(X )/Im CH p-1(y) = CH?(X)/(Z • hp) = 7XHP(X). We shall prove that the last group is nontrivial. The projective quadric X corresponds to the form pE±((dp)a). Since pE represents a , one can find elements f x, ... , f 2p_i € E* such that pE ~ ( a , / , , . . . , f 2p-\) [L]. Comparison of determinants shows that pEY{{dp)a) ~ {

a

, (-1 )Pf x • • •f 2p_x) .

In view of Proposition 2.5 stated below, the elements a , f x, ... , f 2p_l are algebraically independent over / = k{xx, . . . , x 2p_ l) and generate E over / if the choice of was suitable. Therefore, X answers the conditions of 2.3, and so T CHP(X) ^ 0. □ The last step is to state and prove P roposition 2.5. Suppose that l is a field, t0, tx, . . . , tn are independent

variables, E = l(t0, t{ , ... , tn) , and o = ?0Xq + txx \ a------ b tnx 2n for some Xj € /*. There exist / , , . . . , f n £ E* such that (1)

B;

BJ

J/t

Bm

Z,

B J,i
—►• • • ~ c k- 1 ^ Dk - J u „ ^ £>n ~ Cn+1 >—► in Gv#[2(fc - n) - 1] = S(k - n ; ^#)[1] and put X;(z) = 0 for k - n + 1 < i < k . The sequences o-

-

W

-

X, l xM ) -

0

and 0 — * /(* („ )) -

X t{x) -

* |. ( x ("+ l ’* ~ 1)) — 0

are naturally defined and obviously exact for 0 < i < k . Since the simplices

SIMPLICIAL DEFINITION OF ¿-OPERATIONS

17

x {n+uk and x {n+l k_ V) are of the type considered in the lemma, one computes x t{x{n+uk-x)) * A 0A

A A n—l

B k - i- 1

ËlL a ^ n+l

A

A

® D k_ t o

>D k - i »

i copies

^ i ( X (n+l,k-\)^

= A 0 A • ■■AAn_ l ® - f A

An

An

i copies

for O ^ i ^ k - n - l X

( Y ^ +1

A k - n \X

^"1—

V

( r

) c*

A

a

) ~ A k - n (X (n+U k - \ ) ) “ A 0 A

...

a

A

0

A ^ « -1 ®

‘

‘ 0 D n

Q...o f

’

y---------------------'

A:—« copies

and X;.(x(”+1,fc_1)) = obtain an exact sequence

,fc—i)) — 0 for k - n + 1 < / ^ k , and we -A ,X

®Z,(z) - ~ 0

for 0 < i < k . Now consider the diagram -*•

A A ■■■A A n- i ® x i(z) f 1

(x (n+1,Jfc-1)) >->'

Xj(x )

-

Xi(x{n)) t À

Î A

X,(xw ) *,(* k - n . Thus the square X. (x{n+l’k~l)) ^ A 0 A ■■■AAn_ l ® X i(z)

î X,(x)

Î -

X,(xw )

is a pull-back. (We do not know whether the last conclusion can be deduced from Quillen’s axioms of exact category. It is evidently true for projective modules or vector bundles. In the case of an abstract exact category, this can

A. NENASHEV

18

be added to the list of axioms in §3.) Since the nondegeneracy index of the simplices x (n) and z is one less than that of x and the simplex x (n+x,k~l) is of the type considered in the lemma, one can assume by the induction hypothesis that all maps in the sequence (4.1) for these three simplices are defined naturally. Using the pull-back universal property one obtains the sequence (4.1) for x natural in all variables At , Bj , etc. Its exactness is deduced from that of and x(B), hence we are done. 4.3. On higher-dimensional simplices, the map A k can be defined now in an obvious way. §5. Agreement with the operations on Let R be a commutative ring with identity and J i = 3 s {R) be the cate­ gory of finitely generated projective modules over R . Let H be a group. De­ note by ¿P(H; R) the category of finitely generated projective modules over R with the action of H and by K( H \ R) = R)) its Grothendieck group. Each element p e K ( H ; R) determines a homotopy class of maps B H -* \GJt\ . If p = [tr] - [t] with a: H —>Aut F , t : H —►Aut W being repre­ sentations of the group H in the modules F , W g J ? , then the above class is represented by the map BH —►\GJt\ given by p t \— ►( V , W) e GU (FA • • • A V, FA • • • A F® W, ... , W o ■■■o W ) G G( k ; J t)[ 0], (

VAVA---AV FA

A V® W Wo

a(h)/\ ■•■Aoih)

~

>—*■ -oW

FA F A - ' A F - * 0 A

FA r(/i)o •••ot(A)

A V< g>W -» 0

Wo • • • o W -» OJ

€ G{k\ Jt ) [ 1].

SIMPLICIAL DEFINITION OF ¿-OPERATIONS

19

To check that our A-operations agree with those defined in [Hi] and [Kr], it is now sufficient to verify that the diagram BH

\GJT\ = \S{k;jT)\ JA *|

B H —^

\G(k\ J()\

is homotopy commutative. Since the two maps from BH to \G(k; are actually maps from a simplicial groupoid to a simplicial groupoid, it is sufficient to check homotopy on edges. To show that the last statement is trivial, we verify it for k = 3. From the definition of A3 on edges one deduces that the map A3 o p takes any element h e H to the composition of three edges in | FA F A F ■g(^ A1A-1» FA F A F , F A F ® IF 1A1®TW, F A F ® IF IAg!/,l&1. F A F ® IF

F A F ® IF ,

F ® IFo IF T®Üi, F ® IFo IF 1? T.^ ° .r.^> F ® IFo IF

V ® IFo IF,

WoWoW

Wo IFo IF

WoWoW

r(h)o r(h)o r(h)

WoWoW,

where the two middle products V A V A V in the upper row correspond to the sequences of admissible monos 1

a{h)

,

Tr a{h) Tr l

V > —► Fh F h 7V and F w FAF and hence notation such as 1 A 1 A a{h) makes sense (this follows from understanding what are two vertices in S{ 3; J! ) dividing the edge 1/ w H7 TW. W) ( V °(h) . V,

of G Ji into three edges of 5(3; >£)). On the other hand, A3p maps h to the loop V A V A V g(-)A- W - V A V A V , VAV®W

VAV®W,

V S is an arbitrary principal G-bundle, and R Z(G) (resp. RZ(P)) is Grothendieck’s ring of finite-dimensional linear representation of the group scheme G defined over Z . Remark that Serre’s theorem [Se] establishes a natural isomorphism R Z(G) ^

R C(G)

(resp. RZ(P)

R C(P) ),

where R C(G) (resp. RC{P) ) is the usual complex character ring. Using these isomorphisms, Steinberg’s result [St], and the isomorphism mentioned above, we construct an explicit basis for Af-theory of 3?/P (see 5.4, 5.12). This special basis together with Proposition 3.8 gives a result in the Grassmannian case (see 7.1) relating AT-theory of twisted forms of Grassmannian fiber bundles with that of the associated Azumaya algebra and its power. In §8 we state similar results for twisted forms of generalized flag fiber bundles (see 8.1, 8.3, 8.4, 8.7), which may be proved by the methods of the present paper. 1991 Mathematics Subject Classification. Primary 18F25; Secondary 14M15. © 1991 American Mathematical Society 1051-8037/91 $ 1 .0 0 + $.25 per page

21

I. A. PANIN

22

Theorem 1.8 was inspired by the papers [B, BGG, K l, K2], where a de­ scription of some derived categories was given. This theorem was proved in [PI], where Theorem 7.1 was announced. Some of the results of this paper were obtained independently by Y. Srinivas, J. Weyman, and M. Levine [LSW]. Other interesting results were proved recently by A. Suslin [Su] and independently by M. Levine. The author thanks A. A. Beilinson, M. M. Kapranov, A. S. Merkurev, A. L. Smirnov, A. A. Suslin, D.-N. Verma, and others for useful discus­ sions. Theorem 7.1 in the characteristic zero case was obtained together with A. S. Merku/ev. §0. Preliminaries In this paper all schemes are assumed to be quasicompact and locally Noetherian (not necessarily Noetherian or separated). The following propo­ sition was used in [P3] to define a direct image homomorphism (1.1.1). ^

0.1. P roposition. Let f : T ^ > S be aflat projective morphism o f schemes, be a locally free coherent sheaf on T . Then there exists an exact sequence

o f locally free coherent sheaves on T such that R! f f j T ) = 0 for i > 0. P roof. Since / is projective, there exist a projective fiber bundle p: P5( f ) -► S and a closed embedding i: T Ps (fT) such that / = p o i . Denote by tf( 1) the canonical invertible sheaf on PS(J?) and put S ' = i* n 0 (see [H, III.5.2]). It follows from [Mu, II.5.2] that the sheaf f f 5 ? m ) is locally free for such n . Consider the natural homomorphisms

(*,m) Since S is the union of finitely many open affines, there exists an m 0 such that am is an epimorphism for m ^ m 0 [H, III.8.8]. Dualizing (*, m ) and tensoring with s We obtain an exact sequence of locally free coherent sheaves

o _►

&T

/*(/,(-2*®w)v)®

-+jrm-► o.

Tensoring it with the sheaf J ? , we obtain the sequence 0 —* It will be shown now that Rj

■

—>3 am —►0.

(**, m )

= 0 for i > 0, m » 0. Indeed,

= * '/.( /* ( /.( -

ON ALGEBRAIC

-THEORY OF FLAG BUNDLES

23

because f ( ^ f ® m)w is locally free as noted above. Since R l ®JI) = 0 for i > 0 and m ^ n0 (see the beginning of the proof), the sequence ( **, m ) satisfies the conclusion of the proposition for m » 0. □ 0.2. D efinition . A diagram a

a!

^

y

B'

ô

a A B of Abelian groups and their homomorphisms is called a retract diagram of the homomorphism a on the homomorphism a , if it satisfies the following conditions: (1) 0 o y = idA ', (3) a ° y = e ° a ; (2) S o e = idB ; (4) ô oa = ao p .

0.3. Lemma. I f the homomorphism a is an epimorphism (resp. a mo­ nomorphism), then the homomorphism a is also an epimorphism (resp. a monomorphism). In particular, i f a is an isomorphism, then a is also an isomorphism. 0.4. Let A = ® „>0 An be a graded anticommutative ring, B = ® n>0 Bn be a graded anticommutative ^-algebra, /* : A —>B be a structure homo­ morphism, and f t : B -> A be a homomorphism of graded ^-modules. De­ fine the ^ 0-module homomorphisms a : B 0 —* Hom^ (B0, A0) = B* and X: B 0 ^ B 0 —►Hom^ (B , B) in the following way: a(x) - Çx for each x € B0 , where Çx(y) = f ( x ■y ) , and X(b{ ® b2)(x) = f m(b2x) • bx for each b{, b2 e BQ, x e B . 0.5. P roposition. I f there exist a; , bt e B 0 such that l(± ai*bi

= idB

’

\i=i

then the following statements hold: (1) B0 is a finitely generated projective A 0-module, B is a finitely gener­ ated projective graded A-module; (2) b 0 ®AqA ^ B is an isomorphism; (3) a is an isomorphism (see 0.4); (4) each o f the families {a;}"=1, {è.}"=1 generates A0-module B0. P roof. We have the relations x =

idB(x)

=

A( \i= 1

ai

® bA (x) J

=

^ i= 1

/„ ( x • b i ) ' a r

I. A. PANIN

24

Consider two diagrams A"0

An

^An

and B0

B0

A

- ►B

where 8 , p are natural homomorphisms, (p , (/ are the homomorphisms of graded ^-modules given by (pie^ = ai , ¥{x) = Y , f * { x - b i) - ei i= 1

for each x € B ; y>0 = (p\A„ , y/Q= y/\B . The equality x = Y i U x - bi) - a i ;=i

shows that (p o y/ = id and 0 . If y G 9°{h, T ) , then / ^ ( y ) G 9°{S) (see [Mu, II.5.2]). Therefore one has a well-defined exact functor of the direct image K:9>{h, T) -+9°( S). This functor defines homomorphisms Kx{3P{h, Tj) —►K f S ) (/ > 0). In view of [Q, 7.2.7] and Proposition 0.1, the natural inclusion 9°{h, T) 9s (T) induces isomorphisms Ki{9°{h, T)) K f T ) (/ > 0). Composing this isomorphism with the homomorphism K f & l h , Tj) -*■ K f S ) defined above, we obtain a homomorphism which will be denoted by hm: K,{T) ^ W S ) .

(1.1.1)

1.2. The functor h*: 9s(S) —>9°(T) is exact, hence it induces homomor­ phisms h*: K f S ) —►K f T ) . The homomorpism ht is a homomorphism of (5’)-modules, i.e., ht {h*(a) -x) = a ■ht (x) (1-2.1) for any a G Kj ( S) , x e K f T ) . If the square

T — » S is Cartesian and g is a flat morphism, then the base change formula holds: g* ° K = K o ( g ) * : K f T ) -

KfS').

(1.2.2)

1.3. D efinition . Define the A'0(S)-linear pairing ( , ) on K 0(T) by putting (see 0.4 and 0.6): (a, b) = ht (a-b). 1.4. D efinition . Each element z G K0(T x s T) defines homomorphisms z*: Kf T ) —>K{(T)

(i> 0)

I. A. PANIN

26

by the formula z^{x) = p x fp*2 {x) • z ) , where p l , p2: T x s T T are natural projections, x e KfiT) (see also [Ma]). 1.5. It is clear that (zx + z2)t = z x + z 2 . It follows from (1.2.1) that the correspondence z t-> z t defines a group homomorphism K 0( T x s T ) - > E n d KAS)(Kt (T)). 1.6. Lemma. I f a, b e K 0(T) and a® b =f p\{a) ■ p*2 (b) e K0(T x s T ) , then the homomorphism ( a m ) . : Kt{ T ) K ^ T ) is given by the formula ( a B ^ J x ) = h*[h,(b • x)] • a.

( 1 .6 . 1 )

I f x € K 0( T ) , then this formula may be rewritten in the following way: ( a^b) ^( x) = (b, x) - a.

( 1 .6 .2 )

P roof. These statements follow from (1.2.1), (1.2.2), and 1.3. 1.7. R emark. Suppose that the structure sheaf ¿fA(7yS) of the diagonal

A(T/S) c T x s T has a finite locally free resolution on T x s T . Then the sheaf ¿^(Tys) defines an element of K0(T x s T) which will be denoted by t^A(77S)] • ft is easy to see that for i ^ 0 [^A(77S)]* = ^ K :(T) ' 1.8. T heorem. Let f : X —>Y be a flat projective morphism o f schemes. Suppose the structure sheaf ^ ^ x / y ) ° f diagonal A(X/Y) c X x y X has a finite locally free resolution on X x y X . Suppose the following relation holds in K0(X x Y X): n (A /F )i = ^ 2 a i ® bj i=0

for some at , bi e K 0(X) . Then a) K 0(X) is a finitely generated projective K0( Y)-module; b) each o f the families {a,} ”=1 (resp. {¿,}”=1) generates the K 0(Y)-module K 0{X)\ c) the homomorphism a: K f X) —♦ Hom^ (y)(K0( X ) , AT0( 7 )) induced by the pairing ( , ) (see 1.3) is an isomorphism; d) the natural homomorphisms K 0(X) K (y) Kp(Y) - > Kp{X) are isomor­ phisms. B = ® n>0 Kn(X), and let f : A -> B (resp. f t : B -* A) be the inverse image homomorphism (resp. direct image homomorphism). It follows from 1.5 and (1.6.1) that the composition P roof. Put A = ® n^ K n{Y),

K 0(X) ®Ko{Y) K 0(X) -

K0(X x Y X ) ^ EndK' {Y]( Kf X) )

ON ALGEBRAIC ^-THEORY OF FLAG BUNDLES

27

coincides with the homomorphism X: B 0 A^ B 0 ►EndA(B) from 0.4. Relations (1.6.1) and 1.7 show that X^

at ® b^j =i dB .

Hence the theorem now follows from Proposition 0.5. □ 1.9. R emark. The statement 1.8 d) was conjectured by A. A. Suslin. The projectivity of K0(X) over K 0(Y) was noticed by O. T. Izhboldin. 1.10. Corollary. Under the assumptions o f Theorem 1.8, the following statements are equivalent (see 0 .6 ): (1) the family {a(} ”=1 is a free basis o f the K 0(Y)-module KQ( X) ; (2) the family {¿;}" =1 is a free basis o f the K0(Y)-module K 0( X ) ; (3) {at , bj) = Sjj, where StJ is the Kronecker symbol.

Moreover, the family o f graded K 0(Y)-module endomorphisms

gives a family o f mutually orthogonal projections in End^ {Y){ K f X ) ) . §2.

A - t h e o r y o f p r o je c t iv e f ib e r b u n d le s

In this section we reprove the following result of Quillen. 2.1. T heorem [Q]. Let Y be a quasicompact scheme, a locally free sheaf o f rank n on Y , P Y(&) the projective fiber bundle associated with . Then the natural homomorphism

KQ{p y( r ) ) ®*o(y) Kt (Y) -

k ,{p y(r ) )

is an isomorphism and K ^ P y ^ ) ) = ®"J 01 K 0(Y) ■[&{—ij\, where ¿ f(-l) is the tautological locally free sheaf o f rank 1 on P y(«?). P roof. Let p : Py(l?) —*■Y be the natural projection and a \ p*($?*) —* tf(l) be the canonical epimorphism. Put / = ker(a). It is well known that pt