K-Theory of Forms. (AM-98), Volume 98 9781400881413

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Annals of Mathematics Studies Number 98

K-THEORY OF FORMS

BY

AN TH O NY BAK

PR IN C E T O N U N IV E R S IT Y PRESS AND U N IV E R S IT Y O F TO K Y O PRESS

PR IN C ET O N , N EW JE R S E Y 1981

Copyright © 1981 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Wu-chung Hsiang, Robert P. Langlands, John Milnor, and Elias M. Stein Corresponding editors: M. F. Atiyah, Hans Grauert, Phillip A. Griffiths, and Louis Nirenberg

Published in Japan exclusively by University of Tokyo Press; In other parts of the world by Princeton University Press 41 William Street Princeton, New Jersey

Printed in the United States of America by Princeton University Press, Princeton, New Jersey

Library of Congress Cataloging in Publication data will be found on the last printed page of this book

In memory of my father

TABLE OF CONTENTS §1.

3 3 5

INTRODUCTION A. G eneral remarks B. Q uadratic m odules C. H erm itian modules D. The n e c e s s ity for refined d efin itio n s

15

§2.

HYPERBOLIC AND METABOLIC MODULES

17

§3.

AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

24

§4.

K-THEORY GROUPS OF NONSINGULAR MODULES A. The K i-functors B. The K 2 -functors C. R elativ izatio n

60 60 62

§5.

HOMOLOGY EXACT SEQUENCES A. Homology groups and cen tral ex ten sio n s B. The re la tiv e seq u en ce of a homomorphism C. The M ayer-V ietoris seq u en ce of a fibre square D. E xcision

70 70 74 78 85

§6.

K-THEORY IN CATEGORIES WITH PRODUCT A. F ib re product cate g o rie s B. T he relativ e seq u en ce of a product p reserving functor C. The M ayer-V ietoris seq u en ce of a fibre square D. E x cisio n

93 93 97 107 110

§7.

K-THEORY OF NONSINGULAR AND PR O JEC TIV E MODULES A. Approximation sq u ares B. A rithm etic and lo c a liz a tio n , com pletion squares C. Conductor and related sq u ares D. F ib re product categ o ries E. R e stric te d d irect products F. Orders

113 114 126 133 139 145 151

COMPARISON EXACT SEQUENCES A. Change of K i-to rsio n B. Change of K 2 -torsion C. K ernels and co kernels of hyperbolic and m etabolic maps

155 155 161 178

§8.

vii

11

66

viii

T A B L E O F CONTENTS

§9.

SCALING AND MORITA THEORY

180

§10.

REDUCTION MODULO A COMPLETE IDEAL

184

§11.

CHANGE OF FORM PARAMETER A. The group S(r/A) B. The group T(r/A)

190 190 206

§12.

INDUCTION THEORY A. Frobenius modules B. Induction machine

232 232 242

§13.

ALTERN ATE DEFINITIONS OF QUADRATIC MODULES

251

§14.

REMARKS ON NOTATION

256

§15.

WALL’S SURGERY GROUPS

258

BIBLIOGRAPHY

259

SU BJECT INDEX

262

NOTATION INDEX

266

K-Theory of Forms

§1. A.

INTRODUCTION

General remarks This book contains a unified treatment of basic materials for the

theories of quadratic, even hermitian, and hermitian forms and their ‘c l a s s i c a l ’ algebraic K-theories. In the last ten years, there has been a great deal of activity in the areas above, especially in the quadratic ca se which has been influenced by geometric surgery and the problem of com­ puting the surgery obstruction groups. One precise aim of the text is to provide fundamental materials for application in [3], [4], [8], and [9] to the problem above. Accordingly, sp ecial attention is paid to free modules with preferred bases (based modules) which are important to the applica­ tion above. A broader aim is to provide a reference for a cc e s s between the areas above, so that an individual working in one area can learn easily what has been done in a neighboring area and to what extent the techniques carry over. In some c a s e s , the foundational materials carry over on a oneto-one basis so that results in one area can be equivalent to those in another. We take care also to compare K-theory and Witt groups arising from different situations; thus in §11, it is shown how these groups are affected in going from quadratic to even hermitian to hermitian forms and in §8, it is shown how these groups are affected in going from projective modules to free modules to based modules. The procedure above of ‘shift­ ing g ears’ is useful not only for comparison purposes, but also for verifica­ tion purposes, sin ce analogous results in different settings can present different degrees of difficulty to verify. The ingredient used to unify the theories of quadratic and hermitian forms is the form parameter A . However, A serves not only a unifying

3

4

K-THEORY OF FORMS

role, but also solves certain deficiencies in the theories of even hermitian and hermitian forms. For example, in the setting of even hermitian or hermitian forms, there is no reduction theorem modulo a nilpotent ideal; but in the refined context of A-quadratic and A-hermitian modules, there is such a theorem. We comment further on this in §1D. A reader acquainted with the literature will realize that sp ecial ca s e s of many of the results here have appeared already in the literature. Four papers of sp ecial merit in this regard are the foundational papers of H. B ass [11], A. Ranicke [22], and C .T .C .W a ll [36], and the paper of R. Sharpe [26] on the quadratic Steinberg group. I was especially influ­ enced by the paper of H. B ass and I would like to thank him for a c c e s s to his early manuscripts. Next, I would like to comment on the origins of the text and in particu­ lar, on the concept of a form parameter. The concept itself was required to classify [1] normal subgroups of unitary groups of even hermitian forms, but its unification role was not immediately evident to me. This occurred only after C .T .C .W a ll suggested that I try to establish stability theorems for quadratic forms which were analogous to those I had established for even hermitian forms. In subsequent investigations to develop a K-theory of even hermitian and hermitian forms, it became evident that the form parameter was also necessary if one wanted to obtain results, such as Mayer-Vietoris sequences, which were analogous to known results for pro­ jective modules. These requirements plus the convenience mentioned above of shifting gears gave rise to the material presented in this book. The manuscript itself was written during a visit to the Universite de Geneve in 1972-73 and the book was announced in [4]. It is a pleasure for me to express my gratitude to H. B a ss who gave very generously of his time and advice during the preparation of my thesis [l], a portion of which appears in §1B, §2, and §3. His influence is con­ tinually evident in this book. I would also like to thank the Universite' de Geneve and my host there M. Kervaire for their hospitality during the writing of the manuscript and W. -C. Hsiang of the Annals Studies for his patient cooperation in the publication of the book.

5

§1. INTRODUCTION

B.

Quadratic modules The novelty in our definitions is the introduction of the form parameter.

Let A be a ring with involution a n a ;

thus I = a and ab = ba . Let

A e center(A) such that AA = 1 . A form parameter A is an additive sub­ group of A such that 1. { a - A a | a £ A i c A C { a | a e A ,a = - Aai 2. aAa C A for all a e A . We denote the minimum (resp. maximum) choice of A often by min (resp. max).

The pair

(A, A)

is called a form ring. If we wish to emphasize the symmetry A we shall write \ a , A )

in place of (A ,A ). A homomorphism \ a , A ) -» ^ (A ',A ')

of form rings is a homomorphism f : A -> A ' of rings with involution such that f(A) = A' and f(A) C A '.

It is easy to check that all the K-theory

groups we shall construct define functors from the category of form rings to the category of abelian groups. Let M be a right A-module. A sesquilinear form on M is a biadditive map

B : MxM -> A

such that B(ma, nb) = aB(m, n)b . We denote the additive group of sesqui­ linear forms on M by Sesq (M). Sesq (M) has an involution B h B

de­

fined by B(m, n) = B(n, m) . B is called A-hermitian if B = AB , and even \-hermitian if B = C+AC for some C e Sesq (M ). A A-quadratic module is a pair (M, B) where M is a right A-module and B e Sesq (M ). To define a morphism of A-quadratic modules we asso ciate to (M ,B) a A-quadratic form qB : M m

A/A , [B(m, m)]

6

K-THEORY OF FORMS

and a \-hermitian form < , >B : MxM

A.

< , >B = B + AB. Then a morphism (M, B ) -> (1VT, BO of A-quadratic modules is a linear map M -> M' which preserves the associated quadratic and hermitian forms. We say that two A-quadratic modules (M, B) and (M', B ') are equal (and write (M, B ) = (M', B ' ) ) if (M, B) and (M ',B') have the same quadratic and hermitian forms. Note that this does not imply B = B '. C lassically, a quadratic form is a O-quadratic module. This requires of course that A = 1 (because 1 - A eA ) and that the involution be trivial (because a - a e A ).

In order to remove the triviality restriction on the

involution, Tits [30] gives a definition of a quadratic form which is nearly equivalent to a min-quadratic module. P recise equivalence occurs when the underlying modules are finitely generated and projective (see 9 .6). The next result shows that even A-hermitian forms are a sp ecial ca se of A-quadratic forms. T H E O R E M 1.1.

A linear map M -> M' is a morphism (M ,B) ->(M', B ') of

max-quadratic modules if and only if it preserves the a ssociated even A-hermitian forms. Proof. L et f : M -» M' be a linear map such that < m ,n > B = , for all m , n e M . The equation implies that B(m, m )-B '(f(m ), f(m)) = -AB(m, m) + AB'(f(m), f(m )). Thus, by definition, B(m, m )-B '(f(m ), f(m)) e max. Hence, qB(m) = q ,(f(n 0 ). We define the orthogonal sum of two A-quadratic modules by (M, B) 1 (M', BO = (M ©IVT, B © B ') . We say that (M ,B) is nonsingular if M is finitely generated, projective and if the map M -> M* = HomA(M, A ), m t - > B , is an isomorphism.

The most important example of a nonsingular quadratic module is the hyperbolic module.

If P is a finitely generated, projective, right A-module,

we define the hyperbolic module

7

§1. INTRODUCTION

H(P) = ( P ® P * ,B p ) where B p ((p, f ), (q, g)) = f(q) . P * = HomA(P , A) is the right A-module on which the action of A is defined by (f •a )(p ) = a (f(p )). One can check easily that there is a canonical isomorphism H(P©Q) = H(P) 1 H (Q ). H(A) is called the hyperbolic plane. R ecall now the definitions of the algebraic K-theory groups KQ(A) and K1(A) defined in [10, IX §1]. M

The dual operator on right A-modules,

M* (if M is finitely generated, projective then the canonical map

M -» M** , m h- (f b> f(m )), is an isomorphism), and the conjugate transpose operator on m atrices, (a^j) ^

induce respectively involutions, i.e.

actions of Z / 2 Z , on KQ(A) and K j(A ).

Let X and Y be involution

invariant subgroups respectively of KQ(A) and K j(A ).

For convenience,

we shall assume that X contains the cla s s of the free module A and that Y contains the c la s s e s of the matrices -1

and -A . We define the

categories q\ a

as follows.

,A ) x

The objects of Q \ a ,A ) x

are all nonsingular A-quadratic

modules (M ,B) such that the c la s s of M in KQ(A )/X vanishes. Morphisms in Q \ a ,A ) x

are defined analogously to morphisms in Q ^(A ,A ).

The objects of 0^(A , A)^ase(j_Y are all nonsingular A-quadratic modules (M, B) such that M is a free module with a preferred (distinguished) basis e l , * - - , e m such that the mxm-matrix (< e ^ ,e j>

) vanishes in K ^ A V Y .

A morphism f : (M ,B) -» (M', B ') is an isomorphism of A-quadratic modules such that the preferred bases on M and M' have the same number of e le­ ments and such that the matrix determined by f and the preferred bases vanishes in K1(A )/Y . The condition that -1 vanishes in KX(A )/Y guaran­ tees that the operation of orthogonal sum in 0^ (A ,A )^

^_Y is commuta­

tive up to isomorphism. The standard preferred basis for the underlying module A © A * of H(A) is e = ( 1 ,0 ) , f = (0, identity).

If H(A) has

K-THEORY OF FORMS

8

the standard preferred basis then we denote it by H(A)based • H(A)based *s c a ^ ec^ vanishes in K1(A )/Y

based hyperbolic plane. The condition that -A guarantees that H(A)based e Q \ a , A>base M' which preserves the A-hermitian forms. The orthogonal sum of two A-hermitian modules is defined by (M, B) 1 (IVf, BO = (M©M' B © B ') . A A-hermitian module (M, B ) is called nonsingular if M erated, projective and if the map

is finitely gen­

M h> M*= HomA(M,A), m h*B(m, ),

is

an isomorphism. Clearly the expression max-hermitian form has the same meaning as the expression -A-hermitian form. The corresponding result for minhermitian forms is If M is finitely gen erated , projective then the expression

T H E O R E M 1 .3 .

min-hermitian form on M has the same meaning as the expression even -A -hermitian form on M . Proof. Suppose B is a min-hermitian form on M . Choose N such that M©N s

An . Let O denote the trivial form on N and set B ' = B © 0 .

Pick a basis e 1,-* * ,e n for An . The nxn-m atrix (B '(e j,e -)) is -A-hermitian, i.e.

(B '(e j, ej)) = -A ^ B ^ e-, ej)), and since B ' is min-

hermitian, the diagonal coefficients B'(e^, e-) C of categories with product is called cofinal if im age(F) is a cofinal subcategory of C . The purpose of this section is to prove that the hyperbolic and metabolic planes A -H (A ) and A -M (A ) are cofinal respectively in Q (A,A) and H (A ,A ). L et (M ,B) be a quadratic (resp. hermitian module). Two elements m, n e M are called orthogonal if < m ,n > B = 0 (resp. B(m ,n) = 0 ). U be a submodule of M. ment of U j.

L et

Let

= im€M|m is orthogonal to every e le ­

U is called totally isotropic if qB and < , >B are trivial

on U (resp. B is trivial on U ). LEMMA 2 .1 .

Suppose (M ,B) is a A-quadratic (resp, A-hermitian) module.

If M = U + V and if each elem ent of U is orthogonal to each elem ent of V then the canonical map (U, B ly ) 1 (V, B|y )

(M, B) , (u ,v )h » u + v ,

is

an isomorphism. Whenever the hypotheses of 2.1 are satisfied we shall write (M ,B) = U l V. Proof. The principle behind the isomorphism in 2.1 for quadratic modules is the following. If B and B ' e Sesq(M) such that B - B '= C -A C

for some

C , then B and B ' determine the same A-quadratic module; namely, the identity map M -> M defines an isomorphism (M ,B) -> (M'f B ') , qB = qB , and < , >B = < , >B /-

because

To prove the lemma for quadratic modules,

one defines C e Sesq (M) such that if both m, n e U or both m, n e V ,

17

18

K-THEORY OF FORMS

then C(m, n) = 0 , and such that if u e U and v e V then C(u, v) = B(u, v) and C (v ,u ) = 0 .

Then B ^ e B l y + C - A C ^ B .

The proof of 2.1 in the hermitian ca se is clear. Suppose that (M ,B) is a hermitian module and that (U, C)

LEMMA 2 .2 .

is a su bspace. If the map M -» M* , m h> B(m, the map U -> U , u h> C (u , Proof. Clearly U flU ^ = 0 . ueU

),

C O R O L L A R Y 2 .3 .

is an isomorphism then

Furthermore, if m e M then we can choose

su ck that the linear functionals B(m,

Hence, m -u e

),

is an isomorphism if and only if M = U 1 U .

) and B(u, ) agree on U .

. Thus, M = U l U ^ . If (M ,B) is a nonsingular A-quadratic (resp.

A-hermitian) module then a subspace (U, C) of (M ,B) is nonsingular if and only if M = U l U i . C O R O L L A R Y 2 .4 .

Suppose (M ,B) is a A-quadratic (resp. A-hermitian)

module such that the map M ^ M *, mi-*B (resp. m^B(m,

) ) is

an isomorphism. L et (M', B ') be another A-quadratic (resp. A-hermitian) module.

Then any morphism f : (M, B ) -> (1VT, B ') is injective and its image

is an orthogonal summand of (M', B ') . Proof. We prove only the quadratic ca s e . The hermitian c a s e is handled similarly. One begins by noting that fm = 0 ==> m = 0 .

> , = 0 => B

Hence, f is injective. Furthermore, applying

2.2 to the subspace (f(M ), < , > , L B

t (M)

)C (M', B^), one can conclude

that f(M) is an orthogonal summand of (1VT, B y) . LEMMA 2 .5 .

Suppose (M, B ) is a A-quadratic (resp. A-hermitian) module

such that the map M -> M* , m ^ < m ,

>B (resp. mhB(m,

) ) , is an iso ­

morphism. If A is finitely generated as a module over its center and if M is finitely generated over A then any endomorphism of (M, B ) is an automorphism. Proof. By 2.4, any endomorphism f is injective and its image is a direct summand of M. Thus, M = M©N for some N. To complete the proof,

19

§2. H YPER BO LIC AND METABOLIC MODULES

it suffices to show that N = 0 .

If p is a maximal ideal in the cen ter(A ),

let Ap denote A localized at p and let A^ = A ^/Jacobson radical (A^). By a well-known principle [10, III 4 .3 ], N = 0 < = > N ®a A^ = 0 for all p , and by Nakayama’s lemma [10,111 2 .2 ], N ®a A^ = 0 N®a A^ = 0 . But, since A^

is semisimple, it follows from the isomorphism M ®a A^

as M®a A^ 0 N®a A'

that N®a A^ = 0 .

Suppose (M ,B) is a nonsingular A--hermitian module. If U

LEMMA 2 .6 .

is a totally isotropic direct summand of M then (M, B) s Proof. The hermitian form B on

uVu

1 A -M (U ). induces in a canonical way a her­

mitian form on U ^ /U . Since U is a direct summand of M , it follows that U is projective. Hence, U* is projective. Thus, the ex a ct sequence I * I 0 -» U ->M->U -> 0 sp lits, and U is a direct summand of M . Write I ^ M = U ©V . The map : V -> U , v B(v, ) is an isomorphism because B is nonsingular.

Let i : U -> U ** , u h ( f ^ f(u )), be the canonical

identification of U with U ** (remember we make U* and U ** into right A-modules via the involution on A ). Then B(u, ) is an isomorphism. Thus, U © V ^ (U © V )* , u + vh>B(u + v, ) is an iso­ morphism (because U

U* , u h> B(u, ) , is trivial) and (U © V , B|U 0 y )

is nonsingular. By 2 .1 , we can pick an orthogonal complement U/ to U©V. Clearly U' C and hence,

, and U '© U =

. The last equation implies U' = U ^ /U ,

(M, B) = U*^/U 1 (U ® V ,B|u 0 v ) . The assertion of the lemma

follows now from 2 .8 below. LEMMA 2 .7 .

Suppose (M ,B) is a nonsingular A-quadratic module. L et

U 6e a totally isotropic direct summand of M . Suppose that and let B '= B | ^ ,.

= U '© U

Then (M, B ) s

(IT, BO 1 A -H (U ).

Proof. As in the proof of 2 .6 , we can write M =

©V and show that

(U ® V ,B|u 0 v ) is nonsingular. By 2 .2 , we can pick an orthogonal

20

K-THEORY OF FORMS

complement ( U ' B ' ) to (U ® V ,B u e v ) . Clearly, IT C U 1 and U /'®U = U 1 . projection on U" The composite U'-> U "© U -------------------------->U" induces an isomorphism (IT, B ') = (U", B " ) of A-quadratic modules. Hence, (M ,B) s

(U', BO 1

(U ©V ,B|u e v ) . The lemma follows now from 2 .8 below. LEMMA 2 .8 .

Suppose (M, B ) is a nonsingular A -quadratic (re s p .

A-hermitian) module.

Then (M ,B) is A-hyperbolic (resp. A-metabolic)

if and only if M contains a totally isotropic direct summand U such that U = U^“ , in which ca se (M, B ) = A -H (U ) (re s p . A -M (U )). Proof. We consider first the ca s e of quadratic modules. The first task is to find a totally isotropic direct complement to U . Write M = U © V . Every direct complement to U in M is of the form !v + h(v)|v Ui. Choose h such that -B(m , n) = < hm, n >B for all m, n e M (we can do this since < ,

>B is nonsingular, see [19]). Then qB(v + h(v)) = B(v, v)

+ B(hv, v) + B(v, hv) = B(v, v) + (B(hv, v) + A B (v, hv)) + (-A B (v, hv) + B(v, hv)) = B(v, v) + < hv, v >B = 0 , and < v + hv, w + hw >B = < v , w >B + < hv, w >B + < v, hw >B = (B(v, w) + < hv, w >B) + (AB(w, v) + A< hw, v >B) = 0 + 0 = 0. Now we suppose that M = U®V and that V is totally isotropic. Let A -H (U ) = (U ©U*, C ) . Define U © V -^ U © V *, u + v +> u + (v) where 0:V-*U*,

v h *< v ,

>B . Then qc (u + (v)) = 0 (v )u = < v, u >B =

(B(u, v) + AB(v, u)) + (-A B (v, u )+ B (v , u)) = B(u, v) + B(v, u) = qB(u + v) and b + A < v 1, u >b = < v , u 1 > b + < u, vx >b =
B .

We consider next the ca s e of hermitian modules. Write M = U©V . De­ fine U © V -> U © U *, u + v h u + (v) , where : V -*■U* , v +> B(v, ) . Let A -M (U ) = (U © U *,C ) where C (f, g) = B ( 0 _1f, cf>~1g) for all f, g e U*. Then < u + (v), u1 + 0 ( v 1)> c = B(v, Uj) + A B ^ , u) + B(v, v^) = B(v, Uj) + B(u, v^)

+

B(v, v 1)

LEMMA 2.9 .

=

B(u + v, Uj

+ Vj) .

Suppose (M ,B) is a nonsingular A-quadratic (resp.

A-hermitian) module.

Then

21

§2. H YPER BO LIC AND METABOLIC MODULES

(M ,B) 1 (M ,-B ) s

A -H (M ) (resp. A -M (M )).

Proof. The diagonal subspace U = l(m,m |meMi of (M ,B) 1 (M ,-B ) is a totally isotropic direct summand such that U =

. Hence, the lemma

follows from 2 . 8 . C o ro lla ry

2.10.

a) A -H (A ) is cofinal in Q (A ,A )X . b) A -H (A ) with a fixed preferred basis is cofinal in Q(A, A)kase(j_Y > 0(A , A)even_|;)asecj_Y , and Q(A, A)(j-s c r_|)ase(j_Y • c)

The metabolic modules A -M (A ) are cofinal in Q (A ,A )X .

d)

Fix a preferred basis for A ©A* and give this preferred basis to

each A-metabolic module A - M ( A ) .

The resulting based metabolic

modules are cofinal in H(A, A>ba s e d _ Y , H(A, A ) e v e n - b a s e d - Y > and H(A, A )jjs c r.|jase{j_Y • Proof, a) and c ) follow from 2 .9 . b)

We consider the b ased -Y c a s e .

The other ca s e s are handled

similarly. Suppose (M, B) e Q(A, A)^ase(j_Y . Forgetting for a moment the preferred bases for M and A -H (A ), we can find a free A-quadratic module (Mx, B x) such that (M, B ) 1 (M1>B 1) s

1 A -H (A ).

Let a de­

note this isomorphism. Pick a preferred basis for IV^ . With respect to the resulting preferred basis for M©Mj and the preferred basis for n

1 A -H (A ),

o determines an element, say x ,

of K ^ A ).

Let N = M©Mj

but choose for N a preferred basis such that the isomorphism o\ N -* 1 A -H (A ) determines the element x-1

of K ^ A ).

Then o L o \ ((M ,B)

2n

1 (M jjB j)) 1 (I 'L B ^ B j) -> 1 A -H (A ) is an isomorphism of based-Y quadratic modules. d) is proved similarly to b). LEMMA 2.11.

For notational purposes, reca ll that M is a functor S(A, A)

* / a -> H(A,A), (P , a) )-» (P © P , ( A' an isomorphism

V vp

)). ^ '/

The assertion is that there is

22

K-THEORY OF FORMS

M(P, a) 1 M(P, - a ) S M(P, a) i H(P) . Proof. By definition H(P) = M(P, 0 ) .

Identify canonically M (P,a) 1

M ( P , - a ) = M ( P ® P , a ® - a ) and M (P,a) 1 H(P) = M ( P ® P ,a ® 0 ) .

Then

the map

(r4~ ) (~ Y ) : P © P © ( P © P ) * ^ P © P © ( P © P ) * defines an isomorphism M ( P © P , a © - a ) - M ( P ® P ,a ® 0 ) . C O R O L L A R Y 2.12.

P ick two preferred bases for An and give (An)

the corresponding dual ba ses. on An ©(An)

Give one of the resulting preferred bases

to M(An, a ) and give the other preferred basis on An ©(An)

to both M(An, - a ) and H(An) .

Then, in any of the based categories in

Corollary 2 .1 0 d ), there is an isomorphism M(An, a) 1 M(An, - a )

££ M(An, a ) 1 H(An) .

Proof. If p : An ®(An)*® A n ®(An)* -> An ® An ®(An ®An)* ,

-■('; ‘ ,) and if o : An ©An ©(An ©An)* -> An © An ©(An ©An)* ,

then p l op defines an isomorphism M(An ,a ) i M(An, - a ) -> M(An; a) 1 H(An) .

§2. H YPER BO LIC AND METABOLIC MODULES

LEMMA 2.13.

23

Give An and A2n preferred ba ses. With resp ect to these

ba ses, let ft and ^

denote matrices corresponding to nonsingu­

lar K-hermitian forms on resp ectiv ely An and A2n . L et M(An,a ) have a preferred basis as in Corollary 2 .1 2 .

Then in any of the based categories

in Corollary 2 .1 0 d ), there is an isomorphism

M(An, a) 1 (An,/3 ) s

(I \

Proof. The matrix (

ft

V M(An, a) 1 (A n,/3)

(An ®An, ^

) 1 (An, /3 ) .

1 defines an isomorphism

r 1/ (A n® An, f t

1 (An,/3 ).

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

Let \ a , A ) be a form ring.

If cr is a matrix (cr^j) with coefficients

a - e A, let a denote the conjugate transpose of o ,

i.e.

o = transpose

(^ij). If we pick a basis for the free right A-module An and the dual basis

n^

for (A )

n

then we can identify the group Aut (A -H (A )) with a subgroup GQ2n(A,A )

of G L2n(A) called the general A -quadratic group. LEM M A 3.1.

A 2n x 2 n

matrix

^ ^ e G L2n(A) belongs to GQ2n(A,A )

if and only if

» C ?)"-(>’

1)

ii) The diagonal coefficients of ya and 8/3 lie in A . 3.1 A

is an immediate consequence of 3.4 below. matrix o with the properties that o = -Aa

cients of o lie in A

and thediagonal coeffi­

is called A-hermitian. It follows

from

3.1 i) and

ii) that the matrices ya and 8/3 in 3.1 ii) are A-hermitian. C O

R O

L L A R Y

3.2.

a) A 2 n x 2 n

matrix ^

^

e GL2n(A) belongs to

GQ2n(A,A ) if and only if i)

8 a + A/3y = 1 8/3 + Aj8

^ ^

their d iffe re n c e

5/

(^1

0/ *S ^~kerm itian.

M ultiplying out, one obtains the ditterence ab o v e is the matrix (

\Sa—I Since 0

(

ya 3 a -I

I .

3fl/

sa tisfies (i), it follow s that the equations of 3.2 a ) ( i ) hold. Thus,

y/8\

I ya __) = [ _

5/8/

\-A/8y

y/3\ ) and the matrices 3/3/

_ _ ya and 8/3

are max-

27

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

_ hermitian, i.e. ya A-hermitian < = >

/ ya y\S\ Thus, I ) is \ -A 0y W the diagonal coefficients of ya and 5/3 lie in A . __ _ _ = -\va and SB =-\SB .

The equations 3.2 show that if a and 8 are invertible n x n matrices then ^

^

S = a~1 . The matrix ^

belongs to GQ2n(A, A)

is called a hyperbolic matrix and is denoted often by

H(a) =

C A)

The equations 3.2 show that if /3 (resp. y ) is an n x n matrix then (o

0

(resp. ^

belongs to GQ2n(A, A) /3 is A-hermitian

(resp. y is A-hermitian). We let EQ 2n(A ,A )

denote the subgroup of GQ2n(A ,A ) generated by all < : and ( o

T-#. )

? )■ (:

;>

such that /3 and y are as above and e is an elementary EQ2n(A ,A ) is called the elementary A-quadratic

matrix (defined below). group.

EQ 2n(A ,A ) has a set of generators called elementary A-quadratic matrices which we describe next. 1 and n and let a e A .

An n x n

Let i and j be two integers between matrix is called elementary if it is

of the kind Ejj(a) = (

The notation

/*

I

v

.

.

.

| ( i^ j ) .

j denotes the matrix with

a

as ( i , j ) ’th

■ i,

coefficient, the identity as diagonal coefficients, and 0 as all other

28

K-THEORY OF FORMS

coefficien ts.

We c a l l a

matrix e le m e n ta r y

2nx2n

A -q u a d ra tic if it is

one of

/I

a;iJ

H(ei j (a )) =

(i^ j)

-aji

1J

(i^j)

(i^ j)

en+i,j =

a ij

-Aa^

i,n+i

0

(a f A)

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

29

(a n+k(b)] = ein+k(ab)

(c)

[H(£ij(a)), en+i>k(b)] = en+j;k(-a b )

(d)

Ui)n+j(a ),s n + u (b)] = H(£jk(-A I b » [ei,n+j(a )’ £n+j,j(b)] = H(eij(ab))ein + i(-A ab a).

3.5

follows with a little computation from the following matrix identities.

3.6 (a)

(b)

(c)

l + 0 y + /3y/3y

(3y/3\

-y fiy

I-y ft)

)

K-THEORY OF FORMS

30

Let q be an involution invariant ideal of A . We define GQ2n(A ,A ,q ) = ker(GQ2n(A,A ) - GQ2n(A/q, A/q)) where A/q denotes the image of A in A/q . We define EQ2n(A ,A ,q ) to be the normal subgroup of EQ2n(A ,A ) generated by all elementary A-quadratic matrices where the off diagonal coefficients lie in q. GQ2n(A ,A ,q ) (resp.

EQ2n(A ,A ,q )) is called the relative or congruence

subgroup (resp. relative elementary subgroup) of level q . Clearly, GQ2n(A>A, A) = GQ2n(A,A ) and EQ2n(A, A, A) = EQ2n(A, A ) . If G is a group and H C G is a normal subgroup then the m ixed com­ mutator group [G,H] of G and H is the subgroup of G generated by all commutators

[g,h] such that g c G

and h e H .

The group [G,G] is

the commutator subgroup of G . G is called connected or perfect if G = [G, G ] . The next result is deduced routinely from 3.5 and 3.6. Details will be left to the reader. 3.7.

If n > 3 then

(a) EQ2n(A, A ,q ) is generated as a normal subgroup of EQ2n(A,A )

c. ?) - c n where

a and /3 = 0 mod q .

(b) EQ2n(A ,A ,q ) = [EQ2n( A ,A ,q ) ,E Q 2n(A, A)].

In particular,

EQ2n(A ,A ) is p e r f e c t . There is a natural embedding

GQ2n(A ,A ) -> GQ2(n+i)(A- A ) , ^ g ^ i-

f a

0

0\

I 2

L_0

0 .J

\0

0

0 1

/

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

31

We defi ne GQ( A, A , q ) = lim GQ2 n( A , A , q ) n

E Q ( A , A, q) = lim E Q 2 n( A , A , q ) . n

S u ppose

Q u a d r a t i c W h i t e h e a d Lemma 3.8.

(y s) (c T h e n in

GQ4 n( A , A )

d

) ' GQ-(A'A)

we have

(Ha) (H*-) e , .I., - v , j

v

- i'

■

¥

"Y

‘I

^

U^A^A^A“TAAt~/ P ro o f. C O

R O

P ro o f.

Straigh tforw ard c om p u tatio n . L L A

R Y

3 .9.

[GQ(A, A , q ), GQ(A, A ) ] = E Q ( A , A , q ) .

C o n s id e r the a b s o l u t e c a s e ( i . e .

q = A ) first.

left-hand s i d e c o n t a i n s the right-hand s i d e .

C onversely, suppose

p-(c P I I = im ag e of

P

3 . 5 s h o w s th at th e

in G Q 4n( A , A )

32

K-THEORY OF FORMS

I 1 p = / —

^----------------.

vTFT/

3 .8 implies (P 1 1)-1 (I i P ) e EQ4n(A, A ) . Thus, if v e GQ2n(A, A) we obtain (for suitable E , E 1 , E 2 e EQ4n(A, A ))

(77

1 I)(P 1 I) = (n 1 I ) ( 1 1 P )

E = ( U P ) ( U 77)E1E = ( I I ( P tt^ E j E - (P77i I ) E 2E 1E = ( P l l ) ( 7 7 i D E ^ j E . We consider the general ca s e next. We use the relativization procedure described in §4C and identify G = GQ(Acx q, Aix A fi q) with GQ(A, A)

k

GQ(A,A,q) and E = EQ(Atxq, Atx A fl q) with EQ(A, A) tx EQ(A, A, q ) . From the absolute c a s e , we obtain that [G, G] = E . But the standard com­ mutator formulas show that G Q(A ,A ,q)].

[G,G] = [GQ(A, A), GQ(A, A)] ix [GQ(A,A),

Hence EQ(A, A, q) = [GQ(A, A), GQ(A, A, q ) ] .

The next theorem extends a result of Sharpe [26, §5] by eliminating the assumptions that A be a root of unity and that the form parameter A = min. In fact, if one examines Sharpe’s proof, one sees that he does not use the latter assumption.

Our proof simplifies the matrix computations in Sharpe’s

paper.

-mm-MHo Let

&)4n =

0 >4

1 •••• 1

L =f \-7T

77 ) + ( ^ 77B 77/ \-ttB V'

three m atrices in (E) proof of 3.13.

VB77 \ ^ en Qne last then one can can replace rep la ce the the last -77BV'B 77 /

)

C

\ (f \

i by >y I --------------1 I ------------- I . T h is com p letes the

w 7 vr/

^roup G GH A ,,A A ,q , q ) in an alogy with the group Next we define the group H ((A analogy

GQ(A, A, q ) . Let a i> •'*>an f A .

G ive the free right A-module A-m odule A n a b asis and Give basis

s is . give (A n)* the dual basis.

L et Let

M0

o (A ( A n) denote the A -m etab olic A-metabolic i ’ ” *’ n

41

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

module such that with respect to the basis of An ®(An) matrix associated to the A-hermitian form on M0

given above, the

_ (An) is

1’ " ’ n

With respect to this basis, we can identify Aut (M_

_ (An) with a

subgroup GH ... a (A, A) a l' ' n of G L0ri(A) called the general hermitian group of M0 „ (An). a l ' " ‘ 'a n is an involution invariant ideal, we define

If q

GHa l , . ,a n( A ,A ,q ) = ker(GHa a (A, A) -* GHa p .. ,afl( A /q ,A /q )) . a x, ... ,an GH q.

_ ( A ,A ,q ) is called the relative or congruence subgroup of level n If b p *•', b € min then M . (An) = M b . b (An). l n ap ,an a i+Dp >a n+Dn 1’

Hence, up to isomorphism, the group GH

(A ,A ,q ) depends only lr " ’ n mod (min). Suppose that A/min is finite,

on the c la s s e s of say of order N, A/min.

and that a p

For n > N ,

is a set of coset representatives for

we define

GH2n(A, A, q) = GHa i . . . ; a N ( 0 . . . ;0(A ,A ) q ) . n-N

There is a natural embedding GH2n(A ,A ,q ) -> GH2 ^n+1^(A, A, q ) , a

(°

0

0

0\

1 ° °V

y

OS

0

0

0 I

0

1/

42

K-THEORY OF FORMS

We define G H (A ,A ,q) = lim GH2n(A ,A ,q ) .

A functor G : ((form rings)) -> ((groups)) is called E -su rjectiv e if given any form ring (A, A) and an involution invariant ideal q of A , the canonical map (commutator subgroup of G(A, A )) -> (commutator sub­ group of G (A /q , A /q )) is surjective. E-surjectivity will play an important role in formulating the exact sequences of §7D. The functor GQ is E -su rjectiv e.

LEMMA 3 .14.

Proof. By 3 .9 the commutator subgroup of GQ(A, A) (resp. G Q (A /q ,A /q )) is EQ (A ,A ) (resp. EQ (A/q, A /q )).

But every generator of E Q (A /q ,A /q )

lifts to a generator of E Q (A ,A ). Q U ES T IO N .

Is

GH

E -s u r j e c t i v e ?

Try first fields of ch aracteristic 2 .

A question related to the one above is Q U ES T IO N .

Find a reasonable set of generators for the commutator sub­

group of GH. The fact that the commutator subgroup is perfect follows from Lemma 2.11 and Lemma 3.15 below. LEMMA 3.15 (B ass).

L et M be an object in a category with product. L et

Gn(M) = Aut(M 1 ••• 1 M) (n times). -» Gn+1(M), a K f l l l j j .

T here is a natural embedding Gn(M)

If G(M) = lim Gfl(M) then the commutator su b ­

group of G(M) is perfect. Proof. Let N = M l ••• 1M (n times). (a 1 cT 1 1 V

If a, f i e Aut(N) then aT^fi~^afi =

1 ( 0 1 1N 1 p - ' r ' i a 1 a“ X 1 1N)(/S 1 1N 1 0 ~ :l) . But accord­

ing to [10, VII 1.8], a l a _ 1 l l N and jS ± 1N ± j8—1 are commutators in Aut ( N 1 N 1 N ) . L et G be a group. A covering or extension V -> G of G is a group V together with a surjective homomorphism V -> G . The homomorphism

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

43

V -> G is called central if its kernel lies in the ce n te r(V ). A central covering f : V -* G is called universal if given any other central covering f': V'-* G , there is a homomorphism g : V -> V ' such that f = f 'g . If V is perfect then it is easy to show that the homomorphism V -> V' is unique (cf. proof of Lemma 3.24 below). Next, we write down the list of identities promised for EQ2n(A ,A ). Then we show that the identities define the perfect, universal, central covering of E Q (A ,A ). We let r-j(a) , £jj(a) , and

j(a) (1 < i, j < n)

denote respectively an upper right hand, lower left hand, and hyperbolic elementary A-quadratic matrix. L e m m a 3.16.

1.

[rij(a), ^ ( b ) ] = 1

2.

[r-j(a), E-^(b)] = Hejk(-A ab)

3.

(li,ji H {k, i\ = 0 ) (i, j, and

(i £ j)

kyG O JjjG ))] = H6ij(ab )rii(-A a b a )

4.

[rjj(b), £ji(a)] = H e ^ b a )? ^ Aaba)

E l.

He^-fa+b) = He-Ca)He^Cb)

E 2.

[Heij(a), Hek£(b)] = 1

E3.

[H e-(a), Hgj^b)] = Hg^(ab)

L I.

£ij(a+b) = £ij(a)£ij(b)

L2.

^ij(a)^rs( b ) - £ rs(b)£ij(a)

L3.

^ ^ ( - A

L4.

[ ^ ( a ) , Hers(b)] = 1

L 5a.

[£y(a), Hejk(b)] = H(£-j(a)) modulo the relations given in 3.16. in the ca se

Thus, if i = j , it is understood that

r - ( a ) (resp. ?jj(a)), the element a fA

(resp. A ) .

S tQ (A ,A ) is called the A-quadratic Steinberg group. T H E O R E M 3 .17.

The canonical map StQ(A, A) -> EQ (A , A), x y (a ) h* Xjj(a) ,

such that x-j = r-j , I - , or He^j , is a p erfect, universal, central covering. The proof of 3.17 will be postponed till the end of the section. A convenient description due to R. Sharpe [26] of StQ (A ,A ) is given as follows.

L et

St (A)

denote the free group generated by the symbols £jj(a) such that i ^ j and a £ A modulo the relations E-(a)e|j(b) = e -(a + b ), [6 -(a ), ek£(b)] = 1 if i ^ i and j ^ k , and [ei j(a), £j£(b)] = e^fab) if i ^ f .

St (A) is called

the Steinberg group. By a theorem of M. Kervaire [20], the canonical map St (A) -» E(A ), Eyfe) ♦-* Eij(a) , is a perfect, universal, central covering. St(A) has an involution defined by £jj(a) = ej i ( a ) which covers that on E (A ). Let

e

h» e

45

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

Mn(A) denote the subgroup of the ring of n xn

matrices Mn(A) consisting of all

a such that a = -Aa and the diagonal coefficients of a lie in A . There is an embedding Mn(A)

Mn+1(A) , a h- (o

o ) , and we let

M(A) = lim Mn(A) .

T H E O R E M 3 .1 8 .

S t Q (A ,A )

is isomorphic to the following group.

The

generators are the symbols n

c

,)•('

p

o

r

1-)

such that y e M(A) , /3 e M(A) , and e HCe'jjAa))

^ij(a)

i— »^ (aejj-A aejj)

( > )

—

rjj(a )

i—

ijj(a)

i— >rCaejj)

(i^ j)

((a e u ) >K a e j j - A a e j j )

( i / j )

.

One constructs in the obvious way an inverse to the map above.

All the necessary verifications are straightforward. The assertions in 3 .1 0 - 3 .1 3 for E Q (A ,A ) are stated so that there are obvious analogous assertions for StQ (A ,A ).

Furthermore, the proofs of

3 .1 0 - 3 .1 3 are made only with manipulations which are valid in StQ (A ,A ). Thus, the analogous assertions for StQ (A,A) are valid. We record next these assertion s. denote the 2 n x 2 n matrix n = ( \A (n times) and let L et

) © •••©[ 0/ \A

77

C -. TH E O R E M 3.1 9 .

,) Mn(A ). We prove now the assertions for Mn( A ). We begin by defining some notation suggested by the monomorphism Mn(A) -» EQ2n(A, A), y If 1 < i ,

j < n , let e -

h-

^

^ .

denote the nxn-m atrix with 1 in the ( i , j ) ’th

coordinate and 0 in all the other coordinates. For i ^ j and a e A , let ?ij(a) = a e - - A a e j j and for i = j and a e A , let ^ j(a ) = a e - . The elements ^ j(a ) generate Mn(A) and satisfy the following relations. 1. £ij(a+b) = £ij(a) + £ij(b ). 2. V a ) + ers(b) = ers(b) + eij(a ) . 3. ^ ( a ) = ^ ( -A a ) . 4.

[fjjCa), £rs (b)] = 0

if

r H i.jl-

55

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

5a.

[ ^ ( a ) , £jk(b)] = ?ik(ab)

if

i, j,

5b.

Kij(a ),6 ji(b)] = £ji(a b -A a b ).

6.

[ ^ ( 3 ) , eik(b)] = £jk(ab) + ?kk(bab) .

and

k

are distinct.

Relations 4-6 show that Mn(A) is E^n(A)-perfect. Thus, it remains to show that Mn(A) is universal for E n(A)-central coverings of Mn(A ). The symbols ?jj(a) and the relations 1-3 form a set of generators and relations for the group Mn(A ).

Furthermore, the symbols £—(a) and the relations

1 -6 present Mn(A) as an E n(A)-group. We outline the rest of the proof. Let p : Y -> Mn(A) be an E n(A)-central covering. F irs t, we show that Y is abelian. Then for each generator ?lj(a) , we pick a lifting y - ( a ) c Y and define, for i , j , p distinct, £P(a) = [yi (1), e -(a )]. The definition is clearly independent of the lifting 1J

H

HJ

y ip (l). We show that for a fixed p, the ^P-(a)satisfy relations

l-5 a ).

Furthermore, it follows from 5a) that if k , i , j , p , q are distinct then (a) = ^?k(a )^ £k j^ ^ = ^ or some z ^ ker P ) tz^ k(a )> ek j ^ =

‘

Thus, we can define unambiguously ^ j(a ) = ?P.(a) where p is any integer such that i , j , l-5 a ).

and p are distinct. Clearly, the fqj(a) satisfy relations

For i ^ p , we define

= ^pi(a) fypp(a )» Ep i^ ^ anc* s bow that

for a fixed p, the ^ j(a ) and ^P(a ) satisfy relations 1 -6 . Furthermore, from 6, it follows that if i , k , p , q are distinct then ??.(a) = £P.(-a) li pi eki(l)3 = (for some z ( ker p ) ^ . ( - 3 ) \ zl^ (a ), eki( l) ] = P .( a ) . Thus, we can define unambiguously ?^(a) = ?P.(a) where p is any integer such that p ^ i .

Clearly, the £-j(a)’s satisfy relations 1 -6 . Since

M (A) is presented by the relations 1 -6 , it follows that there is an E n(A)-homomorphism Mn(A) -> Y , ^ j(a) h> ^ j(a) , such that the diagram Mn( A )

►Y

Mn(A) commutes.

56

K-THEORY OF FORMS

We fill in now the details of the outline above. Y IS A B E L I A N .

For each

pick a lifting y - ( a ) e Y .

It suffices to show that y y (a )

ykP^)

- Yij(a). Suppose

ip j.

Choose

p £ i, j, k or I and choose z e ker p such that y^-(a) = z [yip(l), £pj(a)]. = z[yip(l>yk^C \ [ykj>(b), ep j ( - a ) ] epj(a)l = (for some

Then y ^ a / 1^ Zjekerp)

ztzjyjpCl), ep j( a ) ] = z[y ip( l ) , £pj( a ) ] = yjjCa). Suppose i = j . or I and choose z

Choose p ^ i, k

e ker p such thaty^-(a) = z ypj(-a )

[yPP(a)*£Pi^1)]■ Then yiiyk£(b>=zyPi(-a)[yPP(a)yk£(b>-^k^epi( - l ) ] e pi(l)] = (for some z x f k e r p ) z y pi( - a ) [ z 1ypp(a), £pi( l) ] = y ^ a ) . ekf^k) or j then y - ( a ) = y jj(a).

Next we shall show that if k p i

Choose p ^ i, j, k or I and choose z e ker p such that

Suppose i ^ j .

y ij(a) = z[yip(l) , £pj(a )].

Then y ^ a ) k^ } = z[yip( l ) k^( }, [ekf(b),

epj(-a )]E pj(a)] = (for some z ; ( k e r p ) z i z ^ y ^ i l ) , epj(a)] = yjj(a) . Suppose i = j . Choose p ^ i , k or I and choose z e ker p such that y ^ (a) = z ypi ( - a )[y pp(a ). EpiC1)]-

Then y ^ a ) ^

' = z ypj( - a ) [ypp(a) kf
spj(l)l = yjj(a) . Th e

?jjP (a ) S a t i s f y R e l a t i o n s l-5 a .

For

i, j,

and p distinct,

we define £ -P (a ) = [yjp(lX epj(a )J an= (for

58

K-THEORY OF FORMS

some z «ker p)£pi(-a )£ pj(-a )[z y pp(a), U ^ l ) , epi(-l)]e pi(l)] = £pi(-a>£pj(-a ) frppte)- 6Pi(1)] [ypp(a)> £pj(1)]^ l ( } = hii^a ^ 'p j(-a ) fypp(a )6 p l ( E pj(1)6pi(1)] = (for some z f ker p ) h ^ a ^ ^ - a ) [z ypp(a )y pi(a )y ii(a), epj ( l ) = hu (a)£pj(-a ) [ypp(a), Epj(l)] [£ip(-Aa), epj(l)] [ y ^ a ), epj( l) ] = (5a, 4) hii(a )h jj(a)£ij(-A a ) = (since a = Aa )h ii(a )h jJ(a )h ij ( a ) . _ _ _ eki(l) Set h = h ^ (a b -A a b ) , and suppose that i , j , k , p are distinct. [hkj(a ), £jk(h)]hkk(ab-Aab )~X e ker p and is fixed by E n(A ). Hence, it = [hjk( - ^ ) £ki(1).Ejk(b)6ki(1)] h - 1

[hkj(a )h ij(a ),[e ki(l),E jk(-b )]e jk(b )]h -1

= [hkj(a)h ij(a),Eji(b)ejk(b )]h -1 = Dikj(a), £jk(b)] [h ^ a ). ejk(b)] [hkj(a), Hji(b)]Ejkh - 1 = [hkj(a), Ejk(b)]hik(ab)hkj(a b )[h ^ a ), eji(b) ] ( hkk(ab- ^ b ) hki(ab-A ib )h ii(ab -A -ib r1 = [hkj(a), 6jk(b)] [h ^ a ), Eji(b)] (h^(ab-A ab)h^(ab-A ab))- 1 . Equating the first term with the last, we establish 5b. 6). Define the element ( a ,b ) ^ e ker p by [h^(a), e^Cb)] = hik(ab)hkk(ha b )(a , b )ik . We shall show that the symbol ( ,

is

biadditive and that (a, b)-^ = (a, b ) - ^ ' . Let (a, b) = (a, b)-^ . (a+ax,b) = [hii(a+ aj ).

~ hik^a+ al ^ “ hkk(b (a+ax)b ) = [hj^a), £ik(b)] +

friiOij), £ik(b)] - hik(ab) - hik(a l b) ~ hkk^bab) ~ hkk^b al b^ = (a >b)+ (a i >b) • (a, b+b1) = [hu (a), Eik(b+b1)] - hik(a(b+b1)) - hkk((bTb7)a(b+b1)) = [h ^ a ), Eik(bi )] + [hh (a), eik(b)]Eik(bl> - hik(ab) - hik(ab1) - hkk(bab) - h ^ C b ja b j) - hkk(babi+bi ab) .

However, [h ^ a ), Eik (b)] lk( ^ = [h ^ a ) lk( 1

£ik^b)^

= [h -(a ), Eik(b)] + [hik(ab j), Eik(b)] + [ h ^ b j a b j , £ik(b )]. The last term is zero (by 4), and [hik(ab1), eik(b)] = (3) [h ^ C ^ a ), sik(b)] = (5b) hkk(biab-A b ia b) = h ^ ( b 1ab+bab1) . Making the appropriate substitutions, we see that (a .b + b p = ([hj^a), e ik (b i)]-hik(abi ) - h kk(bl ab l)) + (tbii(a )> eik(b)] - hik(ab) - hkk(bab)) = (a, b p + ( a ,b 2) . Suppose that i , j , k , p are distinct and that a t A . e = 6jk( -l )E ij(b)Ejk(l), e x = 6jj(b)ejk( l) and e 2 = £jk( l ) -

Set Then

59

§3. AUTOMORPHISM GROUPS OF NONSINGULAR MODULES

hii(a ) £ik(b) = hii( a ) [6« S (A /q ,A /q ),

Q(A,A)

induces functors

and G"": F(A , A) -> F (A /q , A/q ).

Using the notation of [10, VII 1.4], we define Kx(A ,q ) = KX(P(A ),G ) KSX(A, A, q ) = K1(S (A ,A ),G /) KF1( A ,A , q ) = K x(F (A ,A ),G " ) . where F = Q or H . The hyperbolic and metabolic functors induce homomorphisms H : Kx(A ,q ) -> KQX(A, A ,q )

and

M : KSx(A ,A ,q )

KHX(A, A ,q )

K-THEORY OF FORMS

62 and we define

WQ1(A ,A ,q ) = coker H WHj CAj A ^ ) = coker M . The fact that we get the same definitions of KQj , e tc ., as above follows from the general nonsense theorem [10, VII 2 .3 ] and the fact that the A -h y p erb o lic m odules

H(An) a re c o fin a l 2 .1 0 in Q(A, A ) and th e n

A -m e ta b o lic m odules 1 * * N’

n(AN+m) a re c o fin a l 2 .1 0 in *' **’

H(A, A ) . Let Y be an involution invariant subgroup of K1(A ). For convenience, we shall assume that Y contains the c la s s e s of the elements - 1 - A e G L x(A ).

and

If F = H or Q , define KFi(A , A ^ e d - Y

= k e r C K F ^ A ) -> K ^ A V Y ) .

One can show as above that KFl ( A>A>based-Y = Kl ( F (A >A)based-Y) • B.

The K2 -functors R ecall the Steinberg group St(A) and the quadratic Steinberg group

StQ(A,A) defined in §3. By [20] and 3.17, they are the universal, perfect, central extensions of respectively E(A) and E Q (A ,A ). Define K2(A) = ker(St(A) -> E(A )) KQ2(A, A ) = ker (StQ(A, A ) -> EQ (A ,A )) . If A/min is finite then by the sentence preceding 3 .1 5 , we know that [GH(A, A), GH(A, A)] is perfect and thus, by Lemma 3.25 has a univer­ sa l, perfect, central extension, say U (A ,A ).

If A/min is finite, define

KH2(A ,A ) = ker(U (A ,A ) -> [GH(A, A ), GH(A, A )]).

The hyperbolic map E(A ) -> E Q (A ,A ), H : K2(A) - KQ2(A, A) . Define

e h> Q

_“~1) * *nc*uces a map

§4. K-THEORY GROUPS OF NONSINGULAR MODULES

63

WQ2(A ,A ) = co k e r(H :K 2( A ) - K Q 2(A ,A )) . Next, we shall develop a formal construction which will allow us to define KH2 without any restriction on A /m in. If G is a group, let X(G) denote the free group on the symbols x(g) such that g f G .

If p : X(G) - G , x(g)

k

g , let X Q(G) = X(G )/[ker p, X (G )].

It is clear that XQ(G)->G is a universal, central, covering of G .

Let

U(G) = [X 0(G ),X 0(G)] H2(G) = ker (U(G) -> G) . If G is perfect then from Lemma 3.25 and from the proof of 3 .2 5 , it follows that U(G) is up to isomorphism the unique universal, perfect, central extension of G . The rule G t-> X(G) defines in the obvious way a functor ((groups)) -* ((groups)).

It follows that the rule G h- U(G) (resp.

G i-> H2(G)) defines a functor ((groups)) -> ((groups)) (resp. ((groups)) ((abelian groups))). LEMMA 4 .1 .

The functors U and H 2 commute with direct limits [10,1 §8 ].

The proof of 4.1 is left as an easy exercise. We follow now B ass [11] in making the following definition. a category with product 1 form a set.

Let C be

in which the isomorphism cla s se s of objects

Assume also that all the morphisms in C are isomorphisms.

If M is an object, define Gn(M) as in Lemma 3.15.

Let G^(M) and

G'(M) denote the commutator subgroup of Gn(M) and G(M) respectively. Define H2(M) = H2(G j (M)) . The action of G^M) on G^(M) by conjuga­ tion induces an action of Gj(M) on H2 (M).

Let K2(M) = H ^ G ^ M ),

H2(M)) = H2(M )/{ ctx- x Ix £H2 (M), aeG j(M )|. LEMMA 4 .2 .

If a, p : M -»*N are isomorphisms in C then K2 (a) = K2 (/3).

Proof. It suffices to consider the ca s e by definition of K2(M).

M = N , /3 = 1M . But K2 (a) = 1

64

K-THEORY OF FORMS

For convenience, let us assume now that C has a trivial object 0 such that M l 0 ~ M for all objects M of C . L et T ran[C ] denote the category whose objects are the isomorphism c la s se s

[M] of objects M

of C . If [M], [N] e Obj (Tran [C]) , let Morph ([M], [N]) = 1[P] |[P] e Obj (Tran [C]) , [M 1 P ] = [N]|. Composition of morphisms is just 1 . By a result of B ass [ 1 0 ,1 §8], Tran[C ] is a directed category and hence if F

is a functor F : T ran[C ] -> ((abelian groups)), one can form the direct

limit lim F [M]. By Lemma 4 .2 , the functor K9 : C -> ((abelian groups)) [m ]

induces a functor K2 : Tran [C] -> ((abelian groups)). We define K2C = lim K2 [M] .

[m ] The next two results together with 2.10, 2.1 1 , and 2.12 show that the definitions of KQ2 and KH2 given at the beginning of the section are compatible with the one given just above. Moreover, the first result, by itself, shows that the definition above of K2 agrees with that given in

6 .12 . L e m m a 4 .3 .

If K2

(M) = H0(Gn(M), H2( M 1 - 1 M ) )

then lim K2

(M) =

h 2 (G'(M)).

Proof. Since Hq and H2 commute with direct limits, it follows that lim R , „(M) = lim H o(GnCM), H2(G^(M)) = HQ(G(M), H2 (G'(M)) and n n H2(GXM))= lim H2(G^(M)). The canonical map H2(G'(M)) - HQ(G(M), n H2(G/(M)) is clearly surjective. Thus, it suffices to show that the action of G(M) on H2(G'(M)) is trivial. L et a e Gn(M) and let x e H2(G^(M)). L et aQ denote the inner automorphism of Gn(M) defined by a . Let m > 2n . The commutative diagram

65

§4. K-THEORY GROUPS OF NONSINGULAR MODULES

H2(a0 ) H2 (Gp(M ))---------------------------------------------

H2 (G'(M))

H (G^(M ))--------------------------------2 m H„(a_ 1 a - 1 i 1 m ,„ ) O

O

M

- H2(G^(M))

~

implies that if H2(aGi a p' 1 l Mm_2n)y = y (y = image of x in H2 (Gm(M))) then H2(aQ 1 1

m_n)y = y-

However, a 1 a -1 e G2n(M) by

M

[10, VII 1.8] and lim G^(M) is perfect by 3.15 .

Hence, for m suitably

large a 1 a -1 1 1 m_ 2n lies in the commutator subgroup of G^(M ). If V M

is a universal, central covering of G^(M) then y = a 1 a -1 i 1 m_ 2n M

to an element of [V ,V ]. Moreover, conjugation by y on H2(GJn(M)) C [V ,V ] corresponds to H2(a0 l a ~ 1 l l

m_ 2n)-

M

But sin ce H2(Gjn(M)) C

center [V, V], it follows that H2 (aQl a” 1 1 1 m_ 2n)y = yyy 1 = Y • M

C O R O L L A R Y 4.4 (B ass).

If C has a cofinal object A then the canoni­

cal homomorphism below is an isomorphism.

H2(G'(A))

k 2(C)

.

Proof. The corollary follows directly from [1 0 ,1 (8 .6 )]. R ecall the metabolic functor M : S(A, A)

H(A, A) defined in §1C.

Define KH2(A ,A ) = K2H (A,A) WH2 (A ,A ) = coker (K2(M): K2S(A, A) -* K2H(A, A)) . It follows from Corollary 4 .4 that the definition just given of KH2 agrees with the one given previously.

Of course, one can also define

K-THEORY OF FORMS

66

KQ2(A, A) = K^Q(A, A) WQ2(A ,A ) = cok er(K 2H ): K2 P ( A ) -> K2Q(A, A)) and then it follows from Corollary 4.4 and Theorem 3 .1 7 that the defini­ tions just given of KQ2 and WQ2 agree with those given previously. It is easy to check that KQ2 , KH2 , etc. define functors ((form rings)) -> ((abelian groups)). If q is an involution invariant ideal of A, the congruence or relative groups

KQ2(A, A ,q )

e tc. will be defined by the relative procedure described immediately below in §4C.

It should be noted that the congruence KQ2-group defined by

Quillen’s methods is a quotient of the one defined above. The quotient can be proper. As usual, if we want to emphasize the symmetry A with respect to which A is defined then we shall write KQ2^(A ,A ) in place of KQ2(A ,A ), etc. C. Relativization In this section, we adapt the relativization procedure of Stein [28] to our situation.

The significance of the procedure rests in the fact that it

can reduce questions about relative groups to questions about absolute groups. We have had already an example of this in the proof of 3 .9 . D E F IN IT IO N 4 . 4 .

Let

r e s p e c t to the e lem e n t id eal of

(A, A) be a form ring s u c h th at A is defin ed with A c c e n t e r (A ). L e t q b e an involu tion invariant

A . A form ideal of l e v e l q of (A, A) is a pair