Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

914

Max L. Warshauer

The Witt Group of Degree k Maps and Asymmetric Inner Product Spaces

Springer-Verlag Berlin Heidelberg New York 1982

Author

Max L.Warshauer Department of Mathematics, Southwest Texas State University San Marcos, TX ?8666, USA

AMS Subject Classifications (1980): 10 C 05

ISBN 3-540-11201-4 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-11201-4 Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. �9 by Springer-Verlag Berlin Heidelberg 1982 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

TABLE OF CONTENTS

INTRODUCTION ................................................ 1

CONVENTIONS ........................................ ~ .......... II

Chapter I THE WITT RING ...................................... 12

i. Setting and notation .............................. 12

2. Inner products .................................... 14

3. Constructing new inner products out of old ........ 23

4. The symmetry operator ............................... 26

5. The Witt equivalence relation ..................... 33

6. Anisotropic representatives ....................... 41

Chapter II WITT INVARIANTS ................................. 47

i. Prime ideals ....................................... 48

2. Hilbert symbols .................................... 51

3. Rank .............................................. 54

4. Diagonalization and the discriminant .............. 55

5. Signatures ......................................... 66

Chapter III POLYNOMIALS .................................... 70

Chapter IV WITT GROUP OF A FIELD .......................... 79

i. Decomposition by characteristic polynomial ........ 79

2. The trace lemma ................................... 90

3. Computing Witt groups ................................ 100

4. Torsion in W(-k,F) ................................ 105

Chapter V THE SQUARING MAP .................................. 108

I. Scharlau~s transfer ................................ !ii

2. The exact octagon over a field .................... 123

IV

Chapter VI THE BOUNDARY ~..~ ...................... , ...... . .... . 133

i. The boundary homomorphism ........................ 133

2. Reducing to the maximal order ..................... 142

3. Computing the local boundary ~(D,P) for the

maximal order ..................................... 152

4. Computing the cokernel of ~(D} ................... 167

Chapter VII NON MAXIMAL ORDERS ........................... 184

i. Traces and canonical localizers . .................. 184

2. Normal extensions ................................ 192

3. Computing trace for finite fields ................ 198

4. The conductor and T(M~ ........................... 203

Chapter VIII THE GLOBAL BOUNDARY ......................... 207

i. The coupling invariants ........................ ~. 207

2. The boundary is onto ............................. 210

Chapter IX A DETAILED ANALYSIS OF THE OCTAGON ............ 226

i. The involutions .................................. 226

2. The map I : We(k2,F) § We(-k,F) .............. 229 e

3. The map d : We(-k,F) § A(F) .................. 238 s

4. The map Se: We(k,F) § We(k2,F) ............... 241

5. The map me: A(F) + We(k,F) ................... 242

Chapter X THE OCTAGON OVER Z ............................. 249

NOTATION .................................................. 259

REFERENCES ................................................ 264

INDEX ..................................................... 267

INTRODUCTION

In these notes our goal has been to develop the algebraic

machinery for the study of the Witt group W(k,I) of degree k

mapping structures and the Witt group of asymmetric inner product

spaces. We are particularly interested in their relationship which

arises in an exact octagon which is studied both for a field F

and for the integers Z We show that this octagon is the

appropriate generalization of the Scharlau transfer sequence [!~n 201].

We have tried to develop the properties of these Witt groups

in a self-contained and complete manner in order to make this

accessible to a larger audience, When references are given they

generally are specific including page numbers. However, we should

mention two general references. First, The Algebraic Theory of

Quadratic Forms by T.Y. Lam [Lm] develops the Witt group over a

field of characteristic not equal to 2, Second, Note____~s o_~n th___ee W itt

Classification of Hermitian Innerproduct Spaces over a Ring of Algebraic

Integers by P.E. Conner [C] discusses Hermitian forms and the Witt

group over an algebraic number field and ring of integers therein.

Together these should provide any background material the reader

might need.

Although the viewpoint we take is entirely algebraic, we would

be remiss if we did not mention its topological motivation. Much

of this work originated in our efforts to explain and exposit

the important work of N.W. Stoltzfus, Unravellin~ the Integral

Knot Concordance Group [Sf-l]. We shall describe this topological

connection momentarily. First however we need to describe the

algebraic objects at hand.

Our object is to define and study a Witt group W(k,I) of

degree k maps, where k is an integer, D is the underlying ring,

and I is a D-module. This group consists of Witt equivalence

classes of triples (M,B,/) satisfying:

(i) B: M x M + I is an I-valued inner product defined on

the D-module M.

(2) /:M § M is a D-module homomorphism of M satisfying

B(/x,/y) = kB(x,y). We refer to 1 as a map of degree k.

A triple (M,B,/) is metabolic (Witt equivalent to zero) if there

is an /-invariant submodule NC M with N = N ~. Here N ~ is the

orthogonal complement. This enables us to define the Witt

equivalence relation ~ by:

(M,B,/) ~ (MI,BI,/I) iff (M ~ MI, B ~9 -B I, Z ~) /I )

is metabolic. The witt equivalence class of (M,B,/) is denoted

[M,B,/]. The operation of direct sum ~ makes this collection of

equivalence classes into a group W(k,I). The identity is the Witt

equivalence class of metabolic triples.

We also develop the basic properties of asymmetric inner

product spaces (M,B) where no symmetry requirement is placed on the

inner product B. The key to understanding these asymmetric inner

products is a "symmetry operator" s satisfying B(x,y) = B(y,sx)

for all x,y e M. As above we define the notion of metabolic by

(M,B) is metabolic if there is an s-invariant subspace N C M

with N = N ~. This leads us to define Witt equivalence as before,

and there results the asymmetric Witt group.

There is a very interesting relationship between this asymmetric

Witt group and the Witt group of degree k maps. This comes from

the squaring map S: W(k,D) + W(k2,D) given by [M,B,/] ~ [M,B,/2].

For D = F a field an eight term exact sequence is developed from

S. This octagon involves the Witt group of asymmetric inner products

A(F) just described.

As a special case of the octagon we obtain the transfer sequence

of Scharlau, Elman and Lain ~m 20~ and ~,L 23-2~. This appears as

an exact octagon in which several terms vanish. We are able to reinter-

pret the kernel and cokernel of this transfer sequence. Thus, these

Witt group W(k,F) and A(F) are an appropriate generalization of the

classical Witt group W(F), at least in so far as relating to and explai-

ning the Scharlau transfer sequence.

We should like to prove exactness in the octagon over Z. in

order to continue we have a boundary sequence which relates

W(k,Z) to W(k,Q)

O ~ W(k,Z) + W(k,Q) ~ W(k,Q/Z)

When k = • this sequence is short exact. Using this we are able

to prove exactness in the octagon over Z when k = • by comparing

the octagon to the octagon over Q.

AS we have said, in these notes we are developing the algebraic

machinery to study the Witt group of degree k maps and the

asymmetric Witt group. We should discuss the topological motivation

for this work. When k = +i the Witt group W(+I,Z) is the crucial

element determining the bordism groups A n of orientation preserving

diffeomorphisms of n-dimensional closed oriented smooth manifolds.

Medrano introduced the appropriate Witt invariant as follows.

Let f: M 2n + M 2n be an orientation preserving diffeomorphism of a

closed, oriented, smooth 2n-dimensional manifold M 2n. We consider

then the pair (M2n,f) in A2n. The degree +i mapping triple

associated to this pair is

(H n (M2n;z)/torsion , B, f*)

where B(x,y) = e. ((x ~ y) ~ [M2n]) , E, is augmentation, [M 2n]

is the fundamental class, ~ is cup product, ~ is cap product,

f* is the induced map on cohomology. If (M2n,f) bounds, this

triple is metabolic. Thus there is an induced homomorphism

I: A2n + W(+I,Z).

The task of computing the bordism groups A2n. was completed

by Kreck [K] who showed that this Witt invariant was essentially

the only invariant for bordism of diffeomorphisms.

More generally, given a closed oriented 2n-dimensional manifold

together with a map 1 of degree k, the corresponding Witt triple

(M,B,I) as above satisfies B(s = kB(x,y) �9 We are thus

led to examine the Witt group W(k,Z). The Witt group of asymmetric inner

product spaces arises in Quinn's work [Q] on open book decomposition.

The relation of this to the Witt group arising in Kreck's ~ork above

is discussed by Stoltzfus in The Algebraic Relationship Between

Quinn's Invariant of Open Book Decomposition and the Isometric

Structure of the Monodromy [Sf-21. in this he gives a geometric

application of the exact octagon obtained in Chapter X.

The exact octagon (renamed "the eight fold way") has also been

extended to the setting of the surgery obstruction groups by A.Ranicki,

L. Taylor, and B. Williams. For a further discussion of the use

of quadratic forms and the Witt group in topology the reader is

referred to Alexander, Conner, Hamrick, Odd Order Group Actions and

Witt Classification of Inner Products [ACH], and Stoltzfus

Unravelling the Integral Knot Concordance Group [Sf-l~.

We now describe briefly the organization of these notes. Chapter I

lays out the inner product spaces and Witt groups we will be studying.

We continue our study of the Wilt group in Chapter II by describing Witt

invariants which will be used to compute the Witt group in many cases.

These invariants include rank mod 2, discriminant, and signatures.

In Chapter III we study the characteristic and minimal polynomial

of the degree k map l in a degree k mapping structure (M,B,~).

This study is used in Chapter IV where we compute the Witt group

W(k,F) for F a field. This is done by decomposing w(k,F)

according to the characteristic polynomial of s and A(F)

according to the characteristic polynomial of s.

In Chapter V we develop an 8 term exact octagon which relates

W(k,F) to A(F).

A(F)

S W+ 1 W+I (k,F) -~ W+I (k2,F) § (-k,F)

~" w-l(-k F) + W_I(k2,F ) S W_I( k , § ,F)

A(F)

S is the squaring map [M,B,/] § [M,B,/2]. We prove again the

Scharlau, Elman, Lam transfer sequence, and see that the exact octagon

we develop is its appropriate generalization. In order to study this

octagon over Z, we relate W(k,Z) to W(k,Q) by a boundary

sequence in Chapter VI.

Now we place an additional requirement on the degree k mapping

structure [M,B,/] , namely:

(*) 1 satisfies the monic integral irreducible polynomial f(t] .

The resulting Witt group of triples satisfying the additional

requirement (*) is denoted W(k,Z;f) .

The action of l induces a Z[t]/(f(t)) - module structure on

M. To simplify the notation let S = Z[t]/(f(t)). Note that S is

only an order in the Dedekind ring of integers O(E) of the algebraic

,lumber field E = Q[t]/(f(t)) . This order S may not be the maximal

order O(E).

The first step to understanding W(k,Z;f) or W(k,Z;S) (the

same thing) is to study the group w(k,Z;D) for D = O(E) the

maximal order. This group consists of Witt equivalence classes of

inner product spaces (M,B) in which M is a finitely generated

torsion free D-module. This is in contrast to W(k,Z;S) in which

we only insist that the module structure of M lifts to the order

S = Z [t]/(f(t)) .

In Chapter VI we are interested only in W(k,Z;D) and the

resultant boundary sequence for the maximal order. We read this

boundary sequence on the Hermitian level, where the - involution

on E is given by t + kt -I and t -I + k-lt. One uses the

following commutative diagram.

(O) O + H(A-I(D/Z)) § H(E) § H(E/A-I(D/Z))

Ct +t +t

~(D) w(k,Q/Z;D) 0 + w(k,Z;D) ~ w(k,Q;D)

~-l(D/Z) denotes the inverse different of D over Z. The vertical

isomorphisms denoted by t are induced by the trace of E over Q.

Thus the method employed for computing W(k,Z;D) is to study

the corresponding boundary sequence in the isomorphic Hermitian

case. The image of ~(D) is the group H(E/~-I(D/Z)) which is

computed as follows:

t g tr H(E/A-I(D/Z)) § W(k,Q/Z;D) ~ (~ w(k,Fp;D/P) § ~ H(D/P)

Here we sum over all - invariant maximal ideals P in D. The

isomorphism g is induced by selecting a generator I/p for the

p-torsion in Q/Z. The trace map on finite fields from D/P to

F induces the isomorphism tr with the Hermitian groups P

H (D/~).

We use the letter M to denote - invariant maximal ideals in

S. In order to study W(k,Z;S) one must use the following

commutative diagram:

3 (D) 0 -~ H(A-I(D/Z)) + H(E) -~ H(E/A-I(D/Z))

% t r t ~(D) ~ t g ~ tr

0 + W(k,Z;D) § W(k,Q;D) * W(k,Q/Z;D) + E) w(k,Fp;D/P)§ ~) H(D/P)

fl + f2 ~(S) + f3 g tr % tr

0 + W(k,Z;S) § W(k,Q;S) + W(k,Q/Z;S) + E) W(k,Fp;S/M)§ ~) H(s/M)

First one computes W(k,Z;D) for the maximal order by going

to the Hermitian level and reading ~ (D) in the group ~) H(D/P).

Then one forgets the D-module structure and remembers only the

S-module structure via the maps f. to gain a computation for 1

W(k,Z;S).

Thus in Chapter VII we study non-maximal orders. Let us be

explicit here in describing the key problems involved.

At every prime P in D there exists a canonically defined

element p(F) in E/A-I(D/Z) with the following properties.

(i) The map of O(E) + E/A-I(D/Z) given by

+ I p(P) induces an embedding of the residue field.

(2) We also have the map Z ~ Q/Z given by

n § n/p which induces an embedding of

F = Z/pZ + Q/Z. P

The element p(P) is canonical in the sense that it makes the

following diagram commute.

0 (EyP § E/A -I (D/Z)

%tr %t

F § Q/Z P

The horizontal maps were just described.

map induced by the finite field trace.

by the number field trace.

tr again denotes the

t denotes the map induced

Thus we see that it is precisely these elements p(P) which

determine the isomorphism tr -I o g o t identifying H(E/~-I(D/Z))

with (~ H(D/P). If we wish to use the commutative diagram just P =

discussed to compute W(k,Z;S) , we must therefore study those

elements p(P). For it is in terms of these elements that one reads

the local boundary

~(D,P) : H(E) + H(E/~-I (D/Z)) + ~ H (D/P) + H (D/P) . P =

in such a way as to make our diagram commute. The last map is

projection to the pth coordinate.

The first two Sections of Chapter VII are due to Conner. In

these we present his theorems which develop the fundamental properties

of these canonical localizers p(P). We complete our study of

W(k,Z;S) for non-maximal orders by discussing the finite field trace

tr, and the maps f~. l

In Chapter 8 we finish our discussion of the boundary homomor-

phism. This includes the notion of coupling from Stoltzfus [Sf-l]

between various ~(D), and a proof that the boundary

2: W(k,Q) § W(k,Q/Z) is onto when k = •

In Chapter IX the terms and maps in the octagon are studied

in detail. This, together with the information about the boundary

map enables us to prove exactness in the octagon over Z in

Chapter X.

10

The idea to study this problem and the possibilities inherent

in the program we have undertaken comes from Professor P.E. Conner.

It is thus a pleasure to thank him for his help in this project with-

out which the notes would ner have been written. The author feels

fortunate to have had the opportunity to study under Professor Conner,

whom we thank not only for his invaluable ideas, but also for his

patience and understanding.

Further, for many of the ideas herein, he should receive credit.

The author is also grateful to Professor Stoltzfus for numerous

conversations throughout this project. We also thank Professor

Dan Shapiro at Ohio State University for his help and suggestions,

not only on this paper but also when the author was just beginning

to study mathematics. We also express our appreciation to Professors

Cordes and Butts at Louisiana State University; to Professor

A. Liulevicius at the University of Chicago; and to Professor

A. Ross at Ohio State University for their interest in the author

at various stages of his career.

Finally, the author wishes to express his gratitude to his

wife, Hiroko, and parents Dr. and Mrs. Albert Warshauer who have

given their constant support and encouragement.

Chapter I THE WITT RING

In this chapter we define the algebraic structures which are to

be studied. Section 1 begins by describing the setting for our inner

product spaces, and includes a brief discussion of prime ideals and

valuations. In Section 2 we develop the elementary properties of

these inner product spaces. Section 3 shows how to construct new

inner product spaces out of old. In particular the operations of

direct sum and tensor product are discussed. These operations later

become addition and multiplication in the Witt ring.

Since our inner products need not be symmetric we are led to

examine a symmetry operator in Section 4. An inner product space

comprises part of the data of a degree k mapping structure which is

defined in Section 5. A Witt equivalence relation is then defined

on these structures.

Section 6 is concerned with selecting from each Witt equivalence

class a certain "anisotropic" representative. We show that in certain

cases this representative is unique.

i. Setting and notation

We are interested in developing a Witt classification for triples

(M,B,Z) where B: M x M + Z is a Z - valued inner product and

s M § M is a map of degree k. This means B(Ix,ly) = kB(x,y) for

all x, y E M. In order to accomplish this one must investigate

algebraic number fields and the Dedekind ring of integers in these

number fields. Further, we must carefully study the role of non-

maximal orders in the Dedekind ring of integers. We begin by des-

cribing the setting.

13

Let D be a Dedekind domain [O'M 52] together with an involution

- The quotient field of D is E. The involution - extends to a

Galois automorphism of E and we denote the fixed field of - by F.

It may happen that F = E; in fact this is precisely the case when -

is the trivial involution.

We shall also use the symbol O(E) to denote the Dedekind ring

of integers D in E. Of course O(F) = O(E) ~ F. Let S be an order

in D. This means S is a subring of D, containing i, with the same

rank as D [B 88]. We shall be particularly interested in principal

orders S of the form S = Z[t,t-l]/(f(t)) = Z[@], where f(t) is

a monic, integral irreducible polynomial. Thus S is an order in O(E),

where E is the algebraic number field Q[t]/(f(t)). Although our pri-

mary interest is in these principal orders in algebraic number fields,

the theory we shall develop applies to arbitrary orders in a Dedekind

domain.

Let p be a prime ideal in O(E) = D. Then p/~ O(F) = P

will denote the corresponding prime ideal in O(F) and p~ S = M will

be the corresponding prime ideal in S. The image of p under the

involution - is denoted by p . p is also prime, hence maximal since

D is a Dedekind domain [O'M 49]. Let I be a - invariant frac-

tional ideal in D. Since I factors uniquely into a product of

n. prime ideals, I = ~p i I , n i e Z, it follows that I C I if and only

1 if ~ = I.

Associated to a prime P in O(F) or p in O(E) is a discrete

non-Archimidean valuation IIp' respectively IIp [O'M i]. F

lies over P, meaning p/~ O(F) = P if and only if I Ip extends I Ip.

Now 1 Ip is a homomorphism from the units in E, E*, onto a

cyclic, multiplicative subgroup of R +. We may thus form OF(P) and

OE(P), the local rings of integers associated to the primes P and

P. we have

14

OE(P% = {w s E: lWIp _< i} .

We also have O(E) = /p%OECP~.

In the local ring of inteaers OE(P) ~ associated to a prime P

is a maximal ideal

mCP) = { w s E: lWIp < i} .

In fact m(P) is a principal ideal generated by some element ~ s OE(P).

Any such z generating m(P) is called a local uniformizer. Two such

clearly have as quotient a local unit.

It is also useful for what follows to think of the exponential

version of I ! p �9 Following [B,S 23] we denote this by Vp .

Vp : E* + Z is given by:

Vp(X) = n means Ixl p = 171 n P

We now have m(~ = {w s E: ~ (w) > 0} If y ~ F*, ~ (y) is

the exponent to which the prime ideal P is raised in the factorization

of yO(E).

We shall review prime ideals and algebraic number theory results

in Chapter II. Our object here has only been to establish notation

for the inner product spaces which we are now ready to describe.

2. Inner products

Again D = O(E) is a Dedekin~ domain. We consider pairs M,D which

15

satisfy either:

(a) M is a finitely generated torsion free D-module (and hence

projective since D is a Dedekind domain [R-I 85]) with K = I

a - invariant fractional ideal in D or

(b) M is a finitely generated torsion D-module with K = E/I

where E is the quotient field of D and I is a - invariant fractional

ideal in D.

We are interested in studying the D-module HomD(M,K), where

M,K satisfy either (a) or (b) above. The D-module structure of

HomD(M,K) is now given by defining:

df(x) = f(dx) for d s D, x s M, f s HOmD(M,K) .

Definition 2.1 A K-valued inner product space (M,B) over D is

a finitely generated D-module M, together with a non-singular bi-

additive mapping

B: M• M+ K

satisfying B(dx,y) = B(x,dy) = dB(x,y) for all x,y s M, d ~ D.

B is linear in the first variable, conjugate linear in the second

variable. Again, M and K are assumed to satisfy one of the standard

assumptions (a) or (b) above.

It is still necessary to say what it means for B to be non-sin-

gular.

Definition 2.2 The map B: M • M + K is non-singular

16

provided the adjoint map AdRB: M + HOmD(M,K) is a D-module iso-

morphism. By AdRB w_ee are denoting the right adjoint map, namely

AdRB(X) = B(-,x).

The left adjoint map, AdLB is similarly defined by AdLB(X) =

B(x,-). We must conjugate in order to have AdLB(X) e HomD(M,K) ,

ie. to make AdLB(X) D-linear.

We have left out any symmetry requirements on B. This is taken

care of by:

Definition 2.3 An inner product space (M,B) i__ss u Hermitian

provided B satisfies B(x,y) = uB(y,x) for all x,y e M, u fixed u e D.

Since B(x,y) = uuB(x,y), it follows that uu = 1 and u is a unit

in D of norm 1 [S 60]. We see that 1 Hermitian is the usual notion

of Hermitian, while -i Hermitian corresponds to skew-Hermitian.

When the involution - is trivial, meaning the identity, 1

Hermitian is symmetric since B(x,y) = B(y,x). Similarly, we define

skew-symmetric, and u symmetric in the case that - is the identity.

Let us now return to study the module M in case (a), namely M

is a finitely generated projective D-module.

We form the vector space M ~D E = V over E.

Definition 2.4 The rank of a finitely generated torsion free

D-module M i_~s the dimension of the vector space M | E over E, E being

the quotient field of D.

17

Thus viewed M is a D-lattice in V [O'M 209]. Hence, M splits n

as a direct sum M = ~9 A. , n = rank M, where each A. is a fractional i= 1 i I

ideal in D. In fact, [O'M 212], there is the splitting:

M = AlZ 1 ~ Dz 2 ~ ... ~ Dz n

where A 1 = fractional ideal in D;

A 2 = A 3 = ... = A n = D.

{z i} is a basis for V.

Since M splits as a sum of fractional ideals, and Hom D is additive

over direct sums, we are reduced to studying HOmD(A,I), where A

is a fractional ideal in D.

Lemma 2.5 Let A, I be fractional ideals in D, with I a

- invariant fractional ideal. Then the map

T : A-II + HomD(A,I )

given by

x + y (x) with T (x)(c) = cx

is a D-module isomorphism. Here the D-module structure on HomD(A,I)

is as previously defined.

Proof: First observe that T(x) e HomD(A,I). To see this, note

18

that T(x) is clearly D-linear, and T is a D-module homomorphism

since:

T(dx) (C) = c(d--x) = (cd)(~) = T(X) (dc) = (dT(x)) (C).

We must show that T is an isomorphism.

(a) T is i-I: Suppose T(x) = T(y) �9 Then cx = cy, for all

c 8 A. Let c ~ 0, and cancel to obtain x = y, hence x = y.

(b) T is onto: Let f ~ HOmD(A,I). We must show f ~ image T .

Tensoring with E, we extend f to HomD(E,E) , where E is the quotient

field of D. Since the Lemma is clearly true for E, it follows that

f(c) = x0 c, for x0 e E. But x0 c ~ I for all c s A. Hence 2 0 s A-II,

and x 0 ~ ~-i~ = ~-iI. Therefore, f(c) = T(x0), and f ~ image T .

Theorem 2.6 Let M be a finitely generated torsion free D-module,

and I a - invariant maximal ideal in D. Then there is a canonical

D-module isomorphism

r M ~ HomD(HOmD(M,I)I))

given by

(x) (f) = f-U~Y �9

Proof: Recall again the module structure on Y E HOmD(HOmD(M,I),I))

is given by: (dy) (f) = y(df) , where (df) (x) = f(dx). By the remarks

immediately preceding Lemma 2.5, it suffices to prove the theorem for

M = A a fractional ideal.

19

We apply Lemma 2.5 twice to obtain an isomorphism ~ . 1 T

r A = (A-II)I-- + HomD(A-II,I) + HomD(HOmD(A,I),I)). ~ is given by

the composition. We have then:

%(x) (f) = 7(x) (m) where T(m) = f, so f(x) = x m

= x m

= f (x)

as claimed. We again observe that r is a D-module isomorphism

since:

r (f) = f(dx) = (df) (x) = r (df) = ((de) (x)) (f) []

We now wish to establish this result in case (b), namely

when M is a finitely generated torsion D-module, with K = E/I,

and I = ~ as usual. In order to do this we simplify matters by

recalling the decomposition of finitely generated torsion modules

over a Dedekind domain. This is done as follows. Let P be a prime

ideal in D, and let M(P) denote the localization of M at P , so

M(P) is a torsion module over the principal ideal domain (p.i.d.)

D ( ~ .

M(P) = M localized at P = {x E M: P nx = 0 for some n}

We first decompose M as M = ~ M(P). Then using the structure theorem

for modules over a p.i,d., M is isomorphic to a direct sum of cyclic

modules with each cyclic module of the form D/~ some P , i s Z.

Since Hom D is additive over direct sums, without loss of generality

2O

we may assume M is a cyclic module, M = D/P i.

In order to study HomD(M,E/I) , we begin by simplifying E/I.

Suppose the fractional ideal I factors as

I = Pli... Pk k , where the Pi are maximal ideals in D.

Let ~i be a uniformizer for Pi" Then in the notation above,

I(P) = I localized at p = IS -I where S = D -P . Since I =

it is clear then that

P11 ... Pk k ,

i. I(Pj) = z 3D(p).

We can now simplify K = E/I.

Lemma 2.7 E/I = �9 E/I(P). P

Proof: P P

f is clearly well-defined, and a homomorphism.

(a) f is i-i: Suppose f(e + I) = 0. Then e e I(P)

P. Thus e E I, by [O'M 46]. Hence, f is i-i.

(b) f is onto: Consider ~ (a i + I(Pi)) s ~ E/I(P). i P

apply the Strong Approximation Theorem, [O'M 42]. Letting I I i

denote the Pi-adic valuation we can find x e E with Ix - ail i

I I(P i) I i' at the finite set of i when (a) a i 4 I(P i) or (b)

I(P i) @ D(Pi) ; with Ixli ~ 1 otherwise. It follows that

f(x + I) = e(ai + I(Pi)),and f is onto. Q l

Define f: E/I § �9 E/I(P) by e + I + ~ (e + I(P)).

for all

We now

Notice that the only summand of E/I with P-torsion is E/I(P).

21

Thus, when M = D/P i , we may apply Lemma 2.7 to obtain an isomorphism:

HOmD(M,E/I ) = HOmD(M,E/I(P)).

We further identify E/I(P) ~ E/zJD(P). Here I(P) = ~JD(P), where

z is a uniformizer for P as before. Multiplication by z -j then

gives the isomorphism E/zJD(P) = E/D(P)

The module structure of M lifts to D(P) , and we claim there is

a D(P)-module isomorphism T: M(P) § HOmD(p) (M(P) ,E/D(P)). In

order to define T, it suffices to consider the case M(P) = D/~ i, since

M(F) is a direct sum of cyclic modules. Let x s M be a generator

for D/P i. Then define T(x) by:

T(x) (cx) = c/z I where c E D(P), x e M(P).

As in Lemma 2 .5, T(dx) (cx) = d c/~ l = (dT)(x) (cx) defines T over M.

is clearly an isomorphism. Combining these isomorphisms over

all cyclic module summands of M, we obtain:

Lemma 2.8 Let M be a finitely generated torsion D-module. Then

there is a canonical isomorphism M = HomD(M,E/I).

Exactly as Theorem 2.6 was proved, we now have:

Theorem 2.9 Let M be a finitely generated torsion D-module,

I a - invariant fractional ideal. Then there is a canonical D-

module isomorphism:

22

~: M + HomD(HOmD(M,E/I),E/I)

given by

r = f(x) []

For the purposes of the next section, we also need the following

propositions which describe the relationship of | to Hom .

Proposition 2.10 There is a canonical isomorphism

4: HomD(A,HOmD(B,C) + HomD(A~ B, C), where A, B, C are D-modules.

Proof: [R-I 25] Define 9 by (~f)(a~ b) = f(a)(b). The inverse

of ~ is given by (~-lg) (a) (b) = g(a~ b) F~

Preposition 2.11 Let X I, X 2, YI" Y2 be finitely generated

projective D-modules. Then

HomD(XI,Y I) Q HomD(X2,Y 2) ~ HomD(Xl| X 2 , Y1 | Y2 )-

The isomorphism is given by

feD g ~ f g , where (f g) (Xl~ x 2) = f(x I) ~ g(x 2) ,

and f g is extended to Xl@ X 2 bilinearly.

Proof: For X i, Yj free, the assertion is clear using bases.

23

Now, if Xi, Yj are projective, then each is a direct summand of

a free. Hom and ~ are additive over finite direct sums. Hence

the isomorphism for free splits into isomorphisms for the summands.

3. Constructing new inner products out of old

3.1 Direct Sums

Let (M,B) and (MI,B I) be two K-valued inner product spaces. The

easiest way to construct a new inner product space is to form the sum.

(M,B) ~ (MI,B I) + (M ~ M I, B ~ B I)

Here (B ~ BI) ((x,y), (z,w)) = B(x,z) + Bl[Y,W). It is clear that

B ~ B 1 is an inner product since the adjoint map

AdR(B �9 B I) : M ~ M 1 + HomD(M ~ MI,K)

splits as AdRB @ AdRB I.

3.2 Tensor Products

The next operation on an inner product space is ~ . Let (M,B)

and (MI,B I) be two type (a) inner product spaces. In other words,

M and M 1 are both finitely generated projective D-modules. Assume that:

B: M x M + Y1 and BI: M1 • M1 § Y2 "

We have the adjoint isomorphisms:

24

AdRB: M ~ HomD(M,Y l) and AdRBI: M 1 + HomD(MI,Y2).

Taking the tensor product of these, we obtain by Proposition l.ll

AdR(B O B I) =

AdRB Q AdRBI: M ~ M 1 HomD(M,Y l) ~ Hom D(MI,Y 2)

= H~ | MI' Y1 ~ Y2 )"

This shows that the adjoint of B O B 1 is an isomorphism, and hence

(M ~ MI, B | BI) is a Y1 ~ Y2 - valued inner product space. We can

identify Y1 Q Y2 with the product of ideals, YIY2 .

3.3 Scaling an inner product

There is the operation of scaling an inner product.

Let (M,B) be a K-valued inner product space with d e E*.

denotes the units in E, ie. E* = E - {0}, since E is a field.

Clearly, (M,dB)

(dB) (x,y) = d B(x,y).

product, namely

E*

is a dK-valued inner product space, where

We may view this as a special case of tensor

(M,B) • (D,B d) = (M,dB), where Bd(X,y) = dxy.

3.4 Tensoring with the quotient field of D

Given an inner product space (M,B), with M of type (a), we

can form (M,B) ~D E = (M | E, B | i) , where we now denote the

extension of B to the quotient field by B O i. When there is no

confusion, we will write B | 1 = B.

25

3.5 The discriminant inner product space

Let (M,B) be a D-valued inner product space, and suppose

M = A ~ D~... ~ D with n factors, where n is the rank of M. We form

th the n-- exterior power, AnM = A, with inner product

AnB: AnM x AnM + D defined by

AnB(XlA...AXn , y! A ...Ay n) = determinant (aij),

where the matrix (aij) is given by aij = B(xi,xj).

In order to verify that this is an inner product, one again

needs to check that the adjoint AdR(AnB) is an isomorphism, [B 30].

We then call (AnM, AnB) the discriminant inner product space.

Comment 3.6 The adjoint map of B, AdRB or simply Ad B,

Ad B: M + HomD(M,I) is an isomorphism by hypothesis. Taking the

th A n (Hom D n-- exterior power, An(Ad B) : AnM + (M,I)) is an isomorphism.

n-i However, HomD(M,I) -~ ~) HomD(D,I) ~) HomD(A,I)= ~) I �9 A-II.

i=l i=l

An(HomD(M,I)) ~ I ~ ... I ~ A-II. However, I | ... | ~-i I Thus

is not in general isomorphic to HomD(AnM , I) so that AnB is not in

general non-singular.

However, for I = D, the Dedekind ring of integers,

Ad(AnB) = An(Ad B) : A nM ~ HomD(AnM, D)

will be an isomorphism by the above, and (AnM,AnB) is indeed an

inner product space.

We note that M is free as a D-module if and only if the ideal

is principal. Thus the discriminant inner product space yields infor-

mation about the structure of the D-module M.

26

We may apply the operation of 3.4, tensoring with the quotient

field, to the discriminant inner product space. AnB ~ 1 is then

multiplication by a fixed x 0 s E. Thus associated with an inner

product space (M,B) is a pair (x0,A) , where A = AnM, and x 0 is as

described.

This x 0 specifies the adjoint isomorphism,

AdRAnB: A + HomD(A,D) = A-ID, by AnB(al,a2 ) = x0ala 2 ,

with x 0 unique in F*/NE*, where F* is the fixed field F of -,

and NE* denotes the multiplicative group of norms of elements in E*.

Hence, x0A = A-ID~ ie. x0AA = D.

4. The symmetry operator

In Definitions 2.1 and 2.2 of an inner product space, and non-

singular mapping, we made use of the right adjoint operator Ad R. In

this section, we relate the two adjoint operators Ad R and Ad L-

Using Theorems 2.6 and 2.9, this is done as follows.

Theorem 4.1 Let (M,B) be an inner product space of either type

(a) or (b). Let B: M • M + K satisfy B(dx,y) = B(x,dy) = dB(x,y).

Then the right adjoint map AdRB is an isomorphism if and only if the

left adjoint map is.

Proof: Let ~ denote the canonical isomorphism

: M +Hom ~Hom ~M,K) ,K)

27

of Theorems 2.6 and 2.9, given by ~(x)(f) = f(x).

Assume AdRB is an isomorphism. We can thus identify

M = HOmD(M,K) via AdRB. Here y e M is identified with B(-,y)

HomD(M,K).

The isomorphism r is now given by:

AdRB r M + HomD(HOmD(M,K),K) § HomD(M,K)

x § Cx where Cx(y) = B(x,y) ,

in other words r is Ad B. The converse follows similarly. D L

Corollary 4.2 Let (M,B) be an inner product space. Then we

can define a unique D-linear isomorphism s: M + M by the equation

B(x,y) = B(y,sx).

Proof: AdLB(X) = ~ s HomD(M,K). Since B is an inner

product, we define s(x) by :

AdR(SX) = B(x,-) = B(-,sx).

s is an isomorphism by Theorem 4.1. []

Notation 4.3 We shall reserve the letter s for this map which

is related to the symmetry of B as described above. It is precisely

this map s which enables us to work with non-symmetric inner product

spaces.

Let N be a subspace of M. We say that N is s invariant provided

28

s (N) C N.

Proposition 4.4 N i__{s s invariant if and only if sN = N.

Proof: Sufficiency is clear.

In order to prove necessity, suppose s(N) C N. Then we can form

an ascending chain of submodules of M, T. = {m s M: si(m) s N}, 1

Ti+ 1 ~ T i. Since D is Noetherian, and M is finitely generated, M

is Noetherian, IS 47]. Hence, this chain terminates. Suppose

Ti= T N for i ! N.

Claim: T 0 = T 1 = ... = T N, and hence sN = N.

It clearly suffices to show that T i ~ Ti+ 1 implies Ti+ 1 ~ Ti+2.

i+l Suppose then that T i ~ Ti+ I. Let m 0 e Ti+ 1 - T i. Then s (m 0)

N and sl(m 0) # N. s is an isomorphism, so there exists mle M

with sm I = m 0. Hence, mle Ti+ 2. If mle Ti+l, then m0C T i

which is a contradiction. Thus, mls Ti+ 2 - Ti+ I.

We observe that Proposition 4.4 can be proved for any ring R

in place of D, with R not necessarily Noetherian. Namely, let s

be an isomorphism s: M § M of a finitely generated R-modules, with

R not necessarily Noetherian.

Claim: If NCM is a submodule with sN C N, then sN = N.

We sketch the proof using the following form of Nakayama's

Lemma: Assume M is finitely generated. Then M = IM implies

I + Ann(M)= R where Ann(M)denotes the annihilator of M.

Consider the diagram :

S

0 + M + M + 0

+ + ^

S

M/N ~ M/N + 0

29

It suffices to show ~ is I-i. View M/N as an R[x]-module, where x

acts as { Suppose x m 0 = 0. We want to show m 0 = 0. Let I = (x).

Since ~ is onto, M/N = I(M/N). Hence, by Nakayama, above, I + Ann (M/N)

= R[x]. Write 1 = rx + t, where r ~ R[x], t s Ann (M/N). Then

1. m 0 = rx- m 0 + t- m 0 = 0, so m 0 = 0. Thus ~ is i-i. Q

For a subspace N of M, we define

N R = {v 6 M: B(n,v) = 0 for all n E N} .

Thus N R is the kernel of AdRB restricted to N, AdRB: M + HOmD(N,K) .

We call N R the right orthogonal complement of N. The underlying

inner product space is understood. Similarly,

N L = {v E M: B(v,n) = 0 for all n s N} .

Now, let N be s invariant. Since sN = N, it follows that

N L = N R. We denote this common orthogonal complement N ~ .

Note 4.5 N ~ is only defined when N i__ss s invariant.

30

Proposition 4.6 If (M,B) is an inner product space of type (a) ,

meaning M is torsion free, then N L and N R are direct summands of M.

Proof: Consider the exact sequence:

0 § N L § M -~ M/N L -~ 0

It suffices to show that M/N L is torsion free, hence projective over

the Dedekind domain D. Then the sequence splits and N L is a summand.

Suppose to the contrary M/N L has torsion. Then there exists

x ~ M, d ~ 0, d s D, with x ~ N L and dx e N L. So B(dx,y) = dB(x,y) =

0 for all y ~ N. Thus B(x,y) = 0 for all y e N since D is a domain.

Hence x ~ N L , contradiction. Thus M/N L is torsion free. []

We have already remarked that if N is s invariant, then N L = N R-

The above shows that N L and N R are always summands. These two con-

ditions turn out to give the converse.

Proposition 4.7 Let (M,B) be an inner product space with M

torsion free. Let N be a summand of M. Then N is s invariant if and

only if N L = N R.

Proof: As observed before Note 4.5, necessity is clear.

order to prove sufficiency, consider the two exact sequences:

AdLB 0 § N L + M § HomD(N,K) ~ 0

AdRB 0 § (NL) R ~ M + HOmD(NL,K) + 0 .

In

31

AdLB and AdRB are onto because N is a summand. Clearly we have

rank N L = rank (HOmD(NL,K)). Thus, by the two sequences above,

rank (NL) R = rank (HOmD(N,K)) = rank N. However, B(ns = 0

for all ns ~ N L, n E N. Thus N C (N L) R" Since N is a summand,

ranks equal, it follows that N = (NL) R. Similarly, N = (NR) L.

We now wish to show that sN C N = (NL) R. So we compute

B(nl,sn) = B(n,ns = 0 for all ns E N L, since N L = N R by hypothesis.

Thus sN C (N L) R = N as desired. [~

Proposition 4.8 N = N L if and only if N = N R.

Proof: Suppose N = N L. Then B(m,n) = 0 for all m, n e N.

Thus, n ~ N R and N C N R. By Proposition 4.6, N = N L implies N is

a summand. Clearly, as in Proposition 4.7, rank N = rank N R, so that

N = N R-

The converse is similar, r3

Theorem 4.1 stated that for an inner product space (M,B),

both AdRB and AdLB are isomorphisms. This enabled us to define the

symmetry operator s, with B-~-x,y) = B(y,sx) . In a like manner, one

can see:

Proposition 4.9 Fixin~ an inner product space (M,B), let

s M + M, be a D-linear operator. Then there is a unique

s M + M, D-linear, with B(x,s = B(s

Notation: Z* is called the adjoint operator of Z, not to be

confused with the adjoint maps AdRB, AdLB previously defined.

32

Proof: For fixed x, we have the map B(x,s s HOmD(M,K).

Since B is non-singular, Ad L is an isomorphism and we can find a

unique w s M such that:

B(x,s = B(w,-)

Define s = w. Then B(x,Zy) = B(s ~* is clearly a well-

defined, D-linear map with the desired properties. Ad L being an

isomorphism shows ~* is unique. C]

One can similarly define a left adjoint operator of s

by insisting B(ix,y) = B(x,*s . It then follows that if

s s HomD(M,M)

B( (*i) *x,y) = B(x,*iy) = B(s

Since B is non-singular, (~s =s Similarly, *(~*)

these two * operations are inverses of one another.

Further, we note that

=~, so that

B(s = B(x,s

= B(i*y,sx)

= B(y,s

= B(s-llsx,y)

Thus s = s-lis. Also we have

33

B(x,y) = B(y,sx)

= B (sx, sy)

= B (s*sx,y)

-i -i so that s* = s Similarly *s = s

in

We summarize these remarks

Theorem 4.10 The correspondences s + Z* and s + *s of the

al~ebra of linear operators on M are inverses. They satisfy

s-l~s -i ~** = r s* = *s = s

Thus s = s for all s 8 HomD(M,M) if and only if s is central.

An easy calculation also shows (iT)* = T*s so that when s

is central the correspondence s + s gives an anti-involution of

the algebra of linear operators on M.

5. The Witt equivalence relation

Definition 5.1 Let k b_~e ~iven, k s D. A degree k mapping

structure over D is a triple (M,B,s satisfying:

(a) (M,B) is an inner product space over D .

(b) ~: M + M is a D-linear map satisfying

B(s163 = kB(x,y) for all x,y e M .

is called a map o_ff de~ree k. For all future considerations, we

shall assume henceforth that k e Z.

In the case that M is torsion free, and k ~ 0 , it follows

34

that ~ is non-singular. To see this suppose i(x) = 0. Then

B(Zx, Zy) = kB(x,y) = 0. Since B has values in K = I a

- invariant fractional ideal, we cancel k and conclude B(x,y) = 0

for all y s M. However, B is non-singular, so that x = 0, and

is i-I.

The Witt equivalence relation for degree k mapping structures

comes from:

Definition 5.2 A degree k mapping structure (M,B,s is

metabolic if there is a D-submodule N C M satisfying:

(a) N is s invariant

(b) N is s invariant

(c) N = N ~

When (M,B,s is metabolic, an N satisfying (a) , (b) and (c)

above will be called a metabolizer for M. We shall also refer to

the triple (M,B,s as M, when B and s are understood, and

speak of M as being metabolic.

The operation of direct sum on inner product spaces extends to

degree k mapping structures. The notation:

(V,B,Z) e (W,B', s = (V �9 W,B �9 B',s ~ s

It is clear that ~ ~ s is of degree k with respect to B ~ B'

At this point we can introduce a relation -- on degree k mapping

structures by:

(V,B,s ~ (W,B',s when (V ~ W,B ~ -B',s �9 s

35

is metabolic. In what follows, we will show - is an equivalence

relation, called the Witt equivalence relation. This agrees with the

usual notion of Witt equivalence [M-H] when no s is present.

Notation: W+I(k,K) denotes degree k mapping structures

(M,B,s modulo ~ with values in K; together with the additional

requirement that the symmetry operator s is the identity map,

so B is symmetric. The underlying ring D is understood.

Emphasis is given to the range of the inner product, namely K.

Similarly, W-I(k,K) is Witt equivalence classes of triples

(M,B,i) having B skew-symmetric.

When there is no s , so that we are taking inner product spaces

modulo ~ , without condition (a) of 5.2, we denote the symmetric

equivalence classes W+I(K), the skew-symmetric W-I(K). Finally,

when no i is present, with no symmetry requirement at all placed

on B , the resulting Witt group is denoted A(K).

We have also defined the notion of B being u Hermitian. We

write Hu(K) to denote those Witt equivalence classes [M,B] for

which B(x,y) = uB(y,x).

In summary, our notation is:

W +I : B symmetric

-i W : B skew-symmetric

A : no symmetry requirements on B (B asymmetric)

H : B u Hermitian u

If we write w+l(K), we are thinking of pairs (M,B); if we write

W+I(k,K), we are thinking of triples (M,B,s with B(s163 = kB(x,y).

36

The K means that B: M • M ~ K.

We write A(k,K) to denote the group which consists of triples

(M,B,Z), with no symmetry requirements on B.

If we let k range over Z, we can form a graded ring, with

multiplication defined by O ,

A(k,K) • A(k',K') ~ A(kk',KK')

(M,B,I) x (M',B',Z') + (M •D M',B | B',s Q i')

This follows from 3.2.

However, for this paper, we shall only be concerned with the

Abelian group structure arising from direct sum.

Our objective now is to show that is an equivalence relation.

It is easy to see that is reflexive, since (V ~ V,B �9 -B,~ ~ s )

has metabolizer N = {(x,x) : x e V}. Clearly ~ is symmetric.

We must show ~ is transitive.

Again when (M,B,i) is metabolic, we will say M is metabolic,

the B, ~ being understood, and write M ~ 0. The following pro-

position is clear.

Proposition 5.3 ~ is transitive if and only if H ~ 0 and

M ~ H ~ 0 implies M ~ 0.

We call M stably metabolic if there exists H ~ 0 with

M ~ H - 0. We may then restate Proposition 5.3 as saying ~ is

transitive if and only if stably metabolic implies metabolic.

Comment: Once we have show that ~ is transitive, it is also

clear that the following relation, ~ , would have yielded the same

37

relation as Define M 0~ M 1 if and only if there exists

H 0 ~ 0, H 1 ~ 0 with M 0 ~ H 0 isomorphic to M 1 ~ H 1 .

Lemma 5.4 Suppose M is a finitely generated torsion free

D-module. Then (M,B,~) ~ 0 over D if and only i_~f (M,B,I) ~ E ~ 0

over E .

Here E is the quotient field of D (see 3.4).

Proof: Necessity is clear, for if N is a metabolizer for

(M,B,I), then N O D E is a metabolizer for (M,B,s O D E.

To show sufficiency, consider the exact sequence

0 § D + E ~ E/D + 0 .

Tensoring with M, we note that M is embedded into M ~ E as

M | i. Suppose M ~D E has metabolizer N = N ~. Let

N 1 = N /~ (M | i) C M • 1 .

Claim: N 1 = N1 J- in M | 1 = M, so that M ~ 0.

To begin with, N 1 is s,s invariant since N is. It also

is clear that NIC N1 "L. Conversely, if x ~ 1 e N1 J', B(x ~ l,y | i) = 0

for all y | 1 E N I. However, if y | r c N, then y ~ 1 e N I, so

N J- B(x ~) l,y ~) r) = rB(x | l,y ~ i) = 0. Hence x | 1 ~ = N.

Thus x O 1 E N I, and NI~C N I. Thus N 1 = N~and M ~ 1 = M ~ 0 o T'I

38

Using Lemma 5.4, we see that to prove ~ is transitive in the

case that M is torsion free, we can assume that D = E a field,

and K = E .

We thus assume for the rest of the proof of transitivity that

K = E = D a field when M is torsion free, and K = E/I for

M torsion as usual.

In either case K is an injective D-module and we have:

Theorem 5.5 Let (M,B) be an inner product space, with Values

in K = E or K = E/I. If N is Z , s invariant, then

N = (N~ ~

Remark 5.6 This is not true for K = I an arbitrary fractional

ideal or even K = D

Proof: We have the exact sequence

AdRB 0 § N ~ ~ M § HomD(N,K) 0

AdRB is onto since K is an injective D-module.

Applying the Hom functor, we obtain

0 § Horn D (Horn D (N, K) , K) + Horn D (M, K) § Horn D (N ~,K) §

Again, Ext(HomD(N,K),K) = 0 , since K is injective and the last

map is onto.

3g

We can identify HOmD(HOmD(N,K),K) = N by Theorem 2.6. This

clearly yields the commutative diagram:

§ N ~ HomD (M,K) § HOmD (Ni, K) + 0 0

+ + +

Ad B 0 § (NI) ~ + M +R HomD (N J-, K) § 0

The inner product provides an isomorphism of the middle terms, so by

diagram chase [M 50] the inclusion N C (N~) ~ is an isomorphism.

Lemma 5.7 For any two ~ , s invariant submodules R and S of

M, where (M,B) is an inner product space as above, we have:

and

Cl~ (R+ s) ~ = ~ s ~

(2) R ~ + S ~ = (R{N S) ~

Proof: (i) follows from the definition of ~ . TO show (2),

observe that

(R ~+ S~) ~ = (R~)~fA (SX) ~ = R{NS

Thus, taking �9 , R ~ + S ~ = (R ~ S) ~ O

Lemma 5.8 Let (M,B) be an inne___~r produc t space as above.

Suppose that M ~ 0 with metabolizer N. Let L C M satisfy

L C L ~. Then L + N ~L ~ = (L + N N L~) ~.

Remark 5.9 This Lemma shows how to go from a metabolizer N, and

40

a subspace L C LJ- to another metabolizer, namely

which contains the self annihilating subspace L.

L + N ~ L J',

Proof: The assumption that L is

stood, in order that L ~ make sense.

We compute using Lemma 5.7:

(L + (N ~ L ~))~= LJ-{~ (N~ LJ-) ~

= LJ'{~ (N J- + (L/') ~)

= L/'{'~ (N + L) since

= (L~'{'% N) + (LJ'/~ L)

= L + (N ~ L J-) since

Thus L + (N/~L ~) is also a metabolizer.

Z, s invariant is under-

N ~= N and (LJ')/" = L

LCL/" .

E3

Theorem 5.10 (Transitivity of ~ )

M ~ H ~ 0 if and only if M ~ 0 .

Let H ~ 0. Then

Proof: Sufficiency is clear. To .show necessity, let N be a

metabolizer for M ~ H, and H 0 a metabolizer for H. We embed

H 0 and H into M ~ H as 0 ~ H 0, 0 ~ H respectively. Notice

that 0 ~ H 0 C (0 ~ H0 )'L , so that by Lemma 5.8, we may rechoose

such that 0 ~ H 0 C N.

We review our notation. (M,B), (H,B') and (M ~ H,B ~ B')

are the inner product spaces. We will write elements in M ~ H

pairs (x,y) with x e M, y e H.

N

as

Let N O = projection of N onto M = {a s M: (a,h) e N for some h}

4~

Claim: N O is a metabolizer for M.

N O is clearly s s invariant since N is, and projection

commutes with Z, s on M ~ H.

We first show that N O @ H 0 = N. If! (a,h) s N, we claim h e H 0.

Let (0,h I) s 0 e H 0 N = N ~. Then

(B ~ B')((0,h), (0,hl)) = B(0,0) + B'(h,h I)

= B(a,0) + B' (h,h I)

= (B �9 B')((a,h), (0,hl))

= 0

since N = N ~ . Hence B'(h,h l) = 0 for all h I E H0, so that

h ~ H~ = H 0 as claimed. Thus N O ~ H 0 = N.

Clearly, N0 C N~. Conversely, let b s N~. Then by computing

as above (b,0) e N = N, so that b s N O . Hence N 0 = N 0 is a

metabolizer for M. [-]

Thus ~ is an equivalence relation, and we may form the Witt

group consisting of equivalence classes of triples (M,B,s modulo ~.

(M,B,s ~ (MI,BI,s I) provided (M ~ M,B ~ -BI,s @ Z I) ~ 0 .

Notation: [M,B,s will denote the Witt equivalence class of

(M,B, s .

6. Anisotropic representatives

Our final goal of this chapter is to find a representative of

each equivalence class. As long as K = E a field, or E/I in the

42

torsion case, this representative is unique.

We begin by describing the representative we will obtain.

Definition 6.1 A degree k mapping structure (M,B,s i__ss

anisotropic if for any s,s invariant D-submodule N of M,

N~N ~ = 0.

Theorem 6.2 Every Witt equivalence class [M,B,Z] has an

anisotropic representative.

We prove this theorem by way of a sequence of Lemmas which are

of interest in their own right.

Lemma 6.3 Let T be an s,s invariant D-submodule of M, with

T C T~ Then T~/T inherits a quotient degree k mapping structure

(T~/T,B, ~) .

Proof: Let it] denote an element in T~/T . Define

B( [tl], [t2]) = B(tl,t 2)

where tl, t 2 are representatives of [tl], It2] respectively.

is clearly well-defined since T is self-annihilating, ie. T C T .

It is likewise clear that ~, the induced map on T~/T is of degree

k with respect to B, and well-defined.

We must show that B is an inner product, ie. that

AdRB: T~/T § HOmD(T~T,K) is an isomorphism.

43

Applying the functor HomD(-,K) to the exact sequence:

0 § T ~ T ~ + T T ~ 0,

we obtain the embedding;

0 + HomD(T~/T,K) + HomD(T~,K).

1_ L Suppose g s HOmD(TTT,K) § ~ ~ HomD(T--,K). We can lift to

g: M § K since T is a summand by Proposition 4.6 in the torsion

free case and since K is injective in the torsion case.

AdRB: M + HomD(M,K) is an isomorphism. Hence g = B(-,x).

g restricted to T equals O, so x c T ~ �9 Thus (x + t) gives the

same g for all t ~ T. So we may read x s T2"/T.

This procedure defines a map:

HOmD (T/-/T, K) + T J-/T, namely g § ~ + [x].

The inverse of this map is simply

§ ~ (-, [x] ) .

Hence AdRB is an isomorphism and B is an inner product. D

Lemma 6.4 With the same hypotheses as in Lemma 6.3,

M �9 -T~/T is metabolic.

Proof: In the torsion free case, by Lemma 5.4, we may assume

44

K = E a field. Thus, in any case, there is no loss of generality

in assuming the hypotheses of Theorem 5.5, namely that K = E or

K = E/I. Consequently, for N an s s invariant subspace of M,

we have N = (N~) m.

We wish to show M ~ -T T ~ 0. So consider N = {(x,x + T) :x E T}.

N is an i, s invariant subspace, and clearly N C N ~. Let

(a,b + T) e N ~, with b e T J-. We compute

(B �9 -B)((a,b + T),(x,x + T)) =B(a,x) - B(b,x) = 0

for all (x,x+T) E N. Thus B(a - b,x) = 0 for all x s T .

Hence, (a - b) c (T~) ~ = T, since by assumption Theorem 5.5

applies. So [b] = [a] - [(a - b)] = [a] in T~/T , and

(a,b + T) = (a,a + T) c N. Therefore, N~C N, and N is a

metabolizer for M ~ -T /T.

Lemma 6.4 shows then that M ~ T--/Tt whenever TO T ~. In the

torsion free case, we can use that rank M < ~ to conclude, after

successive applications of Lemmas 6.3 and 6.4, that M ~ M 0

where M 0 has no Z,s invariant subspace T with TO T ~.

In other words, M 0 is anisotropic.

For the torsion module case, we repeatedly apply Lemmas 6.3

and 6.4 to obtain sequences:

MD T~T~ ...... D T 2 D T I~ T

Since M is Noetherian, the ascending chain condition implies that

the sequence {T i} terminates. Hence, it follows the sequence {T i}

45

will also terminate.

Since both chains terminate, we have M ~ Tr/-/Tr ; with

r T /T r having no s s invariant submodule Tr+ 1 with Tr+IC Tr+ 1 .

Thus, [M,B,s has an anisotropic representative, TrJ-/Tr , as

claimed. This completes the proof of Theorem 6.2.

This anisotropic representative need not be unique for torsion

free D-modules [M-H 28]. However, for K = E a field, or K = E/I,

we shall see that it is unique.

Theorem 6.5 As long as [M,B,s e w(k,K) satisfies Theorem 5.5,

ie. for K = E or E/I, every Witt equivalence class [M,B,s has

a unique anisotropic representative up to isomorphism.

Proof: Suppose (M,B,s ~ (M',B',s with M and M'

both anisotropic. Let N C M ~ M' be a metabolizer, with respect

to B G -B'. We will show that N is the graph of an isomorphism

f: M § M' which satisfies B'(f(k),f(y)) = B(x,y), s o f = f o s

and s' o f = f o s. Thus f is an isomorphism between (M,B,s

and (M',B',s

Let A = {a ~ M: there exists a I s M' with (a,a I) e N} .

Claim: Given a e A, then there exists a unique a I e M'

with (a,a I) E N.

For suppose (a,a I) and (a,a 2) e N. Then (0,a I - a 2) e N.

Consider the s s' invariant subspace, M 1 of M' generated by

a I - a 2 . Since N is s ~ s' and s ~ ~' invariant, this subspace

46

M 1 will have (0,M I) C N. Hence MIC M~ since N = N ~ . This

is a contradiction to M' being anisotropic unless a I - a 2 = 0

so that a I = a 2 .

Similarly, let B = {a I s M': there exists a e M with

(a,a I) s N} . As above, each a I s B has a unique a 6 M with

(a,a I) ~ N. It follows that N is the graph of a i-i function f .

We claim that A = M and B = M'. To see this we show that

i A~)~ A = 0 in M, hence ( = A = M . So let a ~ A , and consider

(a,0) E M ~ M'. Take any (x,y) s N. Then:

since

(B ~) -B') ((a,0), (x,y)) = B(a,x) = 0

a e A J', x e A.

Thus, (a,0) e ~ = N. By the first claim, this implies a = 0.

A similar argument shows B = M'. It follows that f: M + M'

is an isomorphism.

Let (a,f(a)) e N. Then (la,Z'f(a))s N since N is

�9 ~' invariant. Thus, by definition, (f o s (a) = (s o f) (a).

Similarly, (f o s) (a) = (s' o f) (a).

Finally, consider (x,f(x)) and (y,f(y)) ~ N.

so

(B ~) -B') ((x,f(x)) , (y,f(y)) = 0

B(x,y) - B'(f(x),f(y)) = 0 and B(x,y) = B'(f(x),f(y))

as desired.

Chapter II WITT INVARIANTS

Having defined the Witt ring we are led to examine invariants

which will enable us in many cases to compute at least the group

structure of W(k,F).

Section 1 begins with a preliminary discussion of prime ideals

and some results from algebraic number theory. In Section 2 we

state the basic properties of Hilbert symbols. The reader should

also see O'Meara [O'M] Introduction to Quadratic Forms for a complete

exposition.

Following this introduction, we continue by considering

(M,B) a u Hermitian inner product space over a field E with

involution - and fixed field F. B: M • M ~ E satisfies:

B(x,dy) = uB(dy,x) = dB(x,y) for d c E, x,y ~ M .

In Section 3, we discuss the rank mod 2 of M as a Witt group

invariant. Next, in Section 4, we introduce the discriminant in-

variant, the Witt analog of the determinanat for matrices. Thus,

we review the matrix representation of B and diagonalization in

order to define this invariant. Section 5 is concerned with the

signature invariants, which arise from the real infinite ramified

primes.

These invariants completely determine the Hermitian group

H+I(E) for E an algebraic number field by Landherr's Theorem.

Notation: F 2 = {0,i} = additive group of Z modulo 2.

(field with two elements)

Z* = ~i,-i} = multiplicative group of units in Z.

i. Prime ideals

48

The setting is as in Chapter 1. Again E is an algebraic

number field together with an involution - The fixed field of

- is F . The Dedekind rings of integers in E and F are

denoted by O(E) and O(F) respectively. O(F) = O(E)/~ F.

If P is a prime ideal in O(E), then P = p/~ O(F) will

denote the corresponding prime ideal in O(F).

Conversely, if P is a prime ideal in O(F), by going up,

[A,Mc 63] there exists a prime ideal P in O(E) with

p /~O(E) = P. In fact, there may be several such prime ideals in

O(E) lying over P. The answer is given by considering PO(E), g e.

IS 71]. We factor PO(E) = E pi I The Pi satisfy i=l g

pi/~ O(E) = P. Since the extension [E:F] is of degree 2, Z e.f. i= 1 i l

= 2, where fi = [O(E)/p i : O(F)/P] is the residue field degree.

We thus obtain the following cases:

i.i Split e = i, f = i, g = 2. In this case PO(E) = PP

where p is a prime ideal in O(E) with P ~ P. We say that P

splits in this case.

We may examine the split case in terms of the local completion

of F at P, which we denote F(P).

Write E = F(/o), and suppose /~ satisfies the irreducible

polynomial p/o/F(X) = p(x). Then factor p(x) in F(P) Ix]. The

split case corresponds to p(x) = fl(x) �9 f2(x). The prime spots

Pi dividing P are determined b}, F - monomorphisms 7: E ~ L,

where L is an algebraic closure of F(P). The - involution

interchanges the prime spots, hence, PO(E) = FP. [E(P) :F(P)] = 1

is the local degree.

49

1.2 inert e = i, f = 2. In this case PO(E) = P a prime in

O(E). P = ~ , and we say P remains prime, or is inert.

1.3 Ramified e = 2, f = i. PO(E) = p2, P = P in this case

also. We say P ramifies.

In both the inert and ramified cases, p(x) = p/~/F(X) is

irreducible in F(P) [x], and the local degree [E(P) :F(P)] equals 2.

This describes the situation for finite primes.

We next consider all embeddings T: F § C, where C is the

complex numbers. If y: F + R we call T a real infinite prime.

Otherwise T is called a complex infinite prime. We denote infinite

primes by P Our only concern will be with real infinite primes.

Again, since [E:F] = 2, and the characteristic of these fields

is 0 (not 2), we may write E = F(/~), for ~ s F* unique up

to multiplication by a square in F*. For an infinite prime P ,

there are two cases:

1.4 Split If P is complex infinite, P is split. If

P is real infinite, and ~ > 0 with respect to the ordering

induced by P , we again say P is split. In the case of a real

split prime, P , the ordering of F can be extended to E in

two distinct ways.

1.5 Ramified If P is a real infinite prime, and a < 0 with

respect to the P induced ordering we say P is ramified. In

this case, the P ordering of F can be uniquely extended to an

embedding of E into C in such a way that the imaginary part of

/~ is positive.

50

Let T denote the extension of P to E. Then T is

equivariant with respect to complex conjugation - This means

there is a commutative diagram:

T

E § C

E § C

The map -: E ~ E is the involution, -: C § C is complex

conjugation. There should be no confusion.

Associated to a finite prime P in O(F), or P in O(E),

is a discrete, non-Archimedian valuation I Ip, respectively I Ip"

P lies over P if and only if l Ip extends IIp, as discussed

in Chapter i.

We next describe prime ideals in terms of local uniformizers.

If PO(E) = PP is split, then a local uniformizer ~p ~ OF(P) is

also a local uniformizer for both OE(P) and OE(~). Careful, this

does not mean P and P induce the same valuation. On the con-

trary, OE(P) ~ OE(P). It only says (~) in OE(P) is the unique

maximal ideal.

If PO(E) = P is inert, then a local uniformizer Zp for

OF(P) is a local uniformizer for OE(P) also.

If PO(E) = p2 is ramified, then any local uniformizer z of

OE(P) will have norm z~, and ~ is a local uniformizer for

OF(P). This follows since P = P, hence z and ~ are both

local uniformizers for OE(P). So vp(z~) = 2 = Vp(~p) Thus

z~ is a local uniformizer for OF(P), when P ramifies.

51

For y ~ F* , we summarize:

(i) If P splits, PO(E) = p~. Vp(y) = ~Tp(y)

= v~(y)

(2) If P is inert, PO(E) = P Vp(y) = vp(y)

(3) If P ramifies, PO(E) = p2 2Vp(y) = vp(y)

This is not true for y e E*.

Associated to a prime p , finite or infinite, lying over a

prime P, is the extension of localized completions. The degree

[E(p) :F(p)] is denoted n F . n F = 1 if p is split and 2 otherwise.

2. Hilbert symbols

we begin by recalling the theorem of Hasse.

Theorem 2.1 Let Y c F*. Then y is a norm from E* if

and only if Y e F(P)* is a norm from E(P)*, for all p , finite

and infinite in E [O'M 186].

This condition is trivial over P split, for then F(P) = E(P).

We should like to rephrase this in terms of Hilbert symbols

[O'M 169]. We now state briefly the salient properties of these

symbols.

If a,o s F*, a symbol (a,o)p is defined by: (a,G)p = +I

if and only if a is a norm from F(P) (/o) if and only if there

exists x, y e F(P) satisfying ax 2 + ~y2 = +i.

52

In terms of the prime ideals, we summarize.

2.2 If P splits: (a,a)p = +l

2.3 If P is inert: The local degree np = [E(P):F(P)] = 2.

By [O'M 169], evert local unit is a local norm, and the local

uniformizer ~ s F(P) is not a local norm.

In terms of Hilbert symbols, for a e F*, we have the following.

n Let a = ~ v, for v a local uniformizer and v a local unit.

(a,u) n p = (z,q)p(V,q)

n = (~,~) p

n = (-i)

Vp (a) = ( -1 )

2.4 If P is ramified: Again the local degree is 2. As we

have seen, in this case we may pick a local uniformizer ~p of

P to be a local norm, namely, ~P = ~P ~P ' where ~p is a

uniformizer for E(P). We thus study s local units.

The residue field, OF(P)/m(P) = O(F)/P is isomorphic to the

completion OF(P)/m(P). If u is a local unit, then for the following

we denote by u I the image of u in the residue field, OF(P)/m(P).

Claim: For P ramified, a local unit u is a local norm,

ie. (u,O)p = +i if and only if u I is a square in the residue

field. (characteristic # 2)

53

Proof: If u I is a square in the residue field, we may factor

the polynomial t 2 - u I = f(t) in the residue field as

(t + /Ul) (t - /Ul) .

We are assumlng the characteristic of the residue field is not

2, so these two factors are relatively prime. Hence, we may apply

Hensel's Lemma, and conclude that t 2 - u factors in the completion

F(P). Thus u has a square root in F(P), and (u,O)p = +i.

Conversely, if (u,O)p = +I, then u is a norm from E(P),

say xx = u. We write x as x = wz r, for w a local unit, z a

local uniformizer. Then xx = w~( r~ r) = u. Since u is a local

unit, r = 0. Thus u is a square in OE(P)/m(F), since the induced

involution is trivial there. However, OE(P) /m(P) = OE(P)/m(P),

so that u I is a square in the residue field.

2.5 If P is infinite ramified: (a.a~ = -i if and only

if a < 0 and o < 0. This is clear as the completion with

respect to P is R.

We restate the Theorem of Hasse in terms of symbols.

Theorem 2.6 y r F* is a norm from E* if and only if

(y,O)p = +i at all primes P , finite and infinite.

We also list the important properties of the Hilbert symbols, in

addition to the discussion above.

2.7 (a,~)p = +i for almost all P since at almost all P ,

a and a are both units [O'M 166]. Almost all means all but

finitely many in this case.

54

2.8 Hilbert Reciprocity H(a,o) = +i p P

2.9 Realization: If s(P) ~ Z* is a function defined for

all P satisfying

(i) e(P) = +i if P splits

(2) e(P) = +i at all but finitely many primes

(3) H s(P) = +i P

then there is an f s F* with (f,~)p = s(P).

We again refer to O'Meara [O'M 203].

Note: At non'split primes, np = 2, and ~ is not a square in

F(P).

3. Rank

Let [M,B] s H (E) . We define the rank rood 2 of [M,B] , u

denoted rk[M,B], by

rk[M,B] = 0 if [M:E] is even.

= 1 if [M:E] is odd.

Here [M:E] is the rank of the vector space M over E.

55

Theorem 3.1

homomorphism.

rk: Hu(E) + F 2 is a well-defined group

Proof: The only problem is to show that rk is well-defined.

So, let [M,B] c H (E) have [M,B] = 0. Then there is a u

metabolizer N C M with N = N ~. This yields the exact sequence:

0 + N ~ AdRB § M § HOmE(N,E) + 0

Hence rank M = rank N + rank(HOmE(N,E))

= rank N ~ + rank N

= rank N + rank N

= 2rank N.

Thus [M,B] = 0 implies rk[M,B] = 0. It follows that rk is

well-defined.

Clearly rk is additive, so rk defines a group homomorphism.

we also note that rk is in fact a ring homomorphism. P1

Corollary 3.2 rk: Hu(I) + F 2 defined as above is a well-

defined group homomorphism. Here Hu(I) denotes I-valued

u Hermitian inner products on torsion free D-modules.

Proof: Apply I 5.4 .

4. Diagonalization and the discriminant

In order to discuss the discriminant, we must establish some

notation. We first pick a fixed basis, {e I .... ,en} for M .

56

n Thus, if x E M, we write x = (al,...,a n) to mean x = Z a e. , a s E.

i=l i i i

Associated to the inner product B: M • M + E, there is the

matrix B' = (bij), where bij = B(ei,ej). If x = (a I ..... an),

and y = (bl,...,bn), then in terms of B' we have

(a I ..... an) B'Ibl 1

k~n /

= B (x,y)

which we also write as B(x,y) = xB'y t This follows since B is

linear in the first variable and conjugate linear in the second.

Now [M,B] e Hu(E). Thus B(ei,e j) = uB(ej,e i) so that

b.. = ub �9 It follows that B' satisfies B' = uB 't. 13 31

We now let {e#} n x i=l denote the dual basis to {e i} .

e~ 1 : M + E is defined on a basis of M by e # i (ej) = ~ij' the

Kronecker ~ , and extended linearly to M.

We consider the adjoint map of B, AdRB: M + HomE(M,E) .

AdRB: e i + B(-,ei). We express (AdRB) (e i) as a linear combination of the {e #} This yields j

n (AdRB)e i = Z B(ej e# j=l 'ei) 3

We thus see that the matrix of the adjoint transformation, in terms

of the bases {ei}, {e~} is none other than (bij) = B' = (B(ei,ej)).

We can thus state:

Proposition 4.1 Given a bilinear map B: M x M + E, the

adjoint AdRB: M + HomE(M,E) is an isomorphism if and only if

57

(B(ei,ej)) is an invertible matrix. []

Next, we wish to relate u Hermitian to 1 Hermitian.

Proposition 4.2 Hu(E) = HI(E)

Proof:

we can find x I e E with XlXll = u. x I then clearly yields an

isomorphism: Hu(E) = HI(E), merely by scaling the inner product

-i with x I . In other words, [M,B] e Hu(E) + [M,B I] e HI(E) where

Bl(X,y) = (i/Xl)B(x,y). We must check B 1 is 1 Hermitian.

Bl(X,y) = (i/Xl)B(x,y) = (U/Xl)B(y,x) = (i/Xl)B-q-y,x) = Bl(Y,X).

Conversely, if [M,B I] c HI(E), we must check that B is

u Hermitian, where B(x,y) = XlBl(X,y). We compute: B(x,y) =

XlBl(X,y ) = UXlBl(X,y ) = UXlBl(Y,X ) = U(XlBl(Y,X) ) = uB(x,y)

Since u satisfies uu = l, by Hilbert's Theorem 90,

[]

Remark: We could choose x I = 1 + u.

When the characteristic of E is not 2, we shall see that it

is possible to choose a basis for M so that the matrix B'

of B is diagonalized. We prove this first for [M,B] e HI(E).

For [M,B] e Hu(E), we apply Proposition 4.2 and the above, observing

that the isomorphism given in 4.2 preserves diagonalization.

Now let [M,B] e HI(E). By the trace lemma, B 1 defined on

M by Bl(X,y) = trE/F ~ B(x,y) is a non-singular symmetric

bilinear form on M. Here trE/f denotes the trace map. Since

Bl(X,y) = (I/2) (Bl(X + y,x + y) - Bl(X,X) - BI(y,y) ), and B 1 ~ 0,

it follows that there exists v e M with Bl(V,V) ~ 0. It follows

that B(V,V) ~ 0 also. Extend v to a basis of M, {v,v 2 ..... v n} .

58

Notice that {v,v 2 - (B(v2,v)/B(v,v))v ..... v n - (B(Vn,V)/B(v,v))v }

= {w i} is also a basis for M. The computations:

B(v i - (B(vi,v)/B(v,v))v,v) = B(vi,v} - B(vi,v) = 0

= B(v,v i - (B(vi,v)/B(v,v))v)

show that with respect to {w i} , the matrix of B looks like:

I B (v

Z ,v) 0

~ I Continuing, consider Bl(X,y), for y in the span of {w 2 ..... w n} ,

x s M. Again, since B 1 is non-singular, we can find x with

0 ~ Bl(X,y) = (i/2) (Bl(X + y,x + y) - Bl(X,X) - Bl(y,y) ). Write n

x = Z a.w.. Then it is clear that: i=l i 1

n n

Bl(X,y ) = BI( ~ aiwi,Y] + Bl(alwl,Y) = BI( Z aiwi,Y). i=2 i=2

Thus, we can find v e with B(v,v) # 0. Continuing

inductively, we form {Wl,V .... ), .... and diagonalize B. We may

thus state:

Proposition 4.3 Given [V,B]e HI(E), there is a basis for V

which makes the matrix of B diagonal. (characteristic of E ~ 2) []

59

Remark 4.4 This also holds for H (E) by applying 4.2. u

Remark 4.5 As long as the involution - , on E is non-trivial,

we may prove 4.3 directly even if the characteristic of E is 2.

In order to see this, we must show how to produce a vector v with

B(v,v) ~ 0. Suppose to the contrary that B(v,v) = 0 for all v c M.

Assuming that B is non-singular, so not identically 0, we can find

v, w e M with B(v,w) ~ 0. However B(v+w,v+w) = 0 = B(v,w) + B(w,v).

Thus B(v,w) = -B(w,v). Hence, for any a e E,

aB(v,w) = B(av,w) = -B(w,av) = -aB(w,v)

Since B(v,w) ~ 0, a = a for all ac E, and the involution on E is

trivial. Contradiction.

Once we have such a vector v, we proceed as in 4.3 to produce

an orthogonal basis.

Remark 4.6 Thus, we see that 4.3 holds for Hu(E) , provided

we are not in the situation of a trivial involution or a field of

characterisitc 2. For [M,B] E Hu(E), where the characteristic of

E is 2, we may write B as a direct sum of 1-dimensional forms

and metabolic forms,

C0 :) see [K-I 22].

Diagonalizing an inner product space (M,B) means choosing a

60

basis of M with respect to which the matrix of B is diagonal.

In other words, (M,B) = ~ (Mi,Bi), where M i is a 1-dimensional

vector space. It is natural then to compare the matrices of B

given by different choices of bases for M.

Suppose that {e I , .... e n} and {fl ..... f~

M. We write E = matrix of B with respect to {e.} , and 1

F = matrix of B with respect to {f.} ]

We may express {e i} in terms of {fj} . Suppose e i =

and let C = (cij), C t = transpose of C .

are two bases of

n

Z c..f. , j=l 13 ]

Proposition 4.7 E = CFC t

.th l place

Proof: In terms of {e. } , write e. = (0 ..... 1 ..... 0) , 1 l

e i. = (0, . . . . . 1 ..... 0) and e.Ee~1 3 = eij 4 = B(ei,es).~ The ij component

of CFC t is likewise given by:

(0 ..... 1 ..... 0) CF~ t /

i th place I

(-/ = (Cil ..... Cin)F Cjl Cjn

.th 3 place

= B(Cilf I + + Cinfn ,

cjlf I + + Cjnf n)

= B(ei,e j )

= e. �9 as above. 13

[]

We would now like a Witt group invariant corresponding to the

determinant of a matrix. This invariant should be independent of

the choice of basis, as well as the Witt representative of the given

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Witt equivalence class.

Let [M,B] s Hl(E). Let B 1 and B 2 denote two different

matrices of B. By 4.7, we can write B 1 = CB2Ct, for a non-singular

matrix C. Let det B 1 denote the determinant of B I. Then

det B 1 = det C det B 2 det

= det C det B 2 (det C)

Thus, we can read the determinant of B in F*/NE*, since when B 1

is diagonalized, the diagonal elements must be in F* as B is

Hermitian. Unfortunately, this is not a Witt invariant. For example,

det B need not be in NE* even when IV,B] = 0 as we see below.

B l>o detB ii

which may not be a norm.

We are thus led to define a corresponding notion:

Definition 4.8 Let B 1 be a matrix with coefficients i_nn E,

corresponding to a Hermitian form B . Then dis B = (-l)n(n-l)/2det B !,

where n is the dimension of M, is called the discriminant of the

inner product space (M,B).

Lemma 4.9 If [M,B] = 0 then dis B s NE*.

Proof: Let N be a metabolizer for M. Let {nl,...,n t}

62

be a basis for N. Extend this to a basis for M, say

{n I, .... nt,nt+ I, .... n2t} With respect to this basis, B has matrix

Interchanging the first t colums with the last t columns, we obtain

a matrix

C 0

x

This requires interchanging t 2 columns. Hence B has

Thus,

as claimed.

t 2 det B = (-i) det C �9 det C = (-l)tdet C ~ det C.

dis B = (-l)2t(2t-l)/2(-l)tdetC �9 det

= (-l)t+t(2t-l)det C �9 det

= (-l)t+tdet C �9 det

= det C �9 det C e NE*

[]

It follows that dis is exactly the kind of invariant we seek.

There is still a problem, namely dis is not additive. Hence, we

do not obtain a group homomorphism:

HI(E) + F*/NE*.

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To remedy this problem we invent the group

Q(E) = F*/NE* • F 2 F 2 = {0,i}

[Lm 38]. The binary operation in Q(E) is given by:

ele 2 (dl,e I) �9 (d2,e 2) = ((-i) dld2,e I + e 2)

This is an associative, commutative operation with (1,0) the additive

identity. The additive inverse of (d,e) is ((-l)ed,e).

In fact, Q(E) becomes a ring when one defines multiplication by:

e 2 e 1 (dl,e l) Q (d2,e 2) = (d I d 2 ,ele 2)

The multiplicative identity is (i,i).

Proposition 4.10 The map ~: HI(E) + Q(E) defined by

[M,B] 4 (dis B,rk M) is a group homomorphism.

Proof: Consider [M,B] and [W,B I] in Hl(E). Suppose that

rank M = n and rank W = m. We have then:

~([M,B]) = ((-l)n(n-l)/2det B,n)

([W,B I]) = ((-l)m(m-l)/2det Bl,m )

~([M,B]) �9 ~([W,BI]) = ((-l)nm(-l)n(n-l)/2(-l)m(m-l)/2

det B det Bl,n + m)

~([M,B] + [W,BI]) = ((-i) (n+m) (n+m-l)/2 det B det B I, n + m)

84

But

(_l)nm+n(n-l),2/ + m(m-l)/2 = (-i)

= (-i)

= (-i)

(n2-n+m2-m+2nm)/2

((n+m) 2-(n+m))/2

(n+m) (n+m-l)/2

Thus # indeed gives a well-defined group homomorphism. []

As an exercise, the reader should verify that r is actually

a ring homomorphism.

We next consider the kernel of the rank homomorphism rk,

which we shall call J. Thus J is the subgroup of HI(E)

generated by the even dimensional forms.

Proposition 4.11 HI(E)/J 2 i_~s isomorphic t_~o Q(E).

Proof: J is additively generated by 2-dimensional forms,

. To see this, write ~ - . Thus j2

is additively generated by the forms

~< l,b >= .

Applying ~ to a generator, we obtain:

~= ((-l)6(ab)2,0) = (i,0).

Thus, # induces a map ~ : HI(E)/j2 + Q(E). We now construct

an inverse of ~ ,

Define y : Q(E) § HI(E)/j2 by

(a,0) + < l,-a > modulo j2

(a,l) ~ modulo j2

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It is easy to check that y is a homomorphism, and y o ~ = id

o y = id, where id is the appropriate identity map. []

Proposition 4.11 implies that $ is i-i. Hence,

Corollary 4.12 j2 consists of even dimensional forms [M,B],

with dis B = 1 e F*/NE*, ie. det B = (-i) n(n-l)/2 where n = rank m. ~]

Corollary 4.13 Restricting $ t__oo j/j2, we have j/j2 = F*/NE* . ~

In fact, we may think of this as follows: F*/NE* is embedded

into Q(E) by: d + (d,0). This is a subgroup of index 2. We may

2 represent the non-identity coset by (I,i). (i,i) = (-1,0). Thus,

we have the exact sequence:

q2 1 ~ F*/NE* ~ Q(E) + F 2 § 0

q2 is projection onto the second factor. By the above remarks.

Corollary 4.14 This sequence splits if and only if (i,0) = (-i,0)

i__nn Q(E) if and only if -i is a norm in F*/NE*.

Remark 4.15 This defines the discriminant for HI(E). In

order to define discriminant for H (E), we fix an isomorphism: u

fxl: Hu(E) + HI(E), by Proposition 4.2

B + B 1 Bl(X, Y) = (i/Xl)B(x,y).

66

Then define dis B = dis fxl B = dis B 1. We must note that this

depends on the isomorphism chosen ie. this depends on Xl, where

----1 XlX 1 = U.

Remark 4.16 The discriminant inner product space, Chapter I 3.5,

yields the information crucial to the discriminant invariant above,

namely the determinant of B. Its advantage is that it generalizes

the notion to H(D), for D the Dedekind ring of integers.

5. Signatures

The real infinite primes, P , give rise to the signature invariant

which we now discuss. Suppose then that E = F(/o), and o < 0 with

respect to P Thus P is an infinite ramified prime.

Lemma 5.1 If x e E, then NE/F(X) > 0 with respect t__oo P

Proof: N=N denotes the norm. E/F

N(x) = a 2 - b2o > 0 since o < 0. [~

Write x = a + b/o . Then

Let [M,B] e HI(E). By 4.3, we can find a basis {e i} of

M in which B is diagonalized. We can thus write M = X + ~) X- ,

where B(ei,e i) > 0 for e i e X +, B(ei,e i) < 0 for e i ~ X-.

Now, let v e X +, so v = Za.e.. We compute i I I

B(v,v) = B(Zaiei,Zaiei) = Z B(aiei,aie i)

= ZaiaiB(ei,e i) = ~N(ai)B(ei,ei) > 0

by Lemma 5.1.

67

Similarly, for all v s X-, B(v,v) < 0. We now define:

sgn[M,B] = dim X + - dim X- .

sgn[M,B] is called the signature of [M,B]. In order to show sgn

is well-defined, we first need:

Lemma 5 .2 sgn[M,B] is independent of the basis chosen for M.

Proof: Suppose M has two bases, {ei} , {fi } which make B

diagonal. Let B I, B 2 be the matrices of B with respect to

{ei}, {fi } .

Consider [M,B I] - [M,B 2] which is Witt equivalent to 0

With respect to the basis {ei,f i} of M ~ M, B 1 ~ -B 2 has matrix

It follows that sgn[M ~ M, B 1 ~ -B 2] = sgn B 1 - sgn B 2 . Thus in

order to show sgn B 1 = sgn B2, it clearly is sufficient to show:

Any metabolic space [V,h] has sgn[V,h] = 0 with respect to

an arbitrary basis.

So suppose V = X + ~ X- . Let N be a metabolizer for V,

N = N~ We note that n e N implies h(n,n) = 0. Thus, by the

remarks preceding this theorem, X+/~ N = 0 = X-/~ N. However,

X+/~ N = 0 implies dim N < dim V - dim X + = dim X- . Similarly,

+ dim N < dim X + . Now, 2dim N = dim V, so that dim X- and dim X

are both > (i/2)dim V. However, dim X + + dim X- = dim V.

68

Thus, dim X + = dim X- = (1/2)dim V, and sgn[V,h] = 0 with

respect to any basis.

In the process of this proof, we have shown:

Corollary 5.3 if [V,h] is metabolic, sgn[V,h] = 0. []

It is thus clear that sgn gives a well-defined Witt-invariant,

which is a group homomorphism:

sgn: HI(E) + Z.

It is clear that if [M,B] has finite order, sgn[M,B] = 0,

since every element in Z has infinite order. Thus sgn is non-

trivial only on the non-torsion elements in HI(E) .

We finally recall Landherr's Theorem which explicitly computes

HI(E), for E an algebraic number field [Lh]

Landherr's Theorem 5.4 There is an exact sequence:

r 0 + (4Z) + H(E) + Q(E) + 0

[M,B] = (dis B, rk V) .

The kernel of ~, ker~ , is determined by the real infinite

ramified primes and the corresponding signatures, each of which is

divisible by 4. Here r is the number of real infinite primes.

When r = 0, ~ is an isomorphism.

69

This theorem is important in the boundary computation in

Chapter VI.

By Proposition 4.10, H(E)/J 2 = Q(Z), so we can state:

Corollary 5.5 j2 = 4(Z ~ Z .... �9 Z) [] r times

Remark 5.6 As with the discriminant, we can define a signature

invariant for H (E). This is done by picking an isomorphism u

fxl: Hu(E) § HI(E) as in 4.15. We then