Quadratic Forms Over Semilocal Rings

205 Pages·1978·3.181 MB·English

by  Ricardo Baeza (auth.)

Lecture Notes ni
Mathematics
Edited by .A Dold dna .B Eckmann
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Author
Ricardo Baeza
Mathematisches Institut FB 9
Universit~t des Saarlandes
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AMS Subject Classifications (1970): primary: 10C05, 01 E04, 01 E08
ISBN 3-540-08845-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-08845-8 Springer-Verlag New York Heidelberg Berlin
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Preface
The algebraic theory of quadratic forms originates from the well-known
paper ]W[ of Witt (1937), where he introduced for the first time the
so called Witt ring of a field, i.e. the ring of all quadratic forms
over a field with respect to a certain equivalence relation (compare
§4, chap. I in this work). The study of this ring and related questions
is essentially what we understand by the algebraic theory of quadratic
forms. Thirty years after the appearance of Witt'swork, Pfister succeeded
in his important papers [Pf]I,2,3 in giving the first results on the
structure of the Witt ring of a field of characteristic not 2. Since
Pfister's work appeared twelve years ago, a lot researchs on this subject
have been made. Lam succeeded in writing down many of these researchs
in his fine book [L], which is perhaps today the best source to which
a student of the algebraic theory of quadratic forms may turn for a
comprehensive treatment of this subject. Besides of the theory over
fields a corresponding theory of quadratic forms over more general
domains has been growing up. We cite in particular Knebusch's work on
the related subject of symmetric bilinear forms (see [K]I,..,7).
The present work deals with the algebraic theory of quadratic forms
over semi local rings. We have tried to give a treatment which works
for any characteristic, i.e. we do not make any assumption about 2.
If 2 is not a unit, then in general quadratic forms behave better than
bilinear forms, because the former have much more automorphisms (for
example an anisotropic bilinear space over a field of characteristic 2
has only one automorphism). This fact has been exploited throughout
in this work (see §3,5 in chap. III and §3,4 in chap. IV). Of course
our results cannot go so far as in the field case, because over semi
local rings we do not have to our disposal one of the most powerful
methods of the theory over fields, namely the transcendental method.
For example it would be very interesting to have an elementary proof
(i.e. without transcendental methods) of the "Hauptsatz" of Arason-
Pfister (see[Ar-Pf])or even of Krull-intersection theorem for the Witt
ring.
Our treatment is rather self contained. We only suppose the reader to
be acquainted with the most elementary facts of the theory of quadra-
tic forms over fields (for example as given in Dieudonne's book [D] )I
and with the current results of the theory of Azumaya algebras (see
[Ba])°
VJ
We have used only standard notation. For example ~, ~ , Q, ~ denote
the set of non zero integers, the ring of integers, the field of ra-
tionals and the field of real numbers, respectively. Moreover we use
the following notation: ch = characteristic, dim = dimension, Ker =
kernel, Im = image, @ = direct sum, ~ = direct product, ® = tensor
product, etc.
R. Baeza
Mathematisches Institut, FB 9
Universit~t des Saarlandes, SaarbrHcken
Contents
Chapter .I Quadratic forms over rings I
§I. Definitions I
§2. Operations with quadratic and bilinear forms 5
§3. Subspaces 9
§4. Hyperbolic spaces 15
Chapter II. Invariants of quadratic forms 22
§I. Azumaya algebras 22
§2. Clifford algebras 30
§3. The structure of Clifford algebras 36
§4. Some computations 54
§5. Quadratic spaces of lower dimension 56
Chapter III. The orthogonal group 59
§I. Notations 59
§2. The Eichler decomposition of orthogonal the group 16
§3. Proper automorphisms 64
§4. Witt's cancellation theorem over semi local rings 79
§5. Transversality theorems for quadratic forms 83
Chapter IV. Pfister spaces over semi local rings 89
§I. Similarities 89
§2. Pfister spaces 94
§3. Isotropic Pfister spaces 105
§4. Further results on quadratic Pfister spaces 107
Chapter V. Structure of Witt rings 112
§I. Introduction 112
§2 The discriminant map 114
§3 Some computations 117
§4 Quadratic separable extensions 121
§5 An exact sequence of Witt groups 134
§6 The torsion of Wq(A) and W(A) 143
§7 The local global principle of Pfister 146
§8 Nilpotent elements in Wq(A) and W(A) 156
§9 An explicite description of Wq(A) t 160
§10. On the classification of quadratic spaces 167
§11. The behaviour of Wq(A) by Galois extensions 173
IV
Appendix A. On the level of semi local rings 177
Appendix B. The u-invariant 187
References 193
Index 198
CHAPTER I
Quadratic forms over rings
§ I. Definitions.
Let A be a commutative ring with I. Let P(A) denote the category of
finitely generated projective A-modules with the operations @ = direct
sum and ®A = tensor product (henceforth we shall use the unadorned
tensor product ® instead of ®A if the ring A is fixed). For every
M6P(A) we call M* the dual A-module HomA(M,A) 6P(A).
(1.1) Definition. Let M6P(A). A bilinear form b:M x M~A is called sym-
metric, if b(x,y) = b(y,x) for all x,y6M. The pair (M,b), consisting
of a module M6P(A) and a symmetric bilinear form b on M, will be called
a bilinear module over A. We frequently write b instead of (M,b).
Let (M,b) be a bilinear module over A. For every x£M we define~(x)6M*
by the formula db(X) )y( = b(x,y) for all y6M. Thus we obtain a A-line-
ar map db:M~M*. The bilinear module (M,b) is called non singular or
simply bilinear space, if b d is an isomorphism. An isomorphism bet-
ween two bilinear modules (M1,b )I and (M2,b )2 is given by a linear
isomorphism f:M I ~ 2 M with
I b (x,y) = b2 (f )x( ,f )y( )
for all x,y£M .I Then we write (M1,b )I ~ (M2,b2).
(1.2) Remark. Let M = Ae1@...@Ae n be a free n-dimensional A-module
with a symmetric bilinear form b on it. The form b is determined by
the matrix (bij), where bij = b(ei,ej), because we have
n
b(x,y) = ~ b..x.x.
i,j= I 13 1 3
n n
for x = [ xiei, y = [ Yiei . Conversely, using this formula we obtain
i=I i=I
from every symmetric n x n-matrix (bij) over A a symmetric bilinear form
on M, and the bilinear module (M,b) is non singular if and only if
det(bij) 6 A* = group of units of A.
(1.3) Localisation. Let max(A) be the set of all maximal ideals of A.
For m 6 max(A) and M6P(A) let m M denote the localisation of M in m. Cor-
respondingly we write M(m) for the reduction of M modulo m, i.e.
M(m) = M/mM. Then M m6 P(A m) and consequently m M is a free Am-module
of finite dimension (see [BOil). The map rM:max(A) ~ ~, given by
rM(m) = dim A (Mm), is called the rank-map (or simply the rank) of M.
m
M has constant rank n £ ~, if rM(m) = n for all m6 max(A). Let us now
consider a bilinear module (M,b) over A. The bilinear form b induces
on m M a symmetric bilinear form m b over Am, bm:MmX M m~Am, which is
defin@d by
bmlX,b~l = b(x,lz)
ab
for x,y 6 M, a,b ~m. The bilinear module (Mm,b m) is called the locali-
sation of (M,b) in m. For the induced Am-linear map b d :Mm~Mm ~ we
m
get (d b) b = d , since we may identify (M~) m with m M .
m m
The reduction of (M,b) modulo m is defined analogously, i.e. we have
a bilinear A(m) = A/m-module b(m) : M(m) x M(m) ~A/m, which is given by
b(m) (x,y) = b(x,y) for all x,y 6M(m). For the induced A(m)-linear map
db(m) : M(m) ~M(m) * = M~(m) we have db(m) = db(m). Now using this r~narks
and the fact, that a bilinear module over a local ring is non singular
if and only if its reduction is non singular, we deduce (see [Bo] I,
cIhI., §3)
(1.4) Proposition. Let (M,b) be a bilinear module over A. The following
statements are equivalent:
i) (M,b) is non singular
ii) (Mm,b m) is non singular for all m6 max(A)
iii) (M(m),b(m)) is non singular for all m6max(A)
Now we introduce quadratic forms
(1.5) Definition. A quadratic form on a module M6P(A) is a map q:M~A
with the following properties
i) q(ix) = 12q(x) for all x6 M, i 6A
ii) bq(X,y) = q (x+~) -q )x( -q )y( defines on M a bilinear form bq:MX M~A.
The pair (M,q) is called a quadratic module over A and (M,bq) the asso-
ciated bilinear module. If (M,bq) is non singular we call (M,q) non
singular, or, a quadratic space.
An isomorphism between two quadratic modules (M1,ql) and (M2q )2 is
given by a linear isomorphism I ~ M f 2 : M with the property
q1(x) = q2(f(x))
for all I. M x 6 We then write (M1,ql) ~ (M2,q2). Of course in this
case the associated bilinear modules are isomorph, too.
(1.6) Remark. Let us assume £ 2 A*. Then we have a one to one corres-
pondence between bilinear modules and quadratic modules over A. By
this correspondence the bilinear module (M,b) will be identified with
I
the quadratic module (M,qb), where qb(x) = ~ b(x,x) for all x 6 M, and
conversely, the quadratic module (M,q) will correspond to the bilinear
module (M,bq). One sees immediately qb = q and b = b.
q qb
Consider now a free A-module M = Ae10...OAe n endowed with a quadratic
form q. For every i • j we set aij = bq(ei,e )j and for = j i aii= q(ei).
We obtain a n x n-matrix [aij], which is called the value-matrix of
(M,q) with respect to the basis {el, .... en}. This matrix determines q
n
completely, since for x = [ x.e. M 6
i= I i 1
q(x) = ~ a..x.x.
1~i~j<n 13 i 3
For this reason we shall identify the quadratic form q with his
value-matrix [aij] , i.e. we set q = [aij]. For example if M=Ae • Af
and q(e) = a, q(f) = b, b (e,f) = I, we have
q
q a[ ]I
I b
/ 2a I \
The value-matrix of (M,bq) is ( ), / so that (M,q) is non singular,
~1 2b
if and only if 1-4ab 6 A*. In this case we shall write [a,b] for this
quadratic space. If 2 = o in A, all quadratic spaces [a,b] have the
same associated bilinear space (~ Io>,i .e. in general two non iso-
morphic quadratic spaces may have the same associated bilinear space.
In any case we have the following
(1.7) Proposition. Let (M,q) be a quadratic module. Then there exists
a bilinear form M O x : b M~A with q(x) = bo(X,X) and bq(x,y) = bo(x,y)
+ bo(Y,X) for all x,y 6 M.
Proof. First we suppose that M is free, i.e. M Ae18-..~Ae n. Let
aii = q(e i) and aij = bq(ei,e )j for % j. i Then we define bo:M x M~A
by bo(ei,e k) = aik if i~k and bo(ei,ek) = o if > i k. This bilinear
form fulfills the conditions of the proposition. Now we treat the
general case. Let N 6P(A) be a module with M 8 N m ~ A for some mo
On m A we define a quadratic form q' as follows: q'IM = q and q' IN = o.
Using the above mentioned construction we obtain a bilinear form b o'
associated to q' • Then the form ° b = bo ' IM M x has all required pro-
perties.
As in the bilinear case we define for every m6 max(A) the localisa-
tion (Mm,qm) and the reduction (M(m),q(m)) of a given quadratic mo-
dule (M,q) over A. The analogous result to (1.4) is now
(1.8) Proposition. For every quadratic module (M,q) the following
statements are equivalent
)i (M,q) is non singular
ii) (Mm,qm) is non singular for all m6max(A)
iii) (M(m),q(m)) is non singular for all m6max(A).
(1.9) Remark. If A has characteristic 2, then it holds bq(X,X) =
= 2q(x) = o for every quadratic form q over A, i.e. b is an alter-
q
nating form. Thus if (M,q) is moreover non singular and free, the
dimension of M must be even. More general, the rank of any quadratic
space (M,q) over a ring A with r £ 2 = n m (Jacobson radical
m £ max(A)
of A) is an even function.