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Invent. math. 123, 233-240 (1996)
Inventiones
mathematicae
9 Springer-Verlag 1996
Non unimodular Hermitian forms
Eva Bayer-Fluckiger, Laura Fainsilber
URA 741 du CNRS, Laboratoire de Mathematiques, Facult~ des Sciences Universit~ de
Franche-Comt6, 16, route de Gray, F-25030 Besan~on, France
Ob[atum 12-XII-1994
Introduction
The classical theory of integral quadratic forms deals not only with unimodular
forms, but also with non unimodular ones. However, during the last 30 years
unimodular forms received much more attention. This is partly due to the
development of the algebraic theory of quadratic forms, which concentrates on
forms over fields, where the interesting forms are indeed unimodular, and to
the fact that forms arising from topology are usually unimodular.
There is now an extensive literature on unimodular hermitian forms, even
over quite general rings (see for instance [6], [11]). On the other hand, com-
paratively little is known about non unimodular forms. The purpose of the
present note is to show that the classification of non unimodular forms can
be reduced to a module theoretic question and the classification of unimodular
forms over a different ring, and to give some applications of this method.
The main result is proved in Sect. 4 and 5. In Sect. 6, we illustrate our
technique in the case of principal ideal domains. We then consider algebras
which are free modules of finite type over complete discrete valuation rings.
The above results are applied in Sect. 7 to prove a Witt type cancellation theo-
rem. This was previously known for unimodular forms (Quebbemann, Scharlau,
and Schulte [7]), as well as for some non unimodular forms over hereditary
orders (Riehm [10]). Finally, Sect. 8 contains an analogue of a theorem of
Springer for hermitian forms.
1. Hermitian forms over rings
Let A be a ring, and let -:A ~ A be an involution (that is, an anti-
automorphism of order 2 of A). Let M be a left A-module of finite type.
Let e = il. An
e-hermitian
form over A is a biadditive map h : M x M --~ A
234 E. Bayer-Fluckiger, L. Fainsilber
such that
h(ax, by) = ah(x, y)b
and that
h(y,x) = eh(x,y)
for all
x, y E M
and
all
a,b E A.
An isomorphism ~: : (M,h)---+
(M',h')
is an isomorphism of left
A-modules such that
h'(4)(x), (~(y)) = h(x, y)
for all x, y C M.
Let M* = Hom(M,A). Then M* has a structure of left A-module, given
by
a.f(x) = f(x)~
for all a E A, f E M* and x E M. An e-hermitian form h
induces a morphism of left A-modules M ~ M*, which we also denote by h.
The form
(M,h)
is said to be
unimodular
if and only if h : M ---+ M* is an
isomorphism.
A left A-module M is said to be
reflexive
if the morphism e~t : M ---+ M**,
given by
eM(m)(f) = f(m),
is an isomorphism. For instance, every projective
module is reflexive.
The
orthogonal sum (M,h)|
of two forms
(M,h)
and
(N,g)
is
given by the form (M O N, h | g), where
(h 0 g)(m 9 n, m ~ 9 n ~) = h(m, m') +
g(n,n').
2. Hermitian categories
Let ~ be an additive category. Let ,:c~ __, ~ be a duality functor, i.e. an
additive contravariant functor with a natural isomorphism
(Ec)cc~ :id--~**
such that
E~Ec. = idc*
for all C C ~.
A sesquilinear form
in the category ~ is a pair
(C,H),
where C is an
object of cg and H is a morphism H : C --~ C*. Let ~: = +1. A sesquilinear
form
(M,h)
is said to be
~:-hermitian
if H =
cH*EM.
A form
(C,H)
is said
to be
unimodular
if H : C --~ C* is an isomorphism.
Let
(C,H)
and
(C,H ~)
be two e-hermitian forms. A morphism q5 : (C,H)
(C~,H ~)
is an
isomorphism dp : C ---+ C ~
of the category ~ such that H =
~*H'4,.
We denote by H':(~) the category of e-hermitian forms on ~, and by
H~(~) the full subcategory of unimodular e-hermitian forms on ~.
Example.
Let Jg be the category of finitely generated reflexive left A-modules.
Then ~//g is an additive category. The functor * : M --+ M which sends M to M*
is a duality functor on o~, with EM = e~t (cf. Sect. 1 ). The category of e-
hermitian forms over the ring A is equivalent to H~:(~/). The equivalence is
obtained by sending a form h to the corresponding morphism h : M ~ M*.
3. The category of morphisms of ~/~
A morphism of ~' is a triple
(MbM2,f),
where Ml and M2 are finitely
generated reflexive left A-modules and f : M1 --* M2 is a morphism of left A-
modules. Direct sums are defined in the obvious way. Let us call ~ the category
of morphisms of ~. A
~-morphism (M1,M2,f) ~ (N1,N2,g)
is a pair q5 =
(~bl,q52) of morphisms of left A-modules qSi :
Mi ~ Ni
such that 4~2f = g~bl.
Let tlM*2,M*l,jc*~1 be the
dual
of
(Mb M2, f ).
Set
E(M,,M:, f) = ( eM, , eM, ).
This
defines a duality functor on the category ~?.
Non unimodular Hermitian forms 235
4. Non unimodular hermitian forms
Let H~:(.~) be the category of unimodular e-hermitian forms over 2 (cf. Sect.
2).
We define a functor F from the category H':(.<H) of e-hermitian forms over
the ring A to the category //~,(~), as follows. Let
(M,h)
be an e-hermitian
form, h :M --~ M*. Then
(M,M*,h)
is an object of 2. It is easy to check that
the pair
(eM,~:idM*)
defines a unimodular e-hermitian form over
(M,M*,h).
Therefore,
F(M,h)= ((M,M*,h),(eM,~:idM.))
is an object of Hi(z? ).
Theorem
1.
The functor F is an equivalence of categories between the cate-
gory H~(~) of e-hermitian .forms over A and H[~(~).
Proof
As already pointed out, Y sends an e-hermitian form to an object of
H~(22). In order to prove that .~- is a functor, it remains to be checked that
sends morphisms to morphisms.
Let q~ :
(M,h) --, (N,g)
be a morphism of e-hermitian forms. Then we have
the commutative diagram
M J~ M*
N ~ N*.
Therefore,
(fl/),~p*-l) : (M,M*,h) ---+ (N,N*,g)
is a morphism in 2. Moreover, as ~b**eM = ex~b by naturality of (eM)ME //, it
is straightforward to see that it is also a morphism of H~(~).
Let us define a functor N from H~"(22) to the category of e-hermitian forms
over A. Let (Q,~) E H~:(2), with Q =
(N,N*,g)
and ~ = (~1,~2). Set .C6(Q, ~) =
(N, ~2g). This is an e-hermitian form over A, which is unimodular if and only
if g is bijective.
Let us check that N sends morphisms to morphisms. Indeed, let q5 =
(~)l,q~2)
be a morphism of H~(~ ~b: (P,q) -~ (Q,~), where (P,I/) =
((M,M*,h),(~II,I12)),
(Q,~)= ((N,N*,y),(~I,~2)).
In particular, q5 is a mor-
phism of 22, therefore 9q5~ = q~2h. Moreover, q~ is also a morphism of H,;:(A),
therefore 172 q~T~2~b2.
This implies that
r/2h *~ -I ,
: = (/)lg2q)2~b 2 ,q~bl = ~)l(k_2,q)q~l,
so (bl is a morphism of H%~H).
Finally, it is easy to verify that ~ o ~N is the identity of
H%/#)
and that
,r o (q is isomorphic to the identity of H~(~). Therefore, .~- : H,':(2) -~ H%/I/)
is an equivalence of categories. This completes the proof of the theorem.
5. Hermitian forms corresponding to a given object of oQ
If (M,h) is an e-hermitian form over A, we denote by
q(M,h)
the object
(M,M*,h)
of 2.
236 E. Bayer-Fluckiger, L. Fainsilber
We describe the set of isomorphism classes of ~-hermitian forms corre-
sponding to a given object of ~? in terms of rank one hermitian forms over the
ring of endomorphisms of this object. Let us fix an e,-hermitian form
(Mo, ho),
and let E be the ring of endomorphisms of 22o =
q(Mo, ho)
in ~. Let us also fix
a unimodular e,-hermitian form ~/0 over Q0. Such forms exist, for instance we
~
can take r/0 = (eMo,
~idM,; ).
This form induces an involution 9 E ---+ E, defined
by
j~ --I o
= r/0 f r/0,
for all f E E, where fo denotes the dual of f in 2. Let E • be the group of
units of E. Set
E + = {f E E • ? = f},
and define an equivalence relation on E + by
f =-- f' ~ 3gEE• gfg= f'.
Note that
E+/:-
is the set of isomorphism classes of rank one unimodular
hermitian forms over (E,").
Theorem
2.
The set of isomorphism classes of e-hermitian Jbrms (M, h) such
that q(M, h) = Qo is in bijection with E+/=_.
Proof
Let r/=
(eM,~idM*).
The bijection is given by
(M,h), ~
~o~.
Indeed, it is straightforward to check that r/olr/E E +. Moreover, if
(N,g)
is
another e-hermitian form with
q(N,
g) = Q0, then
(M,h ) ~- (N,g) ~=~ ~loLtl = qol ~,
where ~ =
(ex, eidx, ).
This completes the proof of the theorem.
Note that the results of the last two sections indeed reduce the classification
of arbitrary hermitian forms to a module theoretic question and to the classifi-
cation of unimodular forms over the ring of endomorphisms, as announced in
the introduction.
6. An example: symmetric bilinear forms of given determinant over a prin-
cipal ideal domain
In this section, we assume that A is a principal ideal domain. Let K be the
field of fractions of A. The aim of this section is to illustrate the methods of
Sects. 4 and 5 in this case.
o, C K. Let a denote
Let al,...,an be non-zero elements of A and set
ai, j = a--~
the n-tuple (al ..... an) and let E(a) be the subring of
Mn(A)
defined by
E(a) =
{(ai, jxi, j)i,j E Mn(A)lxi, j E A, Vi,j}.
Non unimodular Hermitian forms 237
Let us define an involution a : E(a) --+ E(a) by
a((a,,j,xi,j),,j) = (x/,i)i,1.
Note that
xj, i
= ai.j(aj.ix/.t),
therefore
(xj,~)i.j C
E(a).
As in Sect. 5, we use the notations
E(a) + =
{fE E(a)• = f},
and
f --= f' ~=~ 3g E E•
a(g)fg = f'.
For any non-zero element d E A, we denote by
A(n,d)
the set of n-tuples
(al ..... an) of non-zero elements of A such that
ai
divides a,+j for all i =
1
..... n - 1 and d = al ...an. If a--
(al ..... an) E A(n,d),
let us denote by aA
the n-tuple of ideals
(alA ..... anA)
generated by aj ..... a~. Let
I(n,d)
be the
set of n-tuples of ideals aA with aE
A(n,d).
Proposition
1.
Let n be a positive integer and let d be a non-zero element
of A. The set of isomorphism classes of symmetric bilinear .forms over A of
rank n and determinant d is in bijection with the set
{(aA, f) I a
E
A(n,d), f C
E(a)+/~}.
Proof
Let M be a free A-module of rank n and let h : M x M --~A be a
symmetric bilinear form of determinant d. Let h : M --~ M* be the associated
morphism. As in Sect. 5, we denote by
q(M,h)= (M,M*,h)
the object of
associated to
(M,h).
By the theorem of elementary divisors, there exists
a = (al ..... an) C A(n,d)
such that
q(M,h) ~- (An,A",diag(a)),
where diag(a) :
A n -~ A n is the diagonal map determined by a = (at ..... an). The isomorphism
class of
q(M,h)
is determined by the n-tuple of ideals aA C
I(n,d).
The ring of endomorphisms of
q(M,h)
is
E = {(e, fl) C
Mn(A)
• M,(A) I a diag(a) = diag(a)fl}.
Let t/=
(eM, idM* ).
The involution of E induced by t/ (cf. Sect. 5) is (e, fl) =
(fit, at). A computation shows that
E = {((a,,jx,,i),,j, (x~,j)i,j) E
E(a) • E(a)}.
Therefore (E,-) _~ (E(a),a). By Ths. 1 and 2, this proves the proposition.
7. Witt's cancellation theorem for E-hermitian forms
Let R be a complete discrete valuation ring. Let n be a uniformiser of R.
Assume that A is an R-algebra of finite rank over R, and that there exists an
element a of the center of A with a + ~ = 1. For instance, this condition is
fulfilled if 2 is invertible in R.
238 E. Bayer-Fluckiger, L. Fainsilber
Theorem
3. Let (M,h), (Ml,hl) and
(m2,h2)
be e-hermitkm forms over A.
Assume that
(Ml,hl ) 9 (m,h ) ~- (m2,h2)
@
(m,h ).
Then we have
(Ml,h~ ) ~- (M2,h2).
In other words, Witt's cancellation theorem holds for arbitrary e-hermitian
forms over A.
As in Sect. 4, let us denote by o) the category of morphisms of reflexive
modules over A. In Sect. 4, we have seen that the category of e-hermitian
forms over A is equivalent to the category H~'(~) of unimodular e-hermitian
forms over the category ~. Hence, it suffices to show that Witt cancellation
holds in the category H,':(~). This is the purpose of the next proposition.
Proposition
2. Let (Q,q), (Ql,~l ) and
(Q2,q2)
be objects of H~:(2?). Assume
that
(Ql,qi)O(Q,q) ~ (Q2,r/2) | (Q,~/).
Then
(Ql,ql) ~ (Qz,r/2).
Proof We deduce the proposition from some results of Quebbemann, Scharlau
and Schulte [7]. Let us start by checking that the category ~ satisfies the
hypotheses (i), (ii) and (iii) of [7].
(i) All idempotents in ~ split. This is clear: if z is an idemponent of Q,
then it is easy to check that ~(Q) is a direct summand of Q.
(ii) Every object Q of ~ has a decomposition Q _~ Ql 0... O Q,., with Q,
indecomposable and End(Q,) local. This follows from [9], Sect. 2, (8), (10)
and (11).
(iii) For every object Q of ~, the ring of endomorphisms End(Q) is J(Q)-
adically complete, where J(Q) is the radical of End(Q). Indeed, there exists
a positive integer k such that j(Q)k C 7zEnd(Q) C J(Q) (cf. [2], 5.22), hence
the J(Q)-adic topology of End (Q) is equivalent to the ~z-adic topology.
Moreover, we are assuming that the center of A contains an element a
such that a + 6 = 1. This element a belongs to the endomorphism ring of any
object of 2?. Therefore every e-hermitian form over 2~ is even. By [7], 3.4, we
obtain the desired cancellation result.
8. Springer's theorem for E-hermitian forms
Let K be a local field, CK its ring of integers, and AK an t~x-algebra of finite
rank with an t~x-linear involution -. Let L be a finite extension of K, Cr its
ring of integers, and AL = AK @e~ ~L with the 6.'L-linear involution that extends
-. As in Sect. 4, let 27K (resp. 27t) be denote the category of morphisms of re-
flexive modules over AK (resp. AL). If Q = (M,N, f) is an object of 27K, we set
QL = (ML,Nt,fL) where Mt = M | CL, NL = N | CL, and fL extends f.
Non unimodular Hermitian forms 239
Lemma 1.
Let Q = (M,N,f) and Q'= (M',N',f') be two object in )QK.
Then QL is isomorphic to Q~ in ~z if and only if Q is isomorphic' to Q'
in dx.
Proof
Let (qS,~b)' QL---+ Q~ be a ~L-isomorphism. Then q~ and ~b are AL-
linear and commute with fL and
f'L.
Let n =
[L'K],
and note that 6L is
a free 6)K-module of rank n. We can also see QL and Q~ as objects in
~X :ME, NL, M~, N[ are A~:-modules, isomorphic to
M '~, N ~, M", N ~"
re-
spectively, and
fc, f~
are A~c-linear. The ~c-isomorphism (~b,~b) is also a
~K-isomorphism from QL --~ Q" to Q~ ~- Q~". So, by the Krull-Schmidt theo-
rem (see for instance [2] or [9]), the summands Q and Q~ are ~K-isomorphic.
A theorem of Springer states that if two quadratic forms over a field k
become isomorphic over an odd degree extension of k, then they are already
isomorphic over k (see for instance [11], 2.5.4). Th. 4. below is an analogue
of this result of Springer.
Theorem 4.
Suppose that L/K is' an odd degree extension, and that the resid-
ual extension is separable. Let (M,h) and (M~,h ~) be two a-hermitian forms
over AK. If (ML,hL) and (M[,h2) are isomorphic over AL then (M,h) and
(M',h ~) are isomorphic over AK.
Proof
Consider the morphisms
q(ML, hL), q(M[,h~),
and
q(M,h), q(M',h').
The first two are ~?L-isomorphic, so by the lemma above, the last two are
~K-isomorphic. Let Q =
q(M,h)
and let E be the ring of ~
of Q. By th. 2, the set of isomorphism classes of e-hermitian forms (M',h')
such that
q(M',h') ~- Q
is in bijection with the set of isomorphism classes of
rank one unimodular hermitian forms over (E, - ). So we have two unimodular
hermitian forms r/, r over E that extend to isomorphic forms over EL = E Qe~
[~'L. By Springer's theorem for unimodular hermitian forms over E (cf. [3]),
we have ~/-~ r/. Hence
(M,h) ~_ (M',h r)
over
AK.
This concludes the proof of
the theorem.
Acknowledgements.
We thank the Swiss National Foundation for Scientific Research for
partial support during the preparation of this paper, and the University of Geneva for its
hospitality.
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