Hermitian Analogue of a Theorem of Springer - Chennai ...

HERMITIAN ANALOGUE OF A THEOREM OF SPRINGER 781
1. SOME PRELIMINARIES
We fix some notation in this section and recall the definition of the Witt
group of hermitian forms.
Let Ž X, O . X be a scheme and let A be a sheaf of Azumaya algebras with
an involution � of the first kind over X, i.e., � : A � A is an anti-automorphism
of the O -algebras with � 2 X
� id. Let E be a locally free sheaf
of O -modules with a right A structure. Let E � X
be the left A-module
Hom Ž E, A . , regarded as a right A-module through � . For � � � Ž X .
A 2 ,a
Ž � , � . -hermitian space over A is a pair Ž E, h . , where
h: E � E � ,
� is an isomorphism of right A-modules, with h � � h Žthe
natural map
�� E � E being regarded as an identification . . We say that a Ž � , � . -
hermitian space Ž E, h. is metabolic if there exists an A-subsheaf W of E,
with E�W locally free, such that W � W � , where W � composite morphism
is the kernel of the
h
� �
E � E � W .
There is an obvious notion of an isometry of Ž � , � . -hermitian spaces over
A. Let CŽ A, � , � . denote the set of isomorphism classes of Ž � , � . -hermitian
spaces over A. The orthogonal sum of two hermitian spaces induces a
Ž . � monoid structure on C A, � , � . Let W Ž A, � . be the quotient of the
Grothendieck group of CŽ A, � , � . by the subgroup generated by classes of
metabolic spaces. This is called the Witt group of Ž � , � . -hermitian spaces
over A. AŽ � , � . -hermitian space is said to be isotropic if there exists a
non-zero subsheaf W of E such that E�W is locally free and W � W � .
Ž . � If X � Spec k , where k is a field and A � A, then W Ž A, � . is the
usual Witt group of Ž � , � . -hermitian spaces over A. We note that in this
case, the notion of metabolic spaces coincides with the usual notion of
hyberbolic spaces. Further if A � k, then � is identity and WŽ A, � . is the
Witt group WŽ k. of quadratic forms over k. AŽ � , � . -hermitian space over
a central simple algebra A is also called an �-hermitian form over Ž A, � . .
For a quadratic extension L of k, we also have the notion of �-hermitian
� forms and the corresponding Witt group is denoted by W Ž L, � . , where �
is the nontrivial automorphism of L over k.
Throughout this paper, k denotes a field of characteristic not equal to 2
and A denotes a finite dimensional central simple algebra over k. Byan
A-module we mean a right A-module. Since A is a central simple algebra
over a field k, we note that all modules over A are projective.