JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. The Johns Hopkins University Press is collaborating with JSTOR to digitize, preserve and extend access to American Journal of Mathematics.Introduction. Let X be any scheme. By an inner product space over X, we mean a locally free sheaf S of ?x-modules together with a symmetric bilinear form s: ; x S -4x which induces an isomorphism S6 -4* . We have an evident notion of isomorphism of inner product spaces over X. If A is a commutative ring and X=SpecA, any inner product space over X can be identified with a pair (P,s), where P is finitely generated projective A-module and s: P X P-*A is a nonsingular symmetric bilinear form, i.e., s induces an isomorphism P->P* (cf.
[7]). We shall call such a pair (P,s) an inner product A-space. Let B be a commutative A-algebra. If (P,s) is an inner product
A-space, then (B OA P, B OA S) is an inner product B-space. Any inner product B-space isomoxphic to one such is said to be extended from A.The quadratic analogue of Serre's conjecture is the affirmation of:
(QS) Suppose A = K [Xl,..., X] is the algebra of polynomials in n variables over a field K. Is every inner product A-space extended from K?This question is motivated by the following evidence:1. Serre's conjecture that projective A-modules are free and hence extended from K has been proved by Quillen and Suslin (cf.[5]). Moreover, this implies immediately that "symplectic A-modules" are extended from K [2, Chapter IV (4.11.12)]. 2. If charK #2, a theorem of Karoubi [8, Theorem 1.1] implies that every inner product A-space is stably isomorphic to one extended from K. Theorem 13.4.3) gives an affirmative answer to (QS) for n= 1. 4. Bass [3] has given an affinnative answer to (QS) for n = 2, K algebraically closed and of characteristic =#2.
A theorem of Harder (cf. [9],
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