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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On inner product spaces over Dedekind domains of characteristic two


Author: Richard C. Wagner
Journal: Proc. Amer. Math. Soc. 93 (1985), 1-9
MSC: Primary 11E88; Secondary 13F05, 15A63
MathSciNet review: 766516
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Abstract: Suppose $ D$ is a Dedekind domain of characteristic 2 and $ (M,\varphi )$ is an inner product space, i.e. $ M$ is a finitely generated projective $ D$ module supplied with a nonsingular symmetric bilinear form $ \varphi $. It is shown that $ (M,\varphi )$ is determined up to isometry by the extension of $ \varphi $ to $ F{ \otimes _D}M$, where $ F$ is the quotient field of $ D$, and the value module $ {\mathcal Q}(M)$ of all $ \varphi (m,m)$ for $ m$ in $ M$. In particular, a hyperbolic space $ {\mathbf{H}}(M)$ is completely determined by the rank of the finitely generated projective module $ M$. As consequences, genera coincide with isometry classes, and if $ {N_1}$ and $ {N_2}$ are isometric nonsingular submodules of $ (M,\varphi )$ such that $ {\mathcal Q}({N_1}^ \bot ) = {\mathcal Q}({N_2}^ \bot )$, then $ {N_1}^ \bot $ and $ {N_2}^ \bot $ are isometric. Also, given an $ F$ inner product space $ (V,\varphi )$ and a $ {D^{(2)}}$ submodule $ P$ of $ D$, a necessary and sufficient condition is given for the existence of a $ D$ inner product space $ (M,\Psi )$ such that $ (FM,\Psi ) \cong (V,\varphi )$ and $ {\mathcal Q}(M) = P$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0766516-5
PII: S 0002-9939(1985)0766516-5
Keywords: Symmetric bilinear forms, value modules, hyperbolic and metabolic spaces, Witt cancellation, genera, Dedekind and discrete valuation rings
Article copyright: © Copyright 1985 American Mathematical Society