Patching and weak approximation in isometry groups

Bayer-Fluckiger, Eva; First, Uriya

doi:10.1090/tran/6921

Patching and weak approximation in isometry groups
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by Eva Bayer-Fluckiger and Uriya A. First PDF

Trans. Amer. Math. Soc. 369 (2017), 7999-8035 Request permission

Abstract:

Let $R$ be a semilocal principal ideal domain. Two algebraic objects over $R$ in which scalar extension makes sense (e.g. quadratic spaces) are said to be of the same genus if they become isomorphic after extending scalars to all completions of $R$ and its fraction field. We prove that the number of isomorphism classes in the genus of unimodular quadratic spaces over (not necessarily commutative) $R$-orders is always a finite power of $2$, and under further assumptions, e.g., that the order is hereditary, this number is $1$. The same result is also shown for related objects, e.g., systems of sesquilinear forms. A key ingredient in the proof is a weak approximation theorem for groups of isometries, which is valid over any (topological) base field, and even over semilocal base rings.

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