2015/04/06 by Eva Bayer-Fluckiger, Bayer-Fluckiger, Eva, Uriya A. First +1
Mathematics · #11E39 #11E41 #16H10 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:11E39 #msc:11E41 #msc:16H10
paper · pdf · doi:10.48550/arxiv.1504.01280
33 pages. Changes from previous version: results concerning hereditary orders in separable algebras now hold for hereditary orders in general algebras, other mild clarifications
arxiv created 2016/01/08 · arxiv updated 2016/01/12
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 (non 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. The appendix proves that the isometry group of a quadratic space over an R-order with involution can be regarded as a smooth affine group scheme under mild assumptions.