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Publication# Orders that are étale-locally isomorphic

Abstract

Let R be a semilocal Dedekind domain with fraction field F. It is shown that two hereditary R-orders in central simple F-algebras that become isomorphic after tensoring with F and with some faithfully flat étale R-algebra are isomorphic. On the other hand, this fails for hereditary orders with involution. The latter stands in contrast to a result of the first two authors, who proved this statement for Hermitian forms over hereditary R-orders with involution. The results can be restated by means of étale cohomology and can be viewed as variations of the Grothendieck–Serre conjecture on principal homogeneous spaces of reductive group schemes. The relationship with Bruhat–Tits theory is also discussed.

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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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