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arXiv · 1804.09527

Orders that are \'Etale-Locally Isomorphic

Eva Bayer-Fluckiger·Uriya A. First·Mathieu Huruguen
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Abstract

Let $R$ be a semilocal Dedekind domain with fraction field $F$. We show that two hereditary $R$-orders in central simple $F$-algebras which become isomorphic after tensoring with $F$ and with some faithfully flat \'etale $R$-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 \'etale cohomology and can be seen as variations of the Grothendieck--Serre conjecture on principal homogeneous bundles of reductive group schemes. Connections with Bruhat--Tits theory are also discussed.

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BibTeXRIS

Eva Bayer-Fluckiger, Uriya A. First, Mathieu Huruguen. 2018-04-25. Orders that are \'Etale-Locally Isomorphic. https://arxiv.org/abs/1804.09527

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