E. Bayer-Fluckiger, U. A. First, M. Huruguen, “Orders that are étale-locally isomorphic”, Algebra i Analiz, 2019, Volume 31, Issue 4,Pages <nobr>1

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Research Papers

Orders that are étale-locally isomorphic

E. Bayer-Fluckiger^a, U. A. First^b, M. Huruguen^a

^a Department of Mathematics, École Polytechnique Fédérale de Lausanne
^b Department of Mathematics, University of Haifa

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.

Keywords: hereditary order, maximal order, Dedekind domain, group scheme, reductive group, involution, central simple algebra.

MSC: 16H10, 16W10, 11E57, 11E72

Received: 09.07.2018

Language: English