Orders that are étale-locally isomorphic

Bayer-Fluckiger, E.; First, U.; Huruguen, M.

doi:10.1090/spmj/1615

Orders that are étale-locally isomorphic
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by E. Bayer-Fluckiger, U. A. First and M. Huruguen

St. Petersburg Math. J. 31 (2020), 573-584

DOI: https://doi.org/10.1090/spmj/1615

Published electronically: June 11, 2020

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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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