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Orders that are étale-locally isomorphic
E. Bayer-Fluckigera,
U. A. Firstb,
M. Huruguena a Department of Mathematics, École Polytechnique Fédérale de Lausanne
b Department of Mathematics, University of Haifa
Аннотация:
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.
Ключевые слова:
hereditary order, maximal order, Dedekind domain, group scheme, reductive group, involution, central simple algebra.
MSC: 16H10,
16W10,
11E57,
11E72 Поступила в редакцию: 09.07.2018
Язык публикации: английский