Orders that are \'Etale-Locally Isomorphic

TL;DR

This paper proves that hereditary orders over a semilocal Dedekind domain are uniquely determined by their local isomorphisms after certain extensions, contrasting with cases involving involutions, and relates these results to broader algebraic geometry conjectures.

Contribution

It establishes a new isomorphism criterion for hereditary orders over semilocal Dedekind domains, extending understanding of their classification and contrasting with involution cases.

Findings

Hereditary orders are uniquely determined by étale-locally isomorphic conditions.

The result does not hold for hereditary orders with involution.

Connections to Grothendieck--Serre conjecture and Bruhat--Tits theory are discussed.

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.