Orders that are \'Etale-Locally Isomorphic

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.