Orders That Are Etale-Locally Isomorphic

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 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 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.