  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.
