Bayer-Fluckiger, E.First, U. A.Huruguen, M.2020-07-082020-07-082020-07-082020-01-0110.1090/spmj/1615https://infoscience.epfl.ch/handle/20.500.14299/169881WOS:000541709700001Let 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.Mathematicshereditary ordermaximal orderdedekind domaingroup schemereductive groupinvolutioncentral simple algebraprincipal homogeneous spacesgroup schemestext::journal::journal article::research article