An isomorphism problem for Azumaya algebras with involution over semilocal Bézout domains.
Let \(R\) be a semilocal integral domain with quotient field \(K\) and \(2\in R^\times\). One of Grothendieck's conjectures states that for a reductive \(R\)-linear group \(G\), any \(G\)-torsor which is trivial over \(K\) is already trivial over \(R\). In 1989, Nisnevich has proved this for regular local rings \(R\) which are either Henselian or one-dimensional. For a complete discrete valuation ring \(R\), this result was already known to Tits. If \(R\) is a regular local algebra over a field, Panin (2005) proved the conjecture for the automorphism group of an \(R\)-algebra with involution. In the paper under review, the authors prove that for a semilocal Bézout domain \(R\), two \(R\)-algebras with involution are isomorphic whenever the corresponding \(K\)-algebras are isomorphic.
scientific article; zbMATH DE number 6384232
scientific article