An isomorphism problem for Azumaya algebras with involution over semilocal Bézout domains. (English) Zbl 1314.16010
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.
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.
Reviewer: Wolfgang Rump (Stuttgart)
MSC:
| 16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 14L15 | Group schemes |
| 20G35 | Linear algebraic groups over adèles and other rings and schemes |
Keywords:
Azumaya algebras with involution; central simple algebras with involution; reductive linear algebraic groups; valuation rings; semilocal Bézout domains; skew-Hermitian spaces; bilinear spaces; multipliersReferences:
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