Involutions of Azumaya Algebras
Uriya A. First
Department of Mathematics, University of Haifa,199 Abba Khoushy Avenue, Haifa 3498838, IsraelBen Williams
Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada
Abstract
We consider the general circumstance of an Azumaya algebra of degree over a locally ringed topos where the latter carries a (possibly trivial) involution, denoted . This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra is Brauer equivalent to an algebra carrying an involution extending , by giving a cohomological condition. We remark that these results are novel even in the case of schemes, since we allow ramified, non-trivial involutions of the base object. We observe that, if the cohomological condition is satisfied, then is Brauer equivalent to an Azumaya algebra of degree carrying an involution. By comparison with the case of topological spaces, we show that the integer is minimal, even in the case of a nonsingular affine variety with a fixed-point free involution. As an incidental step, we show that if is a commutative ring with involution for which the fixed ring is local, then either is local or is a quadratic étale extension of rings.
Cite this article
Uriya A. First, Ben Williams, Involutions of Azumaya Algebras. Doc. Math. 25 (2020), pp. 527–633
DOI 10.4171/DM/756