We consider the general circumstance of an Azumaya algebra A of degree n over a locally ringed topos (X, Ox) where the latter carries a (possibly trivial) involution, denoted A. 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 λ are locally isomor-phic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra A is Brauer equivalent to an algebra carrying an involution extending λ, 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 A is Brauer equivalent to an Azumaya algebra of degree 2n carrying an involution. By comparison with the case of topological spaces, we show that the integer 2n is minimal, even in the case of a nonsingular affine variety X with a fixed-point free involution. As an incidental step, we show that if fi is a commutative ring with involution for which the fixed ring S is local, then either R is local or R/S is a quadratic ´etale extension of rings.
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The authors would like to thank Zinovy Reichstein for introducing them to each other and recommending that they study involutions of Azumaya algebras from a topological point of view. They would like to thank Asher Auel for helpful conversations and good ideas, some of which appear in this paper. They owe an early form of an argument in 5.4 to Sune Precht Reeh. The second author would like to thank Omar Antol??n, Akhil Mathew, Mona Merling, Marc Stephan and Ric Wade for various conversations about equivariant classifying spaces, and Bert Guillou for a reference to the literature on equivariant model structures. The second author would like to thank Ben Antieau for innumerable valuable conversations about Azumaya algebras from the topological point of view, and would like to thank Gwendolyn Billett for help in deciphering [Gir71]. We also thank the referees for many valuable suggestions.
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- Azumaya algebra
- Brauer group
- equivariant homotopy theory
- sheaf cohomology
ASJC Scopus subject areas
- Mathematics (all)