Involutions of Azumaya Algebras

Abstract

We consider the general circumstance of an Azumaya algebra $A$ of degree $n$ over a locally ringed topos $(\mathbf{X}, \mathcal O_\mathbf{X})$ where the latter carries a (possibly trivial) involution, denoted $\lambda$. 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 $\lambda$ are locally isomorphic, 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 $\lambda$, 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 $R$ 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 étale extension of rings.