Involutions of Azumaya algebras

TL;DR

This paper studies involutions on Azumaya algebras over a topos with involution, providing criteria for local isomorphism, Brauer equivalence, and extending involutions, including cases with ramification and nontrivial involutions.

Contribution

It generalizes involution theory of Azumaya algebras to a broad topos context, introducing new cohomological criteria and minimal degree results.

Findings

Provides a criterion for local isomorphism of Azumaya algebras with involution.

Characterizes when an Azumaya algebra is Brauer equivalent to one with an involution.

Shows the minimal degree for such algebras is 2n, even in complex cases.

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, nontrivial 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 \'etale extension of rings.