Azumaya geometry and representation stacks

TL;DR

The paper introduces Azumaya geometry as an extension of classical affine geometry, constructing a sheaf of representation stacks that generalizes quotient stacks and explores their topological properties.

Contribution

It develops Azumaya geometry and shows how to extend classical topologies to this setting, establishing sheaf properties for representation stacks and defining Azumaya representation schemes.

Findings

Azumaya geometry extends classical affine geometry to Azumaya algebras.

Representation stacks form sheaves under extended Grothendieck topologies.

Azumaya representation schemes are affine schemes representing sheaves in the Azumaya setting.

Abstract

We develop Azumaya geometry, which is an extension of classical affine geometry to the world of Azumaya algebras, and package the information contained in all quotient stacks $[\mathrm{rep}_n R\,/\,\mathrm{PGL}_n]$ into a presheaf $\mathrm{Rep}_R$ on it. We show that the classical \'etale and Zariski topologies extend to Grothendieck topologies on Azumaya geometry in uncountably many ways, and prove that $\mathrm{Rep}_R$ is a sheaf for all of them. The restriction to a specific Azumaya algebra $A$ with center $C$ gives us a sheaf in the \'etale topology which is represented by an affine $C$-scheme $\mathrm{rep}_A(R)$, which we call the Azumaya representation scheme of $R$ with respect to $A$.