Azumaya geometry and representation stacks

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 $A$.