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 étale 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 étale 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 $A$.

12 pages; rewritten to improve readability

References in corpus (2)

• Geometry of the arithmetic site
• Azumaya toposes

Cited by in corpus (1)

• Grothendieck topologies on posets