Azumaya geometry and representation stacks

2016/06/25 by Hemelaer, Jens, Bruyn, Lieven Le
#Algebraic Geometry (math.AG) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1606.07885

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 [rep_n R / PGL_n] into a presheaf 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 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 rep_A(R), which we call the Azumaya representation scheme of R with respect to A.

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