Rational points in coarse moduli spaces and twisted representations

TL;DR

This paper explores moduli spaces of algebra representations in Azumaya algebras, focusing on twisted representations and their geometric origins, without stability constraints, over general ground rings.

Contribution

It introduces a framework for studying moduli stacks of twisted representations in Azumaya algebras, extending classical moduli spaces to include non-abelian cohomology insights.

Findings

Moduli stacks include twisted representations on gerbes.

Classical rational points correspond to Azumaya algebra representations.

The approach generalizes classical moduli spaces without stability conditions.

Abstract

We study moduli spaces and moduli stacks for representations of associative algebras in Azumaya algebras, in rather general settings. We do not impose any stability condition and work over arbitrary ground rings, but restrict attention to the so-called Schur representations, where the only automorphisms are scalar multiplications. The stack comprises twisted representations, which are representations that live on the gerbe of splittings for the Azumaya algebra. Such generalized spaces and stacks appear naturally: For any rational point on the classical coarse moduli space of matrix representations, the machinery of non-abelian cohomology produces a modified moduli problem for which the point acquires geometric origin. The latter are given by representations in Azumaya algebras.