Rational points in coarse moduli spaces and twisted representations
Fabian Korthauer, Stefan Schröer
TL;DR
The paper develops a general framework for moduli of representations of associative algebras in Azumaya algebras, focusing on Schur representations and twisted forms. It constructs a quasiaffine space $X$ of Schur representations, a freely-acting group $H$ with quotient $Q$, and shows the true moduli stack of twisted Schur representations $\mathscr{M}$ is equivalent to the quotient stack $[X/G^{\mathrm{op}}/Q]$, with $Q$ as the coarse moduli space. The work introduces the gerbe of splittings, the tautological tilting sheaf $\mathscr{T}_{\mathscr{M}}$, and derives Azumaya algebras and Brauer-class relations on $Q$ that reflect the underlying representation theory. It also analyzes how non-abelian cohomology governs when a point of $Q$ has geometric origin and explains how to modify moduli to obtain geometric origins via twisting, with concrete implications for quiver representations through Grassmannian embeddings. Overall, the paper provides a robust stack-theoretic and Brauer-theoretic treatment of representations in Azumaya algebras, linking moduli, twisted forms, and non-abelian cohomology in a coherent framework.
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.