Cohomology of Moduli of Representations of Monomial Algebras

TL;DR

This paper extends methods from quiver representation theory to compute the cohomology of moduli spaces of representations of monomial algebras with relations of length two, providing an algorithm for their classes in the Grothendieck ring.

Contribution

It introduces a new approach to study moduli spaces of monomial algebra representations, generalizing techniques from quiver theory to higher homological dimension cases.

Findings

Developed an algorithm for calculating classes in the Grothendieck ring

Extended quiver methods to monomial algebras with length-two relations

Provided new insights into the cohomology of these moduli spaces

Abstract

In this paper, we study moduli spaces of representations of certain quivers with relations. For quivers without relations and other categories of homological dimension one, a lot of information is known about the cohomology of their moduli spaces of objects. On the other hand, categories of higher homological dimension remain more mysterious from this point of view, with few general methods. In this paper, we will see how some of the methods used to study quivers can be extended to work for representations of any (noncommutative) monomial algebra with relations of length two. In particular, we will give an algorithm to calculate in many cases the classes of these moduli spaces in the Grothendieck ring of varieties.