Moduli spaces of graded representations of finite dimensional algebras

TL;DR

This paper studies the moduli spaces of graded modules over finite dimensional algebras, showing their frequent existence and explicit construction, especially in the local case, and characterizing modules with coarse moduli spaces.

Contribution

It demonstrates the existence of moduli spaces for graded modules more often than for ungraded ones and provides explicit constructions in the local case.

Findings

Existence of fine moduli spaces for graded modules with simple top.

Construction of moduli spaces from quiver and relations.

Modules with coarse moduli spaces are direct sums of local modules.

Abstract

Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, presented as a path algebra modulo relations; further, assume that $\Lambda$ is graded by lengths of paths. The paper addresses the classifiability, via moduli spaces, of classes of graded $\Lambda$-modules with fixed dimension $d$ and fixed top $T$. It is shown that such moduli spaces exist far more frequently than they do for ungraded modules. In the local case (i.e., when $T$ is simple), the graded $d$-dimensional $\Lambda$-modules with top $T$ always possess a fine moduli space which classifies these modules up to graded-isomorphism; moreover, this moduli space is a projective variety with a distinguished affine cover that can be constructed from quiver and relations of $\Lambda$. When $T$ is not simple, existence of a coarse moduli space for the graded $d$-dimensional $\Lambda$-modules with top $T$ forces these modules to be direct sums of local modules; under the latter condition, a finite collection of isomorphism invariants of the modules in question yields a partition into subclasses, each of which has a fine moduli space (again projective) parametrizing the corresponding graded-isomorphism classes.