Moduli spaces of graded representations of finite dimensional algebras

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.