Classifying representations by way of Grassmannians

TL;DR

This paper characterizes when moduli spaces of finite-dimensional algebra representations with fixed dimension and simple top exist, showing they are projective subvarieties of Grassmannians, and explores their properties and applications.

Contribution

It provides criteria for the existence of moduli spaces, describes their structure as Grassmannian subvarieties, and applies these results to classify algebras with finitely many modules of a given top.

Findings

Moduli spaces exist under specific criteria and are projective subvarieties of Grassmannians.

The radical layering is a key invariant for classifying modules with fixed simple top.

Proper degenerations of local modules do not preserve radical layering.

Abstract

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed dimension $d$ and fixed squarefree top $T$. Next to providing a complete theoretical picture, some of these equivalent conditions are readily checkable from quiver and relations. In case of existence of a moduli space -- unexpectedly frequent in light of the stringency of fine classification -- this space is always projective and, in fact, arises as a closed subvariety ${\mathfrak{Grass}}^T_d$ of a classical Grassmannian. Even when the full moduli problem fails to be solvable, the variety ${\mathfrak{Grass}}^T_d$ is seen to have distinctive properties recommending it as a substitute for a moduli space. As an application, a characterization of the algebras having only finitely many representations with fixed simple top is obtained; in this case of `finite local representation type at a given simple $T$', the radical layering $\bigl( J^lM/ J^{l+1}M \bigr)_{l \ge 0}$ is shown to be a classifying invariant for the modules with top $T$. This relies on the following general fact obtained as a byproduct: Proper degenerations of a local module $M$ never have the same radical layering as $M$.