Closures in varieties of representations and irreducible components

TL;DR

This paper classifies irreducible components of module varieties over truncated path algebras using representation-theoretic invariants and module filtrations, providing new tools for understanding their geometric structure.

Contribution

It introduces a complete classification of irreducible components for truncated path algebras and develops a novel invariant for detecting components in module varieties.

Findings

Classified irreducible components of $ ext{Rep}_d( ext{Lambda})$ for truncated path algebras.

Developed a new upper semicontinuous invariant for module varieties.

Provided a characterization of closures of semisimple sequence varieties.

Abstract

For any truncated path algebra $\Lambda$ of a quiver, we classify, by way of representation-theoretic invariants, the irreducible components of the parametrizing varieties $\mathbf{Rep}_{\mathbf{d}}(\Lambda)$ of the $\Lambda$-modules with fixed dimension vector $\mathbf{d}$. In this situation, the components of $\mathbf{Rep}_{\mathbf{d}}(\Lambda)$ are always among the closures $\overline{\mathbf{Rep}\,\mathbb{S}}$, where $\mathbb{S}$ traces the semisimple sequences with dimension vector $\mathbf{d}$, and hence the key to the classification problem lies in a characterization of these closures. Our first result concerning closures actually addresses arbitrary basic finite dimensional algebras over an algebraically closed field. In the general case, it corners the closures $\overline{\mathbf{Rep}\,\mathbb{S}}$ by means of module filtrations "governed by $\mathbb{S}$", in case $\Lambda$ is truncated, it pins down the $\overline{\mathbf{Rep}\,\mathbb{S}}$ completely. The analysis of the varieties $\overline{\mathbf{Rep}\,\mathbb{S}}$ leads to a novel upper semicontinuous module invariant which provides an effective tool towards the detection of components of $\mathbf{Rep}_{\mathbf{d}}(\Lambda)$ in general. It detects all components when $\Lambda$ is truncated.