Closures in varieties of representations and irreducible components

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.