The geometry of finite dimensional algebras with vanishing radical square

TL;DR

This paper characterizes the irreducible components of module varieties for finite dimensional algebras with radical square zero, providing formulas, geometric insights, and implications for representation type.

Contribution

It offers a complete description of module variety components for radical square zero algebras, linking their properties to hereditary cases and extending existing counts.

Findings

Count of irreducible components in terms of Gabriel quiver

Characterization of modules parametrized by each component

Algebras with dense orbit property have finite representation type

Abstract

Let $\Lambda$ be a basic finite dimensional algebra over an algebraically closed field, with the property that the square of the Jacobson radical $J$ vanishes. We determine the irreducible components of the module variety $\text{Mod}_{\bf d}(\Lambda)$ for any dimension vector $\bf d$. Our description leads to a count of the components in terms of the underlying Gabriel quiver. A closed formula for the number of components when $\Lambda$ is local extends existing counts for the two-loop quiver to quivers with arbitrary finite sets of loops. For any algebra $\Lambda$ with $J^2 = 0$, our criteria for identifying the components of $\text{Mod}_{\bf d}(\Lambda)$ permit us to characterize the modules parametrized by the individual irreducible components. Focusing on such a component, we explore generic properties of the corresponding modules by establishing a geometric bridge between the algebras with zero radical square on one hand and their stably equivalent hereditary counterparts on the other. The bridge links certain closed subvarieties of Grassmannians parametrizing the modules with fixed top over the two types of algebras. By way of this connection, we transfer results of Kac and Schofield from the hereditary case to algebras of Loewy length $2$. Finally, we use the transit of information to show that any algebra of Loewy length $2$ which enjoys the dense orbit property in the sense of Chindris, Kinser and Weyman has finite representation type.