Algebras with irreducible module varieties I

TL;DR

This paper introduces the concept of geometrically irreducible algebras, characterizes those without loops as hereditary, and identifies truncated polynomial rings as the only local cases, proposing a broader classification conjecture.

Contribution

It defines geometrically irreducible algebras and provides characterizations for specific classes, including hereditary and local algebras, along with a conjectural classification.

Findings

Hereditary algebras are geometrically irreducible without loops.

Truncated polynomial rings are the only geometrically irreducible local algebras.

A conjectural classification of all geometrically irreducible algebras is proposed.

Abstract

We call a finite-dimensional K-algebra A geometrically irreducible if for all d, all connected components of the affine scheme of d-dimensional A-modules are irreducible. We show that the geometrically irreducible algebras without loops (this includes all algebras of finite global dimension) are the hereditary algebras. We also show that truncated polynomial rings are the only geometrically irreducible local algebras. Finally, we formulate a conjectural classification of all geometrically irreducible algebras.