On generic bricks over tame algebras

TL;DR

This paper characterizes when a generic module over a tame algebra is a brick, linking it to the existence of one-parameter families of bricks and the property of brick-continuity.

Contribution

It establishes a precise criterion for generic bricks over tame algebras, connecting module theory with algebraic properties like brick-continuity.

Findings

A generic module is a brick if and only if it determines a one-parameter family of bricks.

The algebra admits a generic brick if and only if it is brick-continuous.

Abstract

We prove that if ${\Lambda}$ is a tame finite-dimensional algebra over an algebraically closed field and $G$ is a generic ${\Lambda}-$module, then $G$ is a generic brick if and only if it determines a one-parameter family of bricks with the same dimension. In particular, we obtain that ${\Lambda}$ admits a generic brick if and only if ${\Lambda}$ is brick-continuous.