Biserial algebras and generic bricks

Abstract

We consider generic bricks and use them in the study of arbitrary biserial algebras over algebraically closed fields. For a biserial algebra $\Lambda$, we show that $\Lambda$ is brick-infinite if and only if it admits a generic brick, that is, there exists a generic $\Lambda$-module $G$ with $End_{\Lambda}(G)=k(x)$. Furthermore, we give an explicit numerical condition for brick-infiniteness of biserial algebras: If $\Lambda$ is of rank $n$, then $\Lambda$ is brick-infinite if and only if there exists an infinite family of bricks of length $d$, for some $2\leq d\leq 2n$. This also results in an algebro-geometric realization of $\tau$-tilting finiteness of this family: $\Lambda$ is $\tau$-tilting finite if and only if $\Lambda$ is brick-discrete, meaning that in every representation variety $mod(\Lambda, \underline{d})$, there are only finitely many orbits of bricks. Our results rely on our full classification of minimal brick-infinite biserial algebras in terms of quivers and relations. This is the modern analogue of the recent classification of minimal representation-infinite (special) biserial algebras, given by Ringel. In particular, we show that every minimal brick-infinite biserial algebra is gentle and admits exactly one generic brick. Furthermore, we describe the spectrum of such algebras, which is very similar to that of a tame hereditary algebra. In other words, $Brick(\Lambda)$ is the disjoint union of a unique generic brick with a countable infinite set of bricks of finite length, and a family of bricks of the same finite length parametrized by the ground field.