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 $En{d}_{\Lambda }(G)=k(x)$. 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 $2\le d\le 2n$. 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.

arXiv comment29 pages. This is the version accepted in Mathematische Zeitschrift. Some expositions are improved in the new version