Biserial algebras and generic bricks
Kaveh Mousavand, Charles Paquette
TL;DR
The paper addresses the distribution of bricks in biserial algebras by characterizing brick-infinite algebras through the existence of generic bricks, and it provides a precise numerical criterion tied to the algebra’s rank. It develops a geometric view via representation varieties, linking brick-discreteness to $\tau$-tilting finiteness, and proves a complete classification of minimal brick-infinite biserial algebras as gentle with two possible types: hereditary affine $\widetilde{A}_n$ or generalized barbell algebras. This yields a tame-like spectrum for brick modules: a unique generic brick coexists with a countable set of bricks of finite length and a field-parametrized family of bricks of fixed length. The results illuminate the interplay between generic bricks, domestic/generic-brick-domestic behavior, and the algebro-geometric structure of representation spaces, offering new avenues to study $\tau$-tilting finiteness and brick distribution in tame algebras.
Abstract
We consider generic bricks and use them in the study of arbitrary biserial algebras over algebraically closed fields. For a biserial algebra $Λ$, we show that $Λ$ is brick-infinite if and only if it admits a generic brick, that is, there exists a generic $Λ$-module $G$ with $End_Λ(G)=k(x)$. Furthermore, we give an explicit numerical condition for brick-infiniteness of biserial algebras: If $Λ$ is of rank $n$, then $Λ$ 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 $τ$-tilting finiteness of this family: $Λ$ is $τ$-tilting finite if and only if $Λ$ is brick-discrete, meaning that in every representation variety $mod(Λ, \underline{d})$, 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, $Brick(Λ)$ 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.