Generic modules for the category of filtered by standard modules

Abstract

Here we show that, given a finite homological system $({\cal P},\leq,\{Δ_u\}_{u\in {\cal P}})$ for a finite-dimensional algebra $Λ$ over an algebraically closed field, the category ${\cal F}(Δ)$ of $Δ$-filtered modules is tame if and only if, for any $d\in \mathbb{N}$, there are only finitely many isomorphism classes of generic $Λ$-modules adapted to ${\cal F}(Δ)$ with endolength $d$. We study the relationship between these generic modules and one-parameter families of indecomposables in ${\cal F}(Δ)$. This study applies in particular to the category of modules filtered by standard modules for standardly stratified algebras. This article includes a correction of an error in [8].