Generic modules for the category of filtered by standard modules

Abstract

Here we show that, given a finite homological system $(\mathcal{P},\le ,\{\hat{I}{”}_u{\}}_{u\in \mathcal{P}})$ for a finite-dimensional algebra $\hat{I}›$ over an algebraically closed field, the category $\mathcal{F}(\hat{I}”)$ of $\hat{I}”$-filtered modules is tame if and only if, for any $d\in \mathbb{N}$, there are only finitely many isomorphism classes of generic $\hat{I}›$-modules adapted to $\mathcal{F}(\hat{I}”)$ with endolength $d$. We study the relationship between these generic modules and one-parameter families of indecomposables in $\mathcal{F}(\hat{I}”)$. 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].

49 pages