Generic modules for the category of filtered by standard modules

TL;DR

This paper characterizes when the category of filtered modules over a finite-dimensional algebra is tame, linking it to the finiteness of certain generic modules and exploring their relation to indecomposable modules.

Contribution

It establishes a criterion for tameness of filtered module categories based on the finiteness of generic modules with fixed endolength, extending understanding of module category complexity.

Findings

Tame category of filtered modules iff finitely many generic modules per endolength

Relationship between generic modules and one-parameter families of indecomposables

Application to modules filtered by standard modules in stratified algebras

Abstract

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