Generic modules for the category of filtered by standard modules

TL;DR

The paper develops a robust framework linking generic and tame behavior of Δ-filtered module categories 𝔽(Δ) to interlaced weak ditalgebras and their reductions. By refining triangularity, formalizing bimodule reductions, and constructing P-presentations and admissible equivalences, it shows that 𝔽(Δ) is tame if and only if there are finitely many generic adapted Λ-modules of any fixed endolength, and that not-wildness implies generically tame with explicit parametrizations M ≅ Z_G ⊗_{Γ_G} N. The results generalize prior work on standardly stratified and quasi-hereditary settings, provide a tame-wild dichotomy, and give concrete tools for reducing classification problems to minimal ditalgebras. The approach yields explicit descriptions of generic modules via rational algebras Γ_G and bimodules Z_G, and establishes invariances of tameness across admissible equivalences and Δ-presentations, with broad applicability to filtered category theories in finite-dimensional algebras.

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].

Theorems & Definitions (135)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Definition 1.7
• Theorem 1.8
• Definition 2.1
• Remark 2.2