Algebras with homogeneous module category are tame

TL;DR

This paper characterizes tame algebras by showing that their module categories have homogeneous structures, providing an internal criterion for tameness based on Auslander-Reiten quivers.

Contribution

It proves the inverse of Crawley-Boevey's theorem, offering a new internal description of tameness for algebras via Auslander-Reiten quivers.

Findings

Tame algebras have module categories with homogeneous structures.

Provides an internal criterion for algebra tameness.

Connects tameness with properties of Auslander-Reiten quivers.

Abstract

The celebrated Drozd's theorem asserts that a finite-dimensional basic algebra $\Lambda$ over an algebraically closed field $k$ is either tame or wild, whereas the Crawley-Boevey's theorem states that given a tame algebra $\Lambda$ and a dimension $d$, all but finitely many isomorphism classes of indecomposable $\Lambda$-modules of dimension $d$ are isomorphic to their Auslander-Reiten translations and hence belong to homogeneous tubes. In this paper, we prove the inverse of Crawley-Boevey's theorem, which gives an internal description of tameness in terms of Auslander-Reiten quivers.