Cycle-finite algebras
type: Item
instance of: scholarly article
title: Cycle-finite algebras
zbMATH Open document ID: 0841.16020
DOI: 10.1016/0022-4049(94)00094-Y
published in: Journal of Pure and Applied Algebra
publication date: 1996-07-15T00:00:00Z
review text: Let \(A\) be a finite dimensional algebra over an algebraically closed field, and \(\text{mod }A\) denote the category of its finitely generated right modules. Then \(A\) is said to be cycle-finite if, for every sequence \(M_0 @>f_1>> M_1 @>f_2>> M_2\to\dots @>f_n>> M_n=M_0\) of non-zero non-isomorphisms between indecomposable objects in \(\text{mod }A\), none of the \(f_i\) lies in the infinite radical \(\text{rad}^\infty(\text{mod }A)\) of \(\text{mod }A\). The main result of the paper states that if \(A\) is cycle-finite, then the Auslander-Reiten quiver \(\Gamma(\text{mod }A)\) of \(A\) has a sincere stable tube if and only if \(A\) is tame concealed or tubular. As a consequence, any cycle-finite algebra is of polynomial growth. These results generalise those obtained by the author and the reviewer for multicoil algebras [Math. Scand. 71, No. 1, 31-61 (1992; Zbl 0796.16014)]. In addition, it is shown that the following conditions are equivalent for a cycle-finite algebra \(A\): (a) \(A\) is domestic, (b) \(A\) contains no tubular algebra as a full convex subcategory, (c) \(\text{rad}^\infty(\text{mod }A)\) is nilpotent, (d) all but at most finitely many components of \(\Gamma(\text{mod }A)\) are stable tubes of rank 1.
reviewed by: Ibrahim Assem
Mathematics Subject Classification ID: 16G70
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16G20
zbMATH DE Number: 810251
zbMATH Keywords: tame concealed algebras
zbMATH Keywords: cycles in module categories
zbMATH Keywords: tame algebras of polynomial growth
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: finitely generated right modules
zbMATH Keywords: infinite radical
zbMATH Keywords: sincere stable tubes
zbMATH Keywords: cycle-finite algebras
zbMATH Keywords: polynomial growth
zbMATH Keywords: multicoil algebras
zbMATH Keywords: tubular algebras
zbMATH Keywords: Auslander-Reiten quivers
author: Andrzej Skowroński
MaRDI profile type: Publication
cites work: Minimal representation-infinite coil algebras
cites work: Indecomposable Modules over Multicoil Algebras.
cites work: Q4288677
cites work: On Tame Algebras and Bocses
cites work: Tame Algebras and Generic Modules
cites work: Q3889229
cites work: Tilting Wild Algebras
cites work: Q3204617
cites work: Semi-Stable Components of an Auslander-Reiten Quiver
cites work: Infinite radicals in standard Auslander-Reiten components
cites work: Polynomial growth trivial extensions of simply connected algebras
cites work: Iterated tubular algebras
cites work: Tame algebras and integral quadratic forms
cites work: Q3353193
cites work: Regular Auslander-Reiten Components Containing Directing Modules
cites work: Generalized standard Auslander-Reiten components
cites work: On the Composition Factors of Periodic Modules
cites work: Q4311997
cites work: Q4322118
cites work: Directing modules
full work available at URL: 0022-4049(94)00094-y
OpenAlex ID: W2025933434
Recommended article: Indecomposable Modules over Multicoil Algebras.
Recommended article: Q4288677
Recommended article: Q3135174
Recommended article: Minimal representation-infinite coil algebras
Recommended article: Q4875148
Recommended article: On the structure of periodic modules over tame algebras
Recommended article: Cycles of modules and finite representation type
Recommended article: Cycle-Finite Module Categories
Recommended article: Algebras with cycle-finite derived categories
Recommended article: Cycle-finite algebras of semiregular type
description: scientific article; zbMATH DE number 810251