Indecomposables in derived categories of gentle algebras
type: Item
instance of: scholarly article
title: Indecomposables in derived categories of gentle algebras
zbMATH Open document ID: 1032.16011
DOI: 10.1023/A:1025142023594
author: Viktor Bekkert
author: Héctor A. Merklen
published in: Algebras and Representation Theory
publication date: 2003-10-15T00:00:00Z
review text: The authors parameterize the indecomposable objects of the bounded derived category of modules over a gentle algebra. Gentle algebras were introduced by Assem and Skowroński as a class of tame algebras by some restricted conditions on the quiver and the relations defining the algebra. The authors identify the problem of determining the indecomposable objects with a matrix problem presented by Bondarenko. Though the result of the classification is not really complicated it is still too technical to be stated here. As an application the authors characterize when such a gentle algebra is derived discrete (as defined by Vossieck) or derived finite (meaning that up to shift there are only finitely many isomorphism classes of indecomposable objects). They also obtain as a corollary that gentle algebras are derived tame. That the bounded derived category of a gentle algebra is tame is usually shown using the repetitive algebra and Happel's embedding. Recently Igor Burban showed in his 2003 thesis that also the right bounded derived category is tame.
reviewed by: Alexander Zimmermann
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 1993342
zbMATH Keywords: gentle algebras
zbMATH Keywords: derived categories
zbMATH Keywords: derived tameness
zbMATH Keywords: Happel functor
zbMATH Keywords: categories of modules
zbMATH Keywords: indecomposable modules
zbMATH Keywords: matrix problems
MaRDI profile type: Publication
full work available at URL: a:1025142023594
OpenAlex ID: W106621953
description: scientific article; zbMATH DE number 1993342