Algebras with cycle-finite derived categories
type: Item
instance of: scholarly article
title: Algebras with cycle-finite derived categories
zbMATH Open document ID: 0617.16017
DOI: 10.1007/BF01456336
author: Ibrahim Assem
author: Andrzej Skowroński
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164374
review text: Let A be a finite-dimensional algebra (associative, with an identity) over an algebraically closed field, and let \(D^ b(A)\) denote the derived category of bounded complexes of finite-dimensional right modules. We show that, if A is connected, then every cycle of \(D^ b(A)\) lies in a tube if and only if \(D^ b(A)\) is equivalent, as a triangulated category, to \(D^ b(C)\), where C either is hereditary of Dynkin or Euclidean type, or is a tubular canonical algebra (in the sense of Ringel). Moreover, we prove that this is the case if and only if A can be obtained from C by a finite sequence of tilts and cotilts. In the representation-infinite case, we also give a complete description of such an algebra A in terms of its bound quiver.
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 16B50
zbMATH DE Number: 4000158
zbMATH Keywords: finite-dimensional algebra
zbMATH Keywords: derived category of bounded complexes of finite-dimensional right modules
zbMATH Keywords: tube
zbMATH Keywords: triangulated category
zbMATH Keywords: tilts
zbMATH Keywords: cotilts
zbMATH Keywords: representation-infinite
zbMATH Keywords: bound quiver
zbMATH Keywords: repetitive algebras
zbMATH Keywords: iterated tilted algebras
zbMATH Keywords: tubular algebras
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4000158