Piecewise hereditary algebras
type: Item
instance of: scholarly article
title: Piecewise hereditary algebras
zbMATH Open document ID: 0865.16009
DOI: 10.1007/BF01195702
author: Dieter Happel
author: Idun Reiten
author: Sverre O. Smalø
published in: Archiv der Mathematik
publication date: 1997-07-07T00:00:00Z
review text: In some sense, this article continues research about quasi-tilted algebras [see: ``Tilting in Abelian Categories'', Mem. Am. Math. Soc. 575 (1996; Zbl 0849.16011) by the same authors, and ``Piecewise hereditary algebras'', by \textit{D. Happel, J. Rickard} and \textit{A. Schofield}, Bull. Lond. Math. Soc. 20, No. 1, 23-28 (1988; Zbl 0638.16018)]. The reading of the paper will be facilitated by some knowledge of Happel's book on applications of derived and triangulated categories and of papers of Lenzing-Geigle and others on representations of sheaves. A piecewise hereditary algebra of type \(\mathcal H\) is an \(R\)-Artin algebra \(\Lambda\) such that \({\mathcal D}^b(\Lambda)\) is triangle equivalent to \({\mathcal D}^b({\mathcal H})\), where \(\mathcal H\) is a locally finite, hereditary, abelian category. The main theorem of the paper, stated below, gives a description of those algebras for the important cases where \(\mathcal H\) is \(\text{mod }H\) (\(H\) a hereditary \(R\)-Artin algebra) or \(\text{coh }{\mathbf X}(p_1,\dots,p_r)\) (\(\text{coh }{\mathbf X}(p_1,\dots,p_r)\) a weighted projective curve). Theorem. Let \(\Lambda\) be a piecewise hereditary algebra of type \(\text{mod }H\) or \(\text{coh }{\mathbf X}(p_1,\dots,p_r)\). Then, there exists a chain \(\Lambda=\Lambda_0,\dots,\Lambda_i=\text{End}_{\Lambda_{i-1}}(T_{i-1})^{\text{op}},\dots\) (\(i=1,2,\dots,m\)) where each \(T_i\) is a tilting or a cotilting \(\Lambda_i\)-module and where \(\Lambda_m\) is a quasitilted algebra. Actually, the \(\text{mod }H\) case of this theorem had been proved already in the above cited paper by Happel, Reiten and Smalø, and its proof is obtained by reducing it to this earlier argument.
reviewed by: Héctor A. Merklen
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16D90
zbMATH DE Number: 938523
zbMATH Keywords: Artin algebras
zbMATH Keywords: cotilting modules
zbMATH Keywords: quasi-tilted algebras
zbMATH Keywords: triangulated categories
zbMATH Keywords: representations of sheaves
zbMATH Keywords: piecewise hereditary algebras
zbMATH Keywords: Abelian categories
zbMATH Keywords: weighted projective curves
MaRDI profile type: Publication
cites work: Modules determined by their composition factors
cites work: Algebras with cycle-finite derived categories
cites work: Perpendicular categories with applications to representations and sheaves
cites work: Q3994341
cites work: Piecewise Hereditary Algebras
cites work: Tilted Algebras
cites work: Hereditary noetherian categories with a tilting complex
full work available at URL: https://doi.org/10.1007/bf01195702
OpenAlex ID: W2884561423
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description: scientific article; zbMATH DE number 938523