Hereditary categories with Serre duality which are generated by preprojectives
type: Item
instance of: scholarly article
title: Hereditary categories with Serre duality which are generated by preprojectives
zbMATH Open document ID: 1240.18020
author: Carl Fredrik Berg
author: Adam-Christiaan van Roosmalen
published in: Journal of Algebra
publication date: 2012-01-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.1653
review text: This paper is part of a project of the authors whose goal is to classify hereditary categories with Serre duality initiated by a famous paper of \textit{I. Reiten} and \textit{M. Van den Bergh} [J. Am. Math. Soc. 15, No. 2, 295--366 (2002; Zbl 0991.18009)]. In that paper, Reiten and Van den Bergh classified \(k\)-linear abelian hereditary Ext-finite noetherian categories with Serre duality. They posed the question whether every hereditary category with Serre duality is derived equivalent to a noetherian one. However, \textit{C. M. Ringel} gave a counterexample [Representations of algebras. Vol. I, II. Proceedings of the 9th international conference, Beijing, China, August 21-September 1, 2000. Beijing: Beijing Normal University Press. 396--416 (2002; Zbl 1086.16502)], then Reiten modified the question by asking a classification of hereditary categories with Serre duality which are generated by preprojective objects, but are not necessarily noetherian [\textit{I. Reiten}, ``Hereditary categories with Serre duality'', Representations of algebras. Vol. I, II. Proceedings of the 9th international conference, Beijing, China, August 21-September 1, 2000. Beijing: Beijing Normal University Press. 109--121 (2002; Zbl 1087.18500)]. The authors of the paper under review have attacked the question in three papers: ``The quiver of projectives in hereditary categories with Serre duality'' [J. Pure Appl. Algebra 214, No. 7, 1082--1094 (2010; Zbl 1201.16019)], ``Representations of thread quivers'', \url{arXiv:1003.1317} (2010), and the paper under review. The main result is the following: A \(k\)-linear abelian hereditary Ext-finite categories with Serre duality generated by preprojective objects is derived equivalent to the path algebra of a strongly locally finite thread quiver. It takes too long to repeat the technical definitions here, but one of the key ingredients of the result is the notion of split \(t\)-structures which enables the authors to prove a derived equivalence c…
reviewed by: Guodong Zhou
Mathematics Subject Classification ID: 18E10
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16E35
Mathematics Subject Classification ID: 16G70
zbMATH DE Number: 5994287
zbMATH Keywords: derived equivalence
zbMATH Keywords: hereditary category
zbMATH Keywords: Serre duality
zbMATH Keywords: split t-structure
zbMATH Keywords: thread quiver
MaRDI profile type: Publication
OpenAlex ID: W2964315041
arXiv ID: 1004.1653
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DOI: 10.1016/J.JALGEBRA.2010.11.021
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description: scientific article