Hereditary categories with Serre duality which are generated by preprojectives
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 criterion: Let \(\mathcal{A}\) be an abelian category and let \(\mathcal{H}\) be a full subcategory of \(D^b(\mathcal{A})\) such that \(D^b(\mathcal{A})\) is the additive closure of \(\bigcup_{t\in \mathbb Z}\mathcal{H}[t]\) and \(\mathrm{Hom}(\mathcal{H}[s], \mathcal{H}[t])=0\) for \(t<s\), then \(\mathcal{H}\) is an abelian category derived equivalent to \(\mathcal{A}\). The paper is long and technical. However it solves an important open question.
scientific article