Berg, Carl Fredrik; van Roosmalen, Adam-Christiaan The quiver of projectives in hereditary categories with Serre duality. (English) Zbl 1201.16019 J. Pure Appl. Algebra 214, No. 7, 1082-1094 (2010). A quiver \(Q\) is called strongly locally finite if each vertex has only finitely many neighbors and there is no path of infinite length in \(Q\). The authors characterize in the paper the quivers \(Q\) such that \(\mathbb{Z} Q\) possesses a strongly locally finite section. Using this result they prove, following ideas of I. Reiten and M. Van den Bergh [J. Am. Math. Soc. 15, No. 2, 295-366 (2002; Zbl 0991.18009)], that every Noetherian \(k\)-linear (where \(k\) is an algebraically closed field) Abelian Ext-finite category with Serre duality generated by the preprojective objects is derived equivalent to the category of the representations over a strongly locally finite quiver. Finally, the authors show that a quiver \(Q\) is the quiver of the projectives of an Abelian hereditary category with Serre duality if and only if \(\mathbb{Z} Q\) has a strongly locally finite section. Reviewer: Grzegorz Bobiński (Toruń) Cited in 1 ReviewCited in 1 Document MSC: 16G20 Representations of quivers and partially ordered sets 18E30 Derived categories, triangulated categories (MSC2010) 16E35 Derived categories and associative algebras Keywords:strongly locally finite quivers; Serre dualities; Abelian hereditary categories; bounded derived categories; preprojectives; derived equivalences; stable translation quivers Citations:Zbl 0991.18009 PDFBibTeX XMLCite \textit{C. F. Berg} and \textit{A.-C. van Roosmalen}, J. Pure Appl. Algebra 214, No. 7, 1082--1094 (2010; Zbl 1201.16019) Full Text: DOI arXiv References: [1] Reiten, I.; Van den Bergh, M., Noetherian hereditary abelian categories satisfying Serre duality, J. Amer. Math. Soc., 15, 2, 295-366 (2002), (electronic) MR1887637 (2003a:18011) · Zbl 0991.18009 [2] Michael Ringel, Claus, A ray quiver construction of hereditary abelian categories with Serre duality, (Representations of Algebra, vol. II (2002), BNU Press), 396-416, MR2067392 (2005i:16026) · Zbl 1086.16502 [3] Happel, Dieter, A characterization of hereditary categories with tilting object, Invent. Math., 144, 2, 381-398 (2001), MR1827736 (2002a:18014) · Zbl 1015.18006 [4] van Roosmalen, Adam-Christiaan, Abelian 1-Calabi-Yau categories, Int. Math. Res. Not. IMRN, 6 (2008), Art. ID rnn003, 20. MR2427460 (2009g:18016) · Zbl 1144.18008 [5] van Roosmalen, Adam-Christiaan, Classification of abelian hereditary directed categories satisfying Serre duality, Trans. Amer. Math. Soc., 360, 5, 2467-2503 (2008), MR2373322 (2008m:16034) · Zbl 1155.16018 [6] Bondal, A. I.; Kapranov, M. M., Representable functors, Serre functors, and reconstructions, Izv. Akad. Nauk SSSR Ser. Mat., 53, 6, 1183-1205 (1989), MR1039961 (91b:14013) · Zbl 0703.14011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.