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The quiver of projectives in hereditary categories with Serre duality. (English) Zbl 1201.16019

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.

MSC:

16G20 Representations of quivers and partially ordered sets
18E30 Derived categories, triangulated categories (MSC2010)
16E35 Derived categories and associative algebras

Citations:

Zbl 0991.18009
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References:

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