Constructions of derived equivalences for algebras and rings
type: Item
instance of: scholarly article
title: Constructions of derived equivalences for algebras and rings
zbMATH Open document ID: 1396.18010
author: Changchang Xi
published in: Frontiers of Mathematics in China
publication date: 2018-01-10T00:00:00Z
review text: Derived categories (or more generally, triangulated categories) and derived equivalences, introduced by Grothendieck and \textit{J.-L. Verdier} [Lect. Notes Math. 569, 262--311 (1977; Zbl 0407.18008)], have connections with many branches of mathematics and physics. Rickard's Morita theory for derived categories of rings (see [\textit{J. Rickard}, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436--456 (1989; Zbl 0642.16034); J. Lond. Math. Soc., II. Ser. 43, No. 1, 37--48 (1991; Zbl 0683.16030)]) and Keller's Morita theory for derived categories of differential graded algebras (see [\textit{B. Keller}, Ann. Sci. Éc. Norm. Supér. (4) 27, No. 1, 63--102 (1994; Zbl 0799.18007)]) provide powerful tools to understand derived module categories and equivalences of rings and graded rings. However, the following fundamental problem in the study of derived categories and equivalences still remains: How can we construct derived equivalences for algebras and rings? By Rickard's theory, this problem is reduced to constructing all tilting complexes and understanding their endomorphism rings. Related to the construction problems, it seems that the following three cases are worthy to be considered: \((1)\) Given an algebra or ring \(R\), to find a class of rings that are derived equivalent to \(R\). To solve this problem, it is to classify the tilting complexes over this algebra or rings. In this case, even one knows all tilting complexes over a ring, the next tough challenge encountered is how to work out their endomorphism rings. Here, we just mention a couple of discussions in this direction. For representation finite, standard self-injective algebras, a derived classification is provided in [\textit{H. Asashiba}, J. Algebra 214, No. 1, 182--221 (1999; Zbl 0949.16013)]. For preprojective algebras, a complete description of tilting complexes is given in [\textit{T. Aihara} and \textit{Y. Mizuno}, ``Classifying tilting complexes over preprojective algebras of Dynkin type'', Preprint, \u…
reviewed by: Shengyong Pan
Mathematics Subject Classification ID: 18E30
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 18G15
zbMATH DE Number: 6823664
zbMATH Keywords: derived equivalence
zbMATH Keywords: Frobenius-finite algebra
zbMATH Keywords: recollement
zbMATH Keywords: stable equivalence
zbMATH Keywords: tilting complex
zbMATH Keywords: Yoneda algebra
MaRDI profile type: Publication
full work available at URL: s11464-016-0593-0
OpenAlex ID: W2517723372
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DOI: 10.1007/S11464-016-0593-0
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