Constructions of derived equivalences for algebras and rings
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, \url{arXiv: 1509.07387}]. Partial answers to (1) are gotten for many classes of algebras, namely, one may construct some tilting complexes and determine their endomorphism rings. In this direction, see, for example, [\textit{M. Hoshino} and \textit{Y. Kato}, Commun. Algebra 30, No. 1, 83--100 (2002; Zbl 1002.16004); J. Pure Appl. Algebra 177, No. 2, 159--175 (2003; Zbl 1012.18006)]. \((2)\) Given two rings \(R\) and \(S\), to decide whether they are derived equivalent. In this case, we mention one open problem related to it, namely, the famous Broue Abelian Defect Group Conjecture in the modular representation theory of finite groups, which says that the module categories of a block algebra \(A\) of a finite group algebra and its Brauer correspondent \(B\) have equivalent derived categories if their common defect group is abelian (see [\textit{J. Rickard}, Doc. Math. Extra Vol., 121--128 (1998; Zbl 0919.20007)]). This conjecture is considered as one of the hardest problems in the representation theory of finite groups. Though this conjecture is verified for many cases , it seems far away from being solved completely. This may reflect a little flavor of difficulty of deciding whether two given algebras are derived equivalent. For further information on the developments of this conjecture, we refer the reader to [\textit{R. Rouquier}, in: Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22--30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS). 191--221 (2006; Zbl 1108.20006)]. \((3)\) Given a derived equivalence, to find a new derived equivalence based on the given one. In this direction, \textit{J. Rickard} [J. Pure Appl. Algebra 61, No. 3, 303--317 (1989; Zbl 0685.16016); J. Lond. Math. Soc., II. Ser. 43, No. 1, 37--48 (1991; Zbl 0683.16030)] used tensor products and trivial extensions to produce new derived equivalences, \textit{M. Barot} and \textit{H. Lenzing} [J. Algebra 264, No. 1, 1--5 (2003; Zbl 1060.16011)] employed one-point extensions to transfer certain a derived equivalence to a new one. Up to the present time, however, it seems that not much is available for constructing new derived equivalences based on given ones. In this article, we shall survey some of recent constructions of derived equivalences related to the last two cases. Here, the main idea is to construct derived equivalences from certain exact sequences, or by passing to quotient algebras or pullback algebras of derived equivalent algebra In the paper under review, the author surveys some of recent constructions of derived equivalences related to the last two cases. Here, the main idea is to construct derived equivalences from certain exact sequences, or by passing to quotient algebras or pullback algebras of derived equivalent algebras. The author also gives the relationship between derived equivalences and stable equivalences of Morita type. It is readable and useful for who want to study derived equivalences and related topics.
scientific article; zbMATH DE number 6823664
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