On the finiteness of the derived equivalence classes of some stable endomorphism rings
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Mathematische Zeitschrift
On the finiteness of the derived equivalence classes of some stable endomorphism rings Jenny August1 Received: 1 November 2018 / Accepted: 19 December 2019 © The Author(s) 2020
Abstract We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction f : X → Spec R, where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra Acon known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of Acon and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f . This provides evidence towards a key conjecture in the area. Mathematics Subject Classification Primary 16E35; Secondary 16E65 · 14E30
1 Introduction This paper focuses on a fundamental problem in homological algebra: given a basic algebra A, find all the basic algebras B such that A and B have equivalent derived categories. We will give a complete answer to this question for a class of finite dimensional algebras arising from suitable Frobenius categories. By a well known result of Rickard [33], the above problem is equivalent to first finding all the tilting complexes over A and then computing their endomorphism rings. One approach to the first of these problems is to use mutation; an iterative procedure which produces new tilting complexes from old. The naive hope is that starting from a given tilting complex, all others can be reached using mutation. However, for a general algebra, tilting complexes do not behave well enough for this to work. There are two key problems: (1) The mutation procedure does not always produce a tilting complex. (2) It is often possible to find two tilting complexes which are not connected by any sequence of mutations.
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Jenny August [email protected] Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
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J. August
Both problems have motivated results in the literature; the first prompting the introduction of the weaker notion of silting complexes [3,27] and the second resulting in restricting to a class of algebras known as tilting-discrete algebras [2,4]. In this paper, we will combine these ideas to provide a class of examples of finite dimensional algebras for which the derived equivalence class can be completely determined.
1.1 Algebraic setting and results From an algebraic perspective, the algebras we consider arise in cluster-tilting theory, where the objects of study are rigid objects in some category C and the algebras of interest are their endomorphism algebras. More precisely, let E be a Frobenius category such that its stable category C = E is a k-linear, Hom-finite, Krull–Schmid
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