A functorial approach to categorical resolutions
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https://doi.org/10.1007/s11425-018-1614-3
A functorial approach to categorical resolutions Rasool Hafezi & Mohammad Hossein Keshavarz∗ School of Mathematics, Institute for Research in Fundamental Sciences (IPM ), Tehran 19395-5746, Iran Email: [email protected], [email protected] Received December 11, 2018; accepted October 23, 2019
Abstract
Using a relative version of Auslander’s formula, we give a functorial approach to show that the
bounded derived category of every Artin algebra admits a categorical resolution. This, in particular, implies that the bounded derived categories of Artin algebras of finite global dimension determine bounded derived categories of all Artin algebras. Hence, this paper can be considered as a typical application of functor categories, introduced in representation theory by Auslander (1971), to categorical resolutions. Keywords MSC(2010)
categorical resolutions, Artin algebras, Auslander’s formula 18E30, 16E35, 16G10, 14E15
Citation: Hafezi R, Keshavarz M H. A functorial approach to categorical resolutions. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-018-1614-3
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Introduction
Let X be a singular algebraic variety of finite type over an algebraically closed field. A resolution of e → X, where X e is a non-singular singularities of X is a certain (proper and birational) morphism π : X b b e algebraic variety. As remarked in [12, p. 1, Line −4], the functor D (cohX) → D (cohX) induced by π enjoys some remarkable properties. e and X are related by two natural functors, The bounded derived categories of coherent sheaves on X b b e known as the derived pushforward π∗ : D (cohX) → D (cohX) and the derived pullback functor π ∗ : e such that π ∗ is left adjoint to π∗ . Here Db (cohX) stands for the bounded Dperf (cohX) → Db (cohX), derived categories of coherent sheaves on X and Dperf (cohX) stands for the full subcategory of Db (cohX) consisting of perfect complexes. If X have rational singularities, then the composition π∗ ◦π ∗ is isomorphic to the identity functor [12, p. 2]. Based on this observation, as he mentioned, Kuznetsov [12, p. 2] introduced the notion of a categorical resolution of singularities. By the definition a categorical resolution of a triangulated category D is a e and a pair of functors π∗ and π ∗ satisfying almost similar conditions regular triangulated category D as above (see [12, Definition 3.2]). Recall that a triangulated category T is called regular if it is equivalent to an admissible subcategory of the bounded derived category of a smooth algebraic variety [12, Definition 3.1]. Note that, as remarked in [6, p. 7, Line −1], the pushforward functor π∗ identifies Db (cohX) with the e by the kernel of π∗ . Based on this observation, as they mentioned, Bondal and quotient of Db (cohX) * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019 ⃝
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Orlov [6] defined a categorical desingularization of a triangulated cate
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