Different Exact Structures on the Monomorphism Categories
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Different Exact Structures on the Monomorphism Categories Rasool Hafezi1,2 · Intan Muchtadi-Alamsyah3 Received: 21 January 2020 / Accepted: 24 August 2020 © Springer Nature B.V. 2020
Abstract Let X be a contravariantly finite resolving subcategory of mod-Λ, the category of finitely generated right Λ-modules. We associate to X the subcategory SX (Λ) of the morphism catf
egory H(Λ) consisting of all monomorphisms (A → B) with A, B and Cok f in X . Since SX (Λ) is closed under extensions it inherits naturally an exact structure from H(Λ). We will define two other different exact structures other than the canonical one on SX (Λ), and completely classify the indecomposable projective (resp. injective) objects in the corresponding exact categories. Enhancing SX (Λ) with the new exact structure provides a framework to construct a triangle functor. Let mod-X denote the category of finitely presented functors over the stable category X . We then use the triangle functor to show a triangle equivalence between the bounded derived category Db (mod-X ) and a Verdier quotient of the bounded derived category of the associated exact category on SX (Λ). Similar consideration is also given for the singularity category of mod-X . Keywords Exact categories · Monomorphism category · Functor category · Bounded derived category · Singularity category Mathematics Subject Classification 18E10 · 18E30 · 16G10 · 18A25
The first named author thanks the Institut Teknologi Bandung for providing a stimulating research environment during his visit in ITB. This research is founded by P3MI ITB 2019 and partially supported by a grant from the Institute for Research in Fundamental Sciences (IPM), Tehran, Iran.
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Rasool Hafezi [email protected]
1
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
2
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
3
Algebra Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha No. 10, Bandung 40132, Indonesia
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R. Hafezi, I. Muchtadi-Alamsyah
1 Introduction Let Λ be an Artin algebra, and mod-Λ the category of finitely generated right Λ-modules. The monomorphism (or submodule) category S (Λ) of Λ has as objects all monomorphisms f in mod-Λ and morphisms are given by commutative squares. The representation theory of submodule categories has been studied intensively by Ringel and Schmidmeier [30,31]. We can consider S (Λ) as a subcategory of the morphism category H(Λ) of mod-Λ, that is an abelian category with all morphisms in mod-Λ as objects. For a positive integer n, let Λn = k[x]/(x n ), where k[x] is the polynomial ring in one variable x with coefficients in a field k. Let also Πn denote the preprojective algebra of type An . Two different functors F, G : S (Λn ) → mod-Πn−1 have beeen defined, one by Auslander and Reiten in [3], and the other one recently by Li and Zhang in [24]. They both were studied in [32]. They were used to show that
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