Derived categories of N-complexes

Iyama, Osamu; Kato, Kiriko; Miyachi, Jun-ichi

doi:10.1112/jlms.12084

Cited by 31 publications

(30 citation statements)

References 26 publications

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“…Hence, 2-complexes are usual chain complexes over A. Iyama, Kato and Miyachi studied the derived category of N -complexes of an abelian category A, denoted by D N (A) [29]. In particular, they [29,Corollary 4.15] showed that for any ring B, D N (Mod-B) is equivalent to D(Mod-BA N−1 ) as triangulated categories, where A N−1 is the quiver 1 −→ 2 −→ 3 −→ · · · −→ N − 1.…”

Section: Recollements and N -Complexesmentioning

confidence: 99%

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On the Recollements of Functor Categories

Asadollahi

Hafezi

Vahed

2015

Appl Categor Struct

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This paper is devoted to the study of recollements of functor categories in different levels. In the first part of the paper, we start with a small category S and a maximal object s of S and construct a recollement of Mod-S in terms of Mod-End S (s) and Mod-(S \ {s}) in four different levels. In case S is a finite directed category, by iterating this argument, we get chains of recollements having some interesting applications. In the second part, we start with a recollement of rings and construct a recollement of their path rings, with respect to a finite quiver. Third part of the paper presents some applications, including recollements of triangular matrix rings, an example of a recollement in Gorenstein derived level and recollements of derived categories of N -complexes.

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Section: Recollements and N -Complexesmentioning

confidence: 99%

“…But, recently it had been received considerable attention in representation theory of algebras and model theory; see for instance [19,22,29]. In this direction, Iyama, Kato and Miyachi studied the derived category of N -complexes D N (A) of an abelian category A [29].…”

Section: Introductionmentioning

confidence: 99%

On the Recollements of Functor Categories

Asadollahi

Hafezi

Vahed

2015

Appl Categor Struct

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“…Remark 3.6. For N > 0, it is worth noting that O. Iyama, K. Kato and Jun-ichi Miyachi defined the homotopy category of N -complexes and the derived category of N -complexes in [22]. The homotopy category of N -complexes of projective Λ-modules is triangulated equivalent to the homotopy category of projective T N −1 (Λ)-modules for any algebra Λ, see [22,5].…”

Section: 2mentioning

confidence: 99%

Singularity categories of representations of algebras over local rings

2020

Colloq. Math.

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Let Λ be a finite-dimensional algebra with finite global dimension, R k = K[X]/(X k ) be the Z-graded local ring with k ≥ 1, and Λ k = Λ ⊗K R k . We consider the singularity category Dsg(mod Z (Λ k )) of the graded modules over Λ k . It is showed that there is a tilting object in Dsg(mod Z (Λ k )) such that its endomorphism algebra is isomorphic to the triangular matrix algebra T k−1 (Λ) with coefficients in Λ and there is a triangulated equivalence between Dsg(mod Z/kZ (Λ)) and the root category of T k−1 (Λ). Finally, a classification of Λ k up to the Cohen-Macaulay representation type is given.Let Λ be a finite-dimensional algebra with finite global dimension, R k = K[X]/(X k ) be the Z-graded local ring with X degree 1 where k is a positive integer, and Λ k = Λ ⊗ K R k . Then Λ k is a positively graded Gorenstein algebra. C. M. Ringel and M. Schmidmeier [49] investigate the Gorenstein projective modules over Λ k with Λ hereditary of type A 2 by using submodule categories, and describe its Auslander-Reiten quiver for k < 6. The structure of this kind of submodule categories has been studied by many people, see [48,49,52,53,54] and the references therein, and have been discovered to be related to weighted projective lines, see [28,29,30] and the references therein.Later, C. M. Ringel and P. Zhang [50] prove that the singularity category of Λ 2 is triangulated equivalent to the triangulated orbit category D b (mod(Λ))/Σ for Λ a path algebra KQ, where Q is an acyclic quiver. X.-H. Luo and P. Zhang generalize these works and introduce monic representations to describe the Gorenstein projective modules over A⊗ K KQ (also A⊗ K (KQ/I) with I generated by monomial relations) with A a finite-dimensional algebra [37,38]. Recently, D. Shen describes Gorenstein projective modules over the tensor product of two algebras in terms of their underlying one-sided modules [51], see also [21].In this paper, we mainly consider the singularity categories D sg (mod(Λ k )) and D sg (mod Z (Λ k )) over Λ k . K. Yamaura [58] proved that for a positively graded self-injective algebra, its stable category of the Z-graded modules admits a tilting object. Following him, we prove that the singularity category D sg (mod Z (Λ k )) of the Z-graded modules over Λ k has a tilting object with the same construction, in particular, its endomorphism algebra is isomorphic to the triangular matrix algebra T k−1 (Λ) with coefficients in Λ. In this way, we get that D sg (mod Z (Λ k )) is triangulated equivalent to the bounded derived category D b (mod(T k−1 (Λ))). Viewing Λ k as a Z/kZ-graded algebra naturally and considering the singularity category D sg (mod Z/kZ (Λ k )), we prove that the above triangulated equivalence induces a triangulated equivalence D sg (mod Z/kZ (Λ k )) ≃ R T k−1 (Λ) . Furthermore, when k = 2, this result recovers the result of [50]: D sg (mod(Λ 2 )) ≃ D b (mod(Λ))/Σ if Λ is hereditary. Finally, we classify Λ k up to the Cohen-Macaulay representation type, and give some examples to describe the Auslander-Reiten quivers for some ...

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“…Therefore, Imj * is contained in ImΦ ⊥ B D − (Mod-A). Now, Corollary 1.13 of [IKM2] completes the proof. 4.1.…”

Section: Existence Of Recollementsmentioning

confidence: 67%

Derived equivalences of functor categories

Asadollahi

Hafezi

Vahed

2019

Journal of Pure and Applied Algebra

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Let Mod-S denote the category of S-modules, where S is a small category. In the first part of this paper, we provide a version of Rickard's theorem on derived equivalence of rings for Mod-S. This will have several interesting applications. In the second part, we apply our techniques to get some interesting recollements of derived categories in different levels. We specialize our results to path rings as well as graded rings.The differential∂ n is the induced map on residue classes.We denote the homotopy category of C by K(C); the objects are complexes and morphisms are the homotopy classes of morphisms of complexes. The full subcategory of K(C) consisting of all bounded above, resp. bounded, complexes is denoted by K − (C), resp. K b (C). Moreover, we denote by K −,b (C), the full subcategory of K − (A) formed by all complexes X such that there is an integer n = n(X) with H i (X) = 0, for all i ≤ n.

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Derived categories of N-complexes

Cited by 31 publications

References 26 publications

On the Recollements of Functor Categories

On the Recollements of Functor Categories

Singularity categories of representations of algebras over local rings

Derived equivalences of functor categories

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