On Relative Derived Categories

Asadollahi, Javad; Bahiraei, Payam; Hafezi, Rasool; Vahed, Razieh

doi:10.1080/00927872.2016.1172618

Cited by 13 publications

(5 citation statements)

References 36 publications

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“…Remark 5.6. By use of [ABHV,Corollary 7.6] we observe that over an Gorenstein algebra Λ Proposition 5.5 is independent of the orientation of the quiver Q. Hence in this way we can produce more CM-finite algebras.…”

Section: Case 3: Letmentioning

confidence: 99%

When stable Cohen-Macaulay Auslander algebra is semisimple

Hafezi

2021

Preprint

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Let Gprj-Λ denote the category of Gorenstein projective modules over an Artin algebra Λ and the category mod-(Gprj-Λ) of finitely presented functors over the stable category Gprj-Λ. In this paper, we study those algebras Λ with mod-(Gprj-Λ) to be a semisimple abelian category, and called Ω G -algebras. The class of Ω G -algebras contains important classes of algebras, including gentle algebras. Over an Ω G -algebra Λ, the structure of the almost split sequences in the morphism category H(Gprj-Λ) and the monomorphism category S(Gprj-Λ) of Gprj-Λ is investigated. Among other applications, we provide some results for the Cohen-Macaulay Auslander algebras of Ω G -algebras.

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Section: Case 3: Letmentioning

confidence: 99%

When stable Cohen-Macaulay Auslander algebra is semisimple

Hafezi

2021

Preprint

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“…The functor Ω Λ is the inverse of the suspension functor [1] for the stable category mod Λ. The functor Ω 2 Λ is applied pointwise on the ΛQ-module A.…”

Section: The Quiver a 3 Revisitedmentioning

confidence: 99%

“…In [1, ], the authors proved that there is an equivalence of singularity categories D sg (ΛQ) ∼ = D sg (ΛQ(v)), where Λ is a Gorenstein algebra. As a consequence, they obtain a different proof for the equivalence G-proj ΛQ ∼ = G-proj ΛQ(v).…”

Section: Introductionmentioning

confidence: 99%

“…Gorenstein-projective modules feature prominently in relative homological algebra where they assume the role played by projectives modules in homological algebra [8,9]. Ringel and Zhang exhibit a link to derived categories: If Λ = k[ε] = k[T ]/(T 2 ) is the bounded polynomial ring, the authors show the equivalence G-proj ΛQ ≃ D b (mod kQ)/ [1] in [20].…”

Section: Introductionmentioning

confidence: 99%

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A reflection equivalence for Gorenstein-projective quiver representations

Luo¹,

Schmidmeier²

2022

Preprint

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For Λ a selfinjective algebra, and Q a finite quiver without oriented cycles, the algebra ΛQ is a Gorenstein algebra and the category G-proj ΛQ of Gorenstein-projective ΛQ-modules is a Frobenius category. For a sink v of Q, we define a functor F (v) : G-proj ΛQ → G-proj ΛQ(v) between the stable categories modulo projectives, where Q(v) is obtained from Q by changing the direction of each arrow ending in v. The functor is given by an explicit construction on the level of objects and homomorphisms. Our main result states that F (v) is an equivalence of categories. In the case where the underlying graph of Q is a tree, we deduce that the stable category G-proj ΛQ does not depend on the orientation of Q. Moreover, if Q is a quiver of type A 3 and Λ = k[T ]/(T n ) the bounded polynomial algebra, we use the symmetry of the octahedron in the octahedral axiom to verify that the composition of twelve reflections yields the identity on objects.MSC 2020: 18G25 (relative homological algebra), 16G20 (representations of quivers)

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“…The Gorenstein derived category was studied by Gao and Zhang [GZ]. Recently this category has been studied more in [ABHV,AHV1].…”

Section: Letmentioning

confidence: 99%

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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On Relative Derived Categories

Cited by 13 publications

References 36 publications

When stable Cohen-Macaulay Auslander algebra is semisimple

When stable Cohen-Macaulay Auslander algebra is semisimple

A reflection equivalence for Gorenstein-projective quiver representations

Derived equivalences of functor categories

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