Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

Asadollahi, Javad; Hafezi, Rasool; Vahed, Razieh

doi:10.4153/cjm-2014-018-7

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…For a source k in Q, define µ k (Q) to be the quiver obtained from Q by reversing the direction of every arrow starting in k and keeping the remaining vertices and arrows as in Q. We show the following fact (compare with [11,Theorem 3.19]). Proposition 5.6.…”

Section: Subsection 51: Tilting and Cotilting Equivalencesmentioning

confidence: 99%

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Realisation functors in tilting theory

2017

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Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras.

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Section: Subsection 51: Tilting and Cotilting Equivalencesmentioning

confidence: 99%

“…For infinite quivers, this cannot be achieved through the endomorphism ring of a tilting object (the reflected category cannot be regarded as a unital ring), but rather through the heart of a tilting object. We refer to [11] for a detailed discussion of reflection functors and derived equivalences in the setting of infinite quivers.…”

Section: 1mentioning

confidence: 99%

Realisation functors in tilting theory

2017

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“…Recently the classical reflection functors, defined by Bernšteȋn, Gel ′ fand, and Ponomarev for quiver representations over a filed, was generalized in [L] to quiver representations over arbitrary ground rings. For the case that the ground ring is Noetherian of finite global dimension, the same generalization was proved in [AHV1] as well. This implies that if Q is an oriented tree and if Q ′ is obtained from Q by an arbitrary orientation, then the corresponding path algebras over an given arbitrary Artin algebra are derived equivalent.…”

Section: Cm-finiteness Versus Representation-finiteness and Vice Versamentioning

confidence: 67%

On Cohen-Macaulay Auslander algebras

Hafezi¹

2018

Preprint

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Cohen-Macaulay Auslander algebras are the endomorphism algebras of representation generators of the subcategory of Gorenstein projective modules over CM-finite algebras.In this paper, we study Cohen-Macaulay Auslander algebras over 1-Gorenstein algebras and Ω G -algebras. 1-Gorenstein algebras are those of algebras with global Gorenstein projective dimension at most one and Ω G -algebras are a class of algebras introduced in this paper, including some important class of algebras for example Gentle algebras and more generally quadratic monomial algebras. It will be shown how the results for Gorenstein projective representations of a quiver over an Artin algebra, including the submodule category introduced in [RS], or more generally, the (separated) monomorphism category defined in [LZh2] and [XZZ], can be applied to study the Cohen-Macaulay Auslander algebras.

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On Relative Derived Categories

Asadollahi

Bahiraei

Hafezi

et al. 2016

Communications in Algebra

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The paper is devoted to study some of the questions arises naturally in connection to the notion of relative derived categories. In particular, we study invariants of recollements involving relative derived categories, generalise two results of Happel by proving the existence of AR-triangles in Gorenstein derived categories, provide situations for which relative derived categories with respect to Gorenstein projective and Gorenstein injective modules are equivalent and finally study relations between the Gorenstein derived category of a quiver and its image under a reflection functor. Some interesting applications are provided.

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Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

Cited by 3 publications

References 12 publications

Realisation functors in tilting theory

Realisation functors in tilting theory

On Cohen-Macaulay Auslander algebras

On Relative Derived Categories

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