Tilting Theory and Functor Categories II. Generalized Tilting

Martínez-Villa, Roberto; Ortiz-Morales, Martín

doi:10.1007/s10485-011-9266-z

Cited by 13 publications

(18 citation statements)

References 20 publications

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“…The proof is motivated for the case C = mod-Λ. We refer to [MO,Proposition 3.1] or also [HMa,Lemma 5.3]. For convenience of the reader we provide a proof for the case Gprj-Λ) is a non-retraction as well.…”

Section: The Morphism Category Over ω G -Algebrasmentioning

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: The Morphism Category Over ω G -Algebrasmentioning

confidence: 99%

When stable Cohen-Macaulay Auslander algebra is semisimple

Hafezi

2021

Preprint

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“…We remember that under these conditions finitely presented functors have projective covers [MVO2]. In this part we introduce some special subcategories related to the described filtrations in Theorem 2.7.…”

Section: Assume We Have a Filtrationmentioning

confidence: 99%

“…On the other hand, functor categories were introduced in representation theory by Auslander [A] and used in his proof of the first Brauer-Thrall conjecture [A2] and later used systematically in his joint work with I. Reiten on stable equivalence and many other applications [AR,AR2]. Recently, functor categories were employed by Martínez-Villa and Solberg to study the Auslander-Reiten components of finitedimensional algebras [MVS3] and to develop tilting theory in arbitrary functor categories [MVO1,MVO2].…”

Section: Introduction and Basic Conceptsmentioning

confidence: 99%

The Auslander–Reiten Components Seen as Quasi-Hereditary Categories

Ortiz-Morales

2017

Appl Categor Struct

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Abstract. Quasi-hereditary were introduced by L. Scott [Scott,CPS1,CPS2] in order to deal highest weight categories as they arise in the representation theory of semi-simple complex Lie algebras and algebraic groups, and they have been a very important tool in the study of finite-dimensional algebras. On the other hand, functor categories were introduced in representation theory by M. to study the Auslander-Reiten components of finite-dimensional algebras. The aim of the paper is to introduce the concept of quasi-hereditary category, and we can think of the components of the Auslander-Reiten components as quasi-hereditary categories. In this way, we have applications to the functor category Mod(C), with C a component of the Auslander-Reiten quiver.

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“…They did so, in order to stablish when the category of graded functors is noetherian [32,33,34]. Recently, Martínez-Villa and Ortíz studied in [31,30] tilting theory in arbitrary functor categories. They proved that most of the properties that are satisfied by a tilting module over an Artin algebra also hold true for functor categories.…”

Section: Introductionmentioning

confidence: 99%

A Generalization of the Theory of Standardly Stratified Algebras I: Standardly Stratified Ringoids

Mendoza¹,

Ortiz²,

Sáenz³

et al. 2020

Glasgow Math. J.

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We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

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Tilting Theory and Functor Categories II. Generalized Tilting

Cited by 13 publications

References 20 publications

When stable Cohen-Macaulay Auslander algebra is semisimple

When stable Cohen-Macaulay Auslander algebra is semisimple

The Auslander–Reiten Components Seen as Quasi-Hereditary Categories

A Generalization of the Theory of Standardly Stratified Algebras I: Standardly Stratified Ringoids

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