Characterization of eventually periodic modules in the singularity categories

Usui, Satoshi

doi:10.1016/j.jpaa.2022.107145

Cited by 4 publications

(3 citation statements)

References 35 publications

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“…The study of periodic modules is traced back to [4]. We mention that periodic modules play a role in the singularity category [24].…”

Section: Virtually Periodic Objectsmentioning

confidence: 99%

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A non-vanishing result on the singularity category

Chen¹,

Li²,

Zhang³

et al. 2023

Preprint

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We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomology of differential graded Leavitt algebras.

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“…The study of periodic modules is traced back to [4]. We mention that periodic modules play a role in the singularity category [24].…”

Section: Virtually Periodic Objectsmentioning

confidence: 99%

“…Recall that a module is periodic if its higher syzygy is isomorphic to itself. These modules play a particular role in the singularity category [24]. We propose a slightly more general notion: a module is called virtually periodic if its higher syzygy lies in the extension closure of the module itself; see Definition 2.1.…”

Section: Introductionmentioning

confidence: 99%

A non-vanishing result on the singularity category

Chen¹,

Li²,

Zhang³

et al. 2023

Preprint

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“…When R is a (both left and right) Noetherian local ring and M has finite virtual projective dimension, Avramov [4,Theorem 4.4] gave an upper bound for the degree n and showed that the period p is either 1 or 2. On the other hand, using the notion of Tate cohomology, the author [31,Theorem 2.4] obtained a result on the period p without additional assumption on R and M; however, it follows that Tate cohomology gives no information on the degree n. Moreover, there are further results on the values of n and p: for example, [3,Theorem 1.6], [15, the proof of Corollary 6.4], [19,Theorem 1.2] and [30,Proposition 4.3]. We note that eventually periodic modules are examined in the literature such as [9,14,23,16].…”

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confidence: 99%

Periodic dimensions and some homological properties of eventually periodic algebras

Usui¹

2023

Preprint

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For an eventually periodic module, we have the degree and the period of its first periodic syzygy. This paper studies the former under the name 'periodic dimension'. We give a bound for the periodic dimension of an eventually periodic module with finite Gorenstein projective dimension. This bound tells us that the two dimensions are almost equal. Moreover, making use of the bound, we determine the bimodule periodic dimension of a finite dimensional eventually periodic Gorenstein algebra. Another aim of this paper is to obtain some of the basic homological properties of finite dimensional eventually periodic algebras. We show that a lot of homological conjectures hold for this class of algebras. Further, we use this result to characterize finite dimensional eventually periodic Gorenstein algebras. This characterization explains why we consider their bimodule periodic dimensions. Contents 1. Introduction Conventions and notation 2. Preliminaries 2.1. Gorenstein projective modules 2.2. Gorenstein projective dimensions 2.3. Gorenstein rings 3. Periodic dimensions 3.1. The case of modules 3.2. The case of regular bimodules 4. Homological properties of eventually periodic algebras

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Periodic dimensions and some homological properties of eventually periodic algebras

Usui

2024

Journal of Algebra

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Characterization of eventually periodic modules in the singularity categories

Cited by 4 publications

References 35 publications

A non-vanishing result on the singularity category

A non-vanishing result on the singularity category

Periodic dimensions and some homological properties of eventually periodic algebras

Periodic dimensions and some homological properties of eventually periodic algebras

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