Layer Lengths, Torsion Theories and the Finitistic Dimension

Huard, François; Lanzilotta, Marcelo; Mendoza, Octavio

doi:10.1007/s10485-011-9268-x

Cited by 11 publications

(17 citation statements)

References 9 publications

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“…Similarly, t S (S(2n)) = S(2n) and t S (S(2n + 1)) = S(2n + 1). Since P (2) ∈ F(S), we have t S (P (2)) = 0 by [15,Proposition 5.3]. So t S F t S (P (1)) = t S (S(n + 1) ⊕ P (2) ⊕ S(2n) ⊕ S(2n + 1)) = S(n + 1) ⊕ S(2n) ⊕ S(2n + 1).…”

Section: By Corollary 23(1))mentioning

confidence: 99%

The Extension Dimension of Abelian Categories

Zheng

Huang

2019

Algebr Represent Theor

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Let A be an abelian category having enough projective objects and enough injective objects. We prove that if A admits an additive generating object, then the extension dimension and the weak resolution dimension of A are identical, and they are at most the representation dimension of A minus two. By using it, for a right Morita ring Λ, we establish the relation between the extension dimension of the category mod Λ of finitely generated right Λ-modules and the representation dimension as well as the right global dimension of Λ. In particular, we give an upper bound for the extension dimension of mod Λ in terms of the projective dimension of certain class of simple right Λ-modules and the radical layer length of Λ. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.

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Section: By Corollary 23(1))mentioning

confidence: 99%

The Extension Dimension of Abelian Categories

Zheng

Huang

2019

Algebr Represent Theor

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“…We recall some notions from [10]. Let C be a length-category, that is, C is an abelian, skeletally small category and every object of C has a finite composition series.…”

Section: Igusa-todorov Algebrasmentioning

confidence: 99%

“…For a length-category C, generalizing the Loewy length, Huard, Lanzilotta and Hernández introduced in [10,11] the (radical) layer length associated with a torsion pair, which is a new measure for objects of C. Let Λ be an artin algebra and V a set of some simple modules in mod Λ. Let t V be the torsion radical of a torsion pair associated with V (see Section 4 for details).…”

Section: Introductionmentioning

confidence: 99%

The derived dimensions of $(m,n)$-Igusa-Todorov algebras

Zheng¹

2021

Preprint

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We give an upper bound for the dimension of the bounded derived categories of (m, n)-Igusa-Todorov algebras which is a generalization of n-Igusa-Todorov algebras, where m, n are two nonnegative integers. As an applications, we get a new upper bound for the dimension of bounded derived categories in terms of the projective dimensions of certain of simple modules as well as radical layer length of artin algebra Λ.

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“…In [5,4], Huard, Lanzilotta and Hernández introduced the notion of the radical layer length associated with a torsion pair which is a generalization of the Loewy length. Let Λ be an artin algebra and V a set of some simple modules in mod Λ.…”

Section: Introductionmentioning

confidence: 99%

The derived dimensions and syzygy finite type

Zheng

2021

Preprint

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Let Λ be an artin algebra, and V a subset of all simple modules in mod Λ. Suppose that Λ/ rad Λ has finite syzygy type, then the derived dimension of Λ is at most ℓℓ t V (ΛΛ) + pd V. In particular, if the global dimension of Λ is finite, then the derived dimension of Λ is at most ℓℓ t V (ΛΛ) + pd V. This generalized the famous result which state that the derived dimension of Λ is less than or equal to the global dimension of Λ.

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Layer Lengths, Torsion Theories and the Finitistic Dimension

Cited by 11 publications

References 9 publications

The Extension Dimension of Abelian Categories

The Extension Dimension of Abelian Categories

The derived dimensions of $(m,n)$-Igusa-Todorov algebras

The derived dimensions and syzygy finite type

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