Finitistic dimension and Igusa–Todorov algebras

Wei, Jiaqun

doi:10.1016/j.aim.2009.07.008

Cited by 51 publications

(57 citation statements)

References 16 publications

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“…A natural question is: Is it possible to show that the finitistic dimension of B is finite if s > 2? In this direction, Wei gave in [16,Theorem 2.9] an affirmative answer under some homological conditions. In this section, we shall give a partial answer for the case s > 2 by imposing the condition concerning syzygy-finite algebras, which generalizes some results in [19,15].…”

Section: Left Idealized Extensions and Finitistic Dimensionsmentioning

confidence: 97%

Finitistic dimension conjecture and extensions of algebras

Guo

2019

Communications in Algebra

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An extension of algebras is a homomorphism of algebras preserving identities. We use extensions of algebras to study the finitistic dimension conjecture over Artin algebras. Let f : B → A be an extension of Artin algebras. We denote by fin.dim( f ) the relative finitistic dimension of f , which is defined to be the supremum of relative projective dimensions of finitely generated left A-modules of finite projective dimension. We prove that, if B is representation-finite and fin.dim( f ) ≤ 1, then A has finite finitistic dimension. For the case of fin.dim( f ) > 1, we give a sufficient condition for A with finite finitistic dimension. Also, we prove the following result: Let I, J, K be three ideals of an Artin algebra A such that IJK = 0 and K ⊇ rad(A). If both A/I and A/J are A-syzygy-finite, then the finitistic dimension of A is finite.

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Section: Left Idealized Extensions and Finitistic Dimensionsmentioning

confidence: 97%

Finitistic dimension conjecture and extensions of algebras

Guo

2019

Communications in Algebra

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“…Relative hereditary algebras are special cases of Igusa-Todorov algebras studied in [13]. By the definition, one immediately obtains that torsionless-finite algebras are relative hereditary.…”

Section: Definition 21 Let a Be An Artin Algebra We Call A Relativmentioning

confidence: 98%

“…Namely, we introduce the notion of relative hereditary Artin algebras (i.e., 0-Igusa-Todorov algebras in [13]), which is a generalization of algebras with representation dimension at most 3, and then we study the finiteness of the finitistic dimensions of endomorphism algebras of projective modules over such algebras. We prove the following results.…”

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confidence: 99%

Finitistic dimension conjecture and relative hereditary algebras

Wei¹

2009

Journal of Algebra

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We introduce the notion of relative hereditary Artin algebras, as a generalization of algebras with representation dimension at most 3. We prove the following results. (1) The relative hereditariness of an Artin algebra is left-right symmetric and is inherited by endomorphism algebras of projective modules. (2) The finitistic dimensions of a relative hereditary algebra and its opposite algebra are finite. As a consequence, the finitistic projective dimension conjecture, the finitistic injective dimension conjecture, the Gorenstein symmetry conjecture, the Wakamatsu-tilting conjecture and the generalized Nakayama conjecture hold for relative hereditary Artin algebras and endomorphism algebras of projective modules over them (in particular, over algebras with representation dimension at most 3). We also show that the torsionless-finiteness of an Artin algebra is inherited by endomorphism algebras of projective modules, and consequently give a partial answer to the question if the representation dimension of the endomorphism algebra of any projective module over an Artin algebra A is bounded by the representation dimension of A.

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“…✷ Let Λ be an Artin algebra and n be a non-negative integer. Following [8], we say that Λ is a n-Igusa-Todorov algebra if there exists V ∈ mod Λ such that for every X ∈ mod Λ there is a short exact sequence…”

Section: Igusa-todorov Algebrasmentioning

confidence: 99%

Pullback diagrams, syzygy finite classes and Igusa–Todorov algebras

Bravo

Lanzilotta

Mendoza

2019

Journal of Pure and Applied Algebra

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For an abelian category A, we define the category PEx(A) of pullback diagrams of short exact sequences in A, as a subcategory of the functor category Fun(∆, A) for a fixed diagram category ∆. For any object M in PEx(A), we prove the existence of a short exact sequence 0→K→P →M →0 of functors, where the objects are in PEx(A) and P (i) ∈ Proj(A) for any i ∈ ∆. As an application, we prove that if (C, D, E) is a triple of syzygy finite classes of objects in mod Λ satisfying some special conditions, then Λ is an Igusa-Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa-Todorov.

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Finitistic dimension and Igusa–Todorov algebras

Cited by 51 publications

References 16 publications

Finitistic dimension conjecture and extensions of algebras

Finitistic dimension conjecture and extensions of algebras

Finitistic dimension conjecture and relative hereditary algebras

Pullback diagrams, syzygy finite classes and Igusa–Todorov algebras

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