Finitistic dimension conjecture and conditions on ideals

Wei, Jiaqun

doi:10.1515/form.2011.019

Cited by 12 publications

(15 citation statements)

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“…Also, the arguments of proofs of our results here are different from the earlier ones, though the common idea for all proofs is the use of the Igusa-Todorov function. Note also that, comparing with the results in [21], we do not impose any homological conditions (such as finiteness of projective dimension) on ideals or powers of the radical of B since such conditions seem to be strong for our homological questions.…”

Section: Corollary 14 Suppose That B ⊆ a Is An Extension Of Artin Amentioning

confidence: 90%

Finitistic dimension conjecture and radical-power extensions

Wang

2017

Journal of Pure and Applied Algebra

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The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this paper, we investigate those extensions of Artin algebras in which some radical-power of smaller algebras is a one-sided ideal in bigger algebras. Our results, however, are formulated more generally for an arbitrary ideal: Let B ⊆ A be an extension of Artin algebras and I an ideal of B such that the full subcategory of B/I-modules is B-syzygy-finite. Then: (1) If the extension is right-bounded (for example, pd (A B ) < ∞), I A rad(B) ⊆ B and fin.dim (A) < ∞, then fin.dim (B) < ∞.(2) If I rad(B) is a left ideal of A and A is torsionless-finite, then fin.dim (B) < ∞. Particularly, if I is specified to a power of the radical of B, then our results not only generalize some ones in the literature (see Corollaries 1.2 and 1.4), but also provide some completely new ways to detect algebras of finite finitistic dimensions.

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Section: Corollary 14 Suppose That B ⊆ a Is An Extension Of Artin Amentioning

confidence: 90%

Finitistic dimension conjecture and radical-power extensions

Wang

2017

Journal of Pure and Applied Algebra

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“…with P projective A-module. Now we can bound the projective dimension of A X : The proof of Theorem 1.2 is similar to that of [19,Theorem 3.2] and [17,Theorem 3.1], in which the Igusa-Todorov function is used. However, the difference is that the syzygy shifted sequences is employed in theorem above.…”

Section: Quotient Algebras 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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“…The following algebras are known to be syzygy-finite. (vii) algebras possessing an ideal I of finite projective dimension such that I radR = 0 and R/I is syzygy-finite [24].…”

Section: Syzygy-finite Algebrasmentioning

confidence: 99%

“…(v) algebras with an ideal I of finite projective dimension such that I rad 2 R = 0 (or I 2 radR = 0) and R/I is syzygy-finite [24];…”

Section: Igusa-todorov Algebrasmentioning

confidence: 99%