Recollements of derived categories III: finitistic dimensions

Chen, Hong Xing; Xi, Chang Chang

doi:10.1112/jlms.12026

Cited by 28 publications

(14 citation statements)

References 25 publications

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“…All of these conjectures would be valid if so would be the finitistic dimension conjecture. Several special cases for the conjecture to be true are verified (see, for example, [2,9,11,12,13,14,26,27,7]), but it is not yet fully resolved in general. Actually, up to the present time, not many practical methods, so far as we know, are available to detect algebras of finite finitistic dimensions.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Finitistic dimension conjecture and radical-power extensions

Wang

2017

Journal of Pure and Applied Algebra

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“…The finitistic dimension conjecture has a reduction by recollements. This reduction is further refined in [30], where relations among the finitistic dimensions of three algebras in a recollement are described. To state this refinement precisely, we recall the following definitions from [30].…”

Section: Finitistic Dimension Conjecturementioning

confidence: 99%

“…It is not difficult to extend the definition of homological width and cowidth to any complexes which are quasi-isomorphic to complexes in C b (A-Proj) and C b (A-Inj), respectively. Theorem 6.4 [30]. Let R 1 , R 2 and R 3 be rings.…”

Section: Finitistic Dimension Conjecturementioning

confidence: 99%

Derived equivalences of algebras

2018

Bull. London Math. Soc.

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Derived categories and equivalences between them are the pièce de résistance of modern homological algebra. They are widely used in many branches of mathematics, especially in algebraic geometry and representation theory. In this note, we shall survey some recently developed construction methods of derived equivalences for algebras and rings, with applications to homological conjectures, such as Broué's abelian defect group conjecture and the finitistic dimension conjecture, and to computation of higher algebraic K‐groups of algebras and rings.

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“…For example, Happel studied how the finiteness of the finitistic dimension of algebras in a recollement interacted on each other [11], and some authors discussed the finiteness of the global dimension [23,17,1]. Recently, many experts turn to the study of the homological properties or invariants in the framework of recollements (see [1,3,4,16,19,21]). In this paper, we will investigate the behavior of the homological dimensions under recollements of derived categories of algebras.…”

Section: Introductionmentioning

confidence: 99%