Finitistic dimension of standardly stratified algebras

Ágoston, István; Happel, Dieter; Lukács, Erzsébet; Unger, Luise

doi:10.1080/00927870008826990

Cited by 28 publications

(43 citation statements)

References 5 publications

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“…In [AHLU2] it was shown that the projectively defined finitistic dimension of a properly stratified (and even SSS-) algebra is finite (and even that the injectively defined finitistic dimension of such algebra is finite as well). In this subsection we show that the module H can be used to effectively compute fin.dim(A).…”

Section: H and Findim(a)mentioning

confidence: 99%

Properly Stratified Algebras and Tilting

Frisk¹,

Mazorchuk²

2005

Proc. Lond. Math. Soc.

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We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras, whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties, for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

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Section: H and Findim(a)mentioning

confidence: 99%

Properly Stratified Algebras and Tilting

Frisk¹,

Mazorchuk²

2005

Proc. Lond. Math. Soc.

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“…The finitistic dimension of the algebra is the the finitistic dimension of the whole module category -mod. With this notation, we state two results from [DR,2.2] and [AHLU,3.4]. Proposition 1.8.…”

Section: Properties Of Standardly Stratified Algebrasmentioning

confidence: 97%

“…For an optimal bound, obtained by different methods, as well as for a bound on the injectively defined finitistic dimension, we refer to [AHLU,2.1,3.1].…”

Section: Proof To Prove That D Imentioning

confidence: 99%

Standardly Stratified Algebras and Tilting

et al. 2000

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“…It was shown in [2] that the finitistic dimension conjecture holds for ss-algebras. In that paper, it was shown that pfd(R) 2n − 2, where n is the number of iso-classes of simple R-modules.…”

Section: The Finitistic Dimension Conjecture For Ss-algebrasmentioning

confidence: 98%

An approach to the finitistic dimension conjecture

2008

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Let R be a finite dimensional k-algebra over an algebraically closed field k and mod R be the category of all finitely generated left R-modules. For a given full subcategory X of mod R, we denote by pfd X the projective finitistic dimension of X . That is, pfd X := sup{pd X: X ∈ X and pd X < ∞}.It was conjectured by H. Bass in the 60's that the projective finitistic dimension pfd(R) := pfd(mod R) has to be finite. Since then, much work has been done toward the proof of this conjecture. Recently, K. Igusa and G. Todorov defined in [K. Igusa, G. Todorov, On the finitistic global dimension conjecture for artin algebras, in: Representations of Algebras and Related Topics, in: Fields Inst. Commun., vol. 45, Amer. Math. Soc., Providence, RI, 2005, pp. 201-204] a function Ψ : mod R → N, which turned out to be useful to prove that pfd(R) is finite for some classes of algebras. In order to have a different approach to the finitistic dimension conjecture, we propose to consider a class of full subcategories of mod R instead of a class of algebras. That is, we suggest to take the class of categories F (θ), of θ-filtered R-modules, for all stratifying systems (θ, ) in mod R. We prove that the Finitistic Dimension Conjecture holds for the categories of filtered modules for stratifying systems with one or two (and some cases of three) modules of infinite projective dimension.

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Finitistic dimension of standardly stratified algebras

Cited by 28 publications

References 5 publications

Properly Stratified Algebras and Tilting

Properly Stratified Algebras and Tilting

Standardly Stratified Algebras and Tilting

An approach to the finitistic dimension conjecture

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