Standardly Stratified Extension Algebras

Ágoston, István; Dlab, Vlastimil; Lukács, Erzsébet

doi:10.1081/agb-200060498

Cited by 12 publications

(17 citation statements)

References 9 publications

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“…Theorem 1 extends and generalizes results from [ADL1,ADL2,Ma,MO]. The paper is organized as follows.…”

Section: Introductionsupporting

confidence: 72%

“…The aim of this paper is to define a setup for the study of the Koszul property of stratified algebras and to extend to this setup the main result of [Ma]. We note that Koszul standardly stratified algebras that are not quasihereditary appear naturally in [ADL2,Fr3,KKM]. The most general setup for stratified algebras seems to be the notion of standardly stratified algebras, as introduced by Cline et al [CPS1].…”

Section: Introductionmentioning

confidence: 99%

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Koszul Duality for Stratified Algebras Ii. Standardly Stratified Algebras

Mazorchuk

2010

J. Aust. Math. Soc.

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We give a complete picture of the interaction between the Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras, imposing the condition that standard filtrations of projective modules are finite, and develop a tilting theory for such algebras. Under the assumption on existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both the Ringel and Koszul duals belong to the same class, and that these two dualities on this class commute.2000 Mathematics subject classification: primary 16S37; secondary 16E30, 16G10.

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“…Theorem 1 extends and generalizes results from [ADL1,ADL2,Ma,MO]. The paper is organized as follows.…”

Section: Introductionsupporting

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Koszul Duality for Stratified Algebras Ii. Standardly Stratified Algebras

Mazorchuk

2010

J. Aust. Math. Soc.

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“…Later, the same authors investigated the analogous question for Koszul standardly stratified algebras under the additional assumption that the initial algebra was graded. They generalized the standard Koszul property for this class of algebras in [5], and achieved similar results for this case, using Poincaré and Hilbert matrices.…”

Section: Introductionmentioning

confidence: 74%

“…The algebra A * is standardly stratified if A * A * is filtered by right standard A * -modules. In view of Theorem 1 of [5], if A * is tightly graded, then this is equivalent to the condition that A * A * is filtered by left proper standard A * -modules.…”

Section: The Functor Ext *mentioning

confidence: 99%

Stratified modules over an extension algebra

Lukács¹,

Magyar²

2017

Czech.Math.J.

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“…Extension algebras of standard modules of standardly stratified algebras have been widely studied in [1,2,3,8,13,15,20,22,24,25,26]. Different from the usual approach, we do not assume that the standardly stratified algebra is graded, the degree 0 part is semisimple, and its grading is compatible with the filtration by standard modules.…”

Section: Introductionmentioning

confidence: 99%

A generalized Koszul theory and its relation to the classical theory

2014

Journal of Algebra

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Abstract. Let A = i 0 A i be a graded locally finite k-algebra where A 0 is a finite-dimensional algebra whose finitistic dimension is 0. In this paper we develop a generalized Koszul theory preserving many classical results, and show an explicit correspondence between this generalized theory and the classical theory. Applications in representations of certain categories and extension algebras of standard modules of standardly stratified algebras are described. IntroductionThe classical Koszul theory plays an important role in the representation theory of graded algebras. However, there are a lot of structures (algebras, categories, etc) having natural gradings with non-semisimple degree 0 parts, to which the classical theory cannot apply. Particular examples of such structures include tensor algebras generated by non-semisimple algebras A 0 and (A 0 , A 0 )-bimodules A 1 , extension algebras of finitely generated modules (among which we are most interested in extension algebras of standard modules of standardly stratified algebras [8,15]), graded modular skew group algebras, category algebras of finite EI categories [14,27,28], and certain graded k-linear categories. Therefore, it is reasonable to develop a generalized Koszul theory to study representations and homological properties of above structures.In [11,18,19,29] several generalized Koszul theories have been described, where the degree 0 part A 0 of a graded algebra A is not required to be semisimple. In [29], A is supposed to be both a left projective A 0 -module and a right projective A 0 -module. However, in many cases A is indeed a left projective A 0 -module, but not a right projective A 0 -module. In Madsen's paper [19], A 0 is supposed to have finite global dimension. This requirement is too strong for us since in many applications A 0 is a self-injective algebra or a direct sum of local algebras, and hence A 0 has finite global dimension if and only if it is semisimple, falling into the framework of the classical theory. The theory developed by Green, Reiten and Solberg in [11] works in a very general framework, and we want to find some conditions which are easy to check in practice. The author has already developed a generalized Koszul theory in [17] under the assumption that A 0 is self-injective, and used it to study representations and homological properties of certain categories.The goal of the work described in this paper is to loose the assumption that A 0 is self-injective (as required in [17]) and replace it by a weaker condition so that the generalized theory can apply to more situations. Specifically, since we are interested in the extension algebras of modules, category algebras of finite EI categories, and graded k-linear categories for which the endomorphism algebra of each object is a finite dimensional local algebra, this weaker condition should be satisfied by self-injective algebras and finite dimensional local algebras. On the other hand, we also expect that many classical results as the Koszul duality can be preserved. Moreove...

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Standardly Stratified Extension Algebras

Cited by 12 publications

References 9 publications

Koszul Duality for Stratified Algebras Ii. Standardly Stratified Algebras

Koszul Duality for Stratified Algebras Ii. Standardly Stratified Algebras

Stratified modules over an extension algebra

A generalized Koszul theory and its relation to the classical theory

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