Some homological properties of the category 𝒪, II

Mazorchuk, Volodymyr

doi:10.1090/s1088-4165-10-00368-7

Cited by 28 publications

(63 citation statements)

References 44 publications

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“…Proof. Claims (i) and (ii) follow from Proposition 30(iii) and (iv) and Proposition 35 in combination with [Ma3,Theorems 17 and 20]. Claims (iii) and (iv) follow from Proposition 23(i) and (iii) in combination with Theorem 43(i).…”

Section: Projective Dimensions Of Structural Modulesmentioning

confidence: 92%

“…Hence in each such right cell we can choose x to be the corresponding Duflo involution. In this case, the proof of [Ma3,Lemma 23] implies that the extension groups in equation (18)…”

Section: Consequently Both the Functionmentioning

confidence: 99%

“…In the preliminaries we give some overview of the literature on this subject. The first two papers [Ma1,Ma3] in the present series initiated the study of the projective dimension of these structural objects for O, by determining them for the principal block O 0 of the original (i.e. non-parabolic) category O.…”

Section: Introductionmentioning

confidence: 99%

“…Note that we use the convention that the graded length of a module concentrated in a single degree is zero. Some other papers, for example [Ma3], use the convention that the graded length of a module concentrated in a single degree is one.…”

Section: Introductionmentioning

confidence: 99%

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Some homological properties of category 𝒪. IV

Coulembier

Mazorchuk

2016

Forum Mathematicum

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We study projective dimension and graded length of structural modules in parabolic-singular blocks of the BGG category O. Some of these are calculated explicitly, others are expressed in terms of two functions. We also obtain several partial results and estimates for these two functions and relate them to monotonicity properties for quasi-hereditary algebras. The results are then applied to study blocks of O in the context of Guichardet categories, in particular, we show that blocks of O are not always weakly Guichardet.MSC 2010 : 16E30, 17B10

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Section: Projective Dimensions Of Structural Modulesmentioning

confidence: 92%

“…Hence in each such right cell we can choose x to be the corresponding Duflo involution. In this case, the proof of [Ma3,Lemma 23] implies that the extension groups in equation (18)…”

Section: Consequently Both the Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Some homological properties of category 𝒪. IV

Coulembier

Mazorchuk

2016

Forum Mathematicum

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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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On First Extensions in $$\mathcal {S}$$-Subcategories of $$\mathcal {O}$$

Mazorchuk

2022

Springer Proceedings in Mathematics &Amp; Statistics

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Some homological properties of the category 𝒪, II

Cited by 28 publications

References 44 publications

Some homological properties of category 𝒪. IV

Some homological properties of category 𝒪. IV

A generalized Koszul theory and its relation to the classical theory

On First Extensions in $$\mathcal {S}$$-Subcategories of $$\mathcal {O}$$

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