A generalized Koszul theory and its relation to the classical theory

Abstract

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...

Representations of modular skew group algebras

“…Since (ΛG) 0 in general is not semisimple, the classical Koszul theory described in [4,18] cannot apply, and we have to rely on generalized Koszul theories not requiring the semisimple property of (ΛG) 0 ; for example, [9,12,13,15,24]. Let Λ = i 0 Λ i be a locally finite graded k-algebra generated in degrees 0 and 1, i.e., Λ i Λ j = Λ i+j and dim k Λ i < ∞ for i, j 0.…”

“…For proofs and other details, please refer to [12,13]. For proofs and other details, please refer to [12,13].…”

“…We characterize skew group algebras with finite global dimension or finite representation type, and classify the representation types of transporter categories for p = 2, 3. Thus we use a generalized Koszul theory developed in [12,13], which does not demand the semisimple property. Under this assumption, we show that if ΛS is of finite representation type (for the situation where p 5 and Λ is not a local algebra) or of finite global dimension, then the action of S on E must be free.…”

“…Thus we use a generalized Koszul theory developed in [12,13], which does not demand the semisimple property. Therefore, it is reasonable to characterize its Koszul property.…”

A Generalized Koszul Property for Skew PBW Extensions

“…Many interesting algebras with properties similar to a Koszul algebra do not satisfy the assumption that A 0 is semisimple. There do already exist several generalized Koszul theories where the degree 0 part A 0 of a graded algebra A is not required to be semisimple, see [5], [8], and [24]. Each Koszul ring A defined by Woodcock in [24] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module.…”