On a common generalization of Koszul duality and tilting equivalence

Abstract

We propose a new definition of Koszulity for graded algebras where the degree zero part has finite global dimension, but is not necessarily semi-simple. The standard Koszul duality theorems hold in this setting. We give an application to algebras arising from multiplicity free blocks of the BGG category O.Forgetting the grading of Λ, we denote by Mod Λ the category of (ungraded) Λ-modules. If A is a finite dimensional k-algebra, then we denote by mod A the category of finitely generated (ungraded) A-modules.The duality D can also in a straightforward way be defined for morphisms of graded modules, and we get a duality functor D : lfd Λ → lfd Λ op . This functor restricts to dualities D : gr Λ → cgr Λ op and D : cgr Λ → gr Λ op . Restricting further to finite dimensional modules concentrated in degree zero, we get a duality D : mod Λ 0 → mod Λ op 0 . If Λ = Λ 0 , then this is the usual duality D : mod Λ → mod Λ op for finite dimensional algebras.Given a graded Λ-module M , the jth graded shift of M, denoted M j , is the module with graded parts (M j ) i = M i−j and module structure inherited from M . There is a functorial isomorphism D(M j ) ≃ DM −j . 1.2. Graded extensions. The category Gr Λ is abelian, and one can define the usual homological functors. The following well-known result can for instance be found in [NV, Corollaries 2.4.4 and 2.4.7]. Lemma 1.2.1. Let Λ = i≥0 Λ i be a graded algebra, and let M and N be graded Λ-modules. (a) If M is finitely generated, then Hom Λ (M, N ) ≃ j∈Z Hom Gr Λ (M, N j ) (b) If M is finitely generated and has a projective resolution such that all syzygies are finitely generated, then Ext i Λ (M, N ) ≃ j∈Z Ext i Gr Λ (M, N j )for all i ≥ 0.With some finiteness conditions, the duality D and graded extensions combine well. A reference for the following proposition is [Mad, Proposition 1.3].Proposition 1.2.2. Let Λ = i≥0 Λ i be a graded algebra, let M and N be graded locally finite dimensional Λ-modules, and let D denote the above duality. If M is bounded below or N is bounded above, then Ext iGr Λ (M, N ) ≃ Ext i Gr Λ op (DN, DM ) for all i ≥ 0.

A generalized Koszul theory and its application

“…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 [11,15,16,23]. Each Koszul algebra A defined by Woodcock in [23] is supposed to satisfy that A is both a left projective A 0 -module and a right projective A 0 -module.…”

“…Indeed, even the category algebra kE of a standardly stratified finite EI category E (studied in [22]) does not satisfy this requirement: kE is a left projective kE 0 -module but in general not a right projective kE 0 -module. In Madsen's paper [16], A 0 is supposed to have finite global dimension. But for a finite EI category E, this happens in our context if and only if kE 0 is semisimple.…”

Quasi-hereditary algebras and generalized Koszul duality

“…We call the pair (Γ, DT ) the Koszul dual of (Λ, T ). Since T is graded self-orthogonal, we can construct a certain bigraded bimodule X and a functor between unbounded derived categories of graded modules [Mad,Section 3]. When Λ is T -Koszul, the functor G T restricts to an equivalence in different ways and we mention one version here.…”

“…[Mad,Theorem 4.2.1] Let Λ = i≥0 Λ i be a graded algebra with gldim Λ 0 < ∞. Suppose Λ is a Koszul algebra with respect to a module…”

A generalized Koszul theory and its relation to the classical theory

“…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.…”