On a common generalization of Koszul duality and tilting equivalence

TL;DR

This paper introduces a generalized notion of Koszulity applicable to graded algebras with a finite global dimension degree zero part, extending classical duality results and applying to certain blocks in category O.

Contribution

It defines a new form of Koszulity for broader classes of graded algebras and demonstrates that classical duality theorems still hold in this context.

Findings

Standard Koszul duality theorems extend to the new setting.

Application to multiplicity free blocks of category O.

Broader class of algebras with Koszul properties.

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 $\mathcal O$.