Koszul duality for stratified algebras I. Quasi-hereditary algebras

TL;DR

This paper explores the relationship between Koszul and Ringel dualities in quasi-hereditary algebras with linear tilting (co)resolutions, showing they are Koszul, closed under dualities, and commute.

Contribution

It provides a complete characterization of the interaction between Koszul and Ringel dualities for a specific class of quasi-hereditary algebras, establishing their Koszul property and duality closure.

Findings

Algebras with linear tilting (co)resolutions are Koszul.

The class of these algebras is closed under both dualities.

Koszul and Ringel dualities commute on this class.

Abstract

We give a complete picture of the interaction between Koszul and Ringel dualities for quasi-hereditary algebras admitting linear tilting (co)resolutions of standard and costandard modules. We show that such algebras are Koszul, that the class of these algebras is closed with respect to both dualities and that on this class these two dualities commute. All arguments reduce to short computations in the bounded derived category of graded modules.