Koszul duality for stratified algebras II. Standardly stratified algebras

TL;DR

This paper explores the deep relationship between Koszul and Ringel dualities in graded standardly stratified algebras, establishing conditions under which these dualities commute and characterizing a specific class of Koszul algebras.

Contribution

It introduces a class of graded standardly stratified algebras with finite standard filtrations and develops their tilting theory, showing dualities commute within this class.

Findings

Algebras in the class are Koszul.

Ringel and Koszul duals are within the class.

Dualities commute on this class.

Abstract

We give a complete picture of the interaction between Koszul and Ringel dualities for graded standardly stratified algebras (in the sense of Cline, Parshall and Scott) admitting linear tilting (co)resolutions of standard and proper costandard modules. We single out a certain class of graded standardly stratified algebras imposing the condition that standard filtrations of projective modules are finite, and develop the tilting theory for such algebras. Under the assumption of existence of linear tilting (co)resolutions we show that algebras from this class are Koszul, that both Ringel and Koszul duals belong to the class, and that these two dualities on this class commute.