Quasi-hereditary algebras and generalized Koszul duality

TL;DR

This paper establishes a new sufficient condition for standard Koszul algebras to be Koszul with respect to a certain module, and explores the duality and equivalences between related algebraic structures.

Contribution

It introduces an easily applicable criterion for standard Koszul algebras to be Koszul with respect to elta and demonstrates the Koszul duality between the algebra and its extension algebra.

Findings

Koszul duality between elta and the algebra's extension algebra

Equivalence of derived categories of finitely generated graded modules

Extension algebra of elta is Koszul in the classical sense

Abstract

We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to $\Delta$. If a quasi-hereditary algebra $\L$ is Koszul with respect to $\Delta$, then $\L$ and the Yoneda extension algebra of $\Delta$ are Koszul dual in a sense explained below, implying in particular that their bounded derived categories of finitely generated graded modules are equivalent. We also prove that the extension algebra of $\Delta$ is Koszul in the classical sense.