Generalized Koszul Algebra and Koszul Duality

TL;DR

This paper extends classical Koszul theory to a broader class of graded rings with noetherian semiperfect degree zero parts, establishing duality results and characterizations for generalized Koszul rings and modules.

Contribution

It introduces a generalized Koszul theory for $ $-graded rings with noetherian semiperfect degree zero parts, unifying and extending classical Koszul duality results.

Findings

Generalized Koszul duality for $ $-graded rings and modules.

Equivalence of generalized Koszulity with classical Koszulity of Ext rings.

Characterization of generalized Koszul rings with finite global dimension as generalized AS regular.

Abstract

We define generalized Koszul modules and rings and develop a generalized Koszul theory for $\mathbb{N}$-graded rings with the degree zero part noetherian semiperfect. This theory specializes to the classical Koszul theory for graded rings with degree zero part artinian semisimple developed by Beilinson-Ginzburg-Soergel and the ungraded Koszul theory for noetherian semiperfect rings developed by Green and Martin{\'e}z-Villa. Let $A$ be a left finite $\mathbb{N}$-graded ring generated in degree $1$ with $A_0$ noetherian semiperfect, $J$ be its graded Jacobson radical and $S=A/J$. By the Koszul dual of $A$ we mean the Yoneda Ext ring $\underline{\text{Ext}}_A^\bullet(S,S)$. If $A$ is a generalized Koszul ring and $M$ is a generalized Koszul module, then it is proved that the Koszul dual of the Koszul dual of $A$ is $\text{Gr}_J A$ and the Koszul dual of the Koszul dual of $M$ is $\text{Gr}_J M$. If $A$ is a locally finite algebra, then the following statements are proved to be equivalent: $A$ is generalized Koszul; the Koszul dual $\underline{\text{Ext}}_A^\bullet(S,S)$ of $A$ is (classically) Koszul; $\text{Gr}_J A$ is (classically) Koszul; the opposite ring $A^{op}$ of $A$ is generalized Koszul. It is also proved that if $A$ is generalized Koszul with finite global dimension then $A$ is generalized AS regular if and only if the Koszul dual of $A$ is self-injective.