Generalized Koszul Algebra and Koszul Duality

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.