A generalized Koszul theory and its relation to the classical theory

Let $A = \bigoplus_{i \geqslant 0} A_i$ be a graded locally finite $k$-algebra such that $A_0$ is an arbitrary finite-dimensional algebra satisfying a certain splitting condition. In this paper we develop a generalized Koszul theory preserving many classical results. Moreover, we define a quotient graded algebra $\bar{A} = \bigoplus_{i \geqslant 0} \bar{A}_i$ and show that $A$ is a generalized Koszul algebra if and only if $\bar{A}$ is a classical Koszul algebra and a projective $A_0$-module. We also describe an application of this theory to the extension algebras of standard modules of standardly stratified algebras.