A generalized Koszul theory and its relation to the classical theory

TL;DR

This paper develops a generalized Koszul theory for graded algebras with arbitrary finite-dimensional degree-zero parts, linking it to classical Koszul theory through quotient algebras and applications to stratified algebra extensions.

Contribution

It introduces a generalized Koszul framework for broader classes of graded algebras and establishes a connection to classical Koszul algebras via quotient constructions.

Findings

A generalized Koszul theory is developed for graded algebras with arbitrary $A_0$.

A characterization of generalized Koszul algebras via classical Koszul algebras and projectivity is provided.

Applications to extension algebras of standard modules in stratified algebras are demonstrated.

Abstract

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.