Graded and Koszul categories

TL;DR

This paper extends the concepts of Koszul algebras, modules, and duality from finite-dimensional algebras to additive graded categories, providing foundational generalizations for broader mathematical contexts.

Contribution

It introduces a framework for associating Koszul theory with any finite-dimensional algebra via additive graded categories, expanding the scope of Koszul duality.

Findings

Generalization of Koszul algebras to additive categories

Extension of linear modules and duality concepts

Preliminaries for future applications in various mathematical fields

Abstract

Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there is always a naturally associated Koszul theory. To obtain this, the notions of Koszul algebras, linear modules and Koszul duality are extended to additive (graded) categories over a field. The main focus of this paper is to provide these generalizations and the necessary preliminaries.