A Representation theoretic perspective of Koszul theory

TL;DR

This paper explores a new connection between Koszul theory and representation theory, providing combinatorial descriptions, new classes of Koszul algebras, and constructing derived Koszul functors with duality properties.

Contribution

It introduces a combinatorial approach to Koszul complexes, constructs Koszul functors inducing derived equivalences, and extends Koszul duality in a representation-theoretic framework.

Findings

Describes local Koszul complexes and quadratic duals combinatorially

Constructs two Koszul functors inducing derived equivalences

Establishes almost split triangles with specific functors in derived categories

Abstract

We discover a new connection between Koszul theory and representation theory. Let $\La$ be a quadratic algebra defined by a locally finite quiver with relations. Firstly, we give a combinatorial description of the local Koszul complexes and the quadratic dual $\La^!$, which enables us to describe the linear projective resolutions and the colinear injective coresolutions of graded simple $\La$-modules in terms of $\La^!$. As applications, we obtain a new class of Koszul algebras and a stronger version of the Extension Conjecture for finite dimensional Koszul algebras with a noetherian Koszul dual. Then we construct two Koszul functors, which induce a $2$-real-parameter family of pairs of derived Koszul functors between categories derived from graded $\La$-modules and those derived from graded $\La^!$-modules. In case $\La$ is Koszul, each pair of derived Koszul functors are mutually quasi-inverse, one of the pairs is Beilinson, Ginzburg and Soergel's Koszul duality. If $\La$ and $\La^!$ are locally bounded on opposite sides, then the Koszul functors induce two equivalences of bounded derived categories: one for finitely piece-supported graded modules, and one for finite dimensional graded modules. And if $\La$ and $\La^!$ are both locally bounded, then the bounded derived category of finite dimensional graded $\La$-modules has almost split triangles with the Auslander-Reiten translations and the Serre functors given by composites of derived Koszul functors.