Koszul pairs and applications

TL;DR

This paper introduces the concept of Koszul pairs involving rings and corings over a semisimple ring, establishing criteria for Koszulity, and explores applications including Hochschild (co)homology and twisted tensor products.

Contribution

It defines almost-Koszul pairs, proves their equivalence to Koszul rings via associated corings, and applies this framework to Hochschild (co)homology and tensor product constructions.

Findings

A connected ring is Koszul iff it admits a compatible coring forming a Koszul pair.

The twisted tensor product of two Koszul rings is itself Koszul.

Applications include a generalization of Fröberg's theorem.

Abstract

Let $R$ be a semisimple ring. A pair $(A,C)$ is called almost-Koszul if $A$ is a connected graded $R$-ring and $C$ is a compatible connected graded $R$-coring. To an almost-Koszul pair one associates three chain complexes and three cochain complexes such that one of them is exact if and only if the others are so. In this situation $(A,C)$ is said to be Koszul. One proves that a connected $R$-ring $A$ is Koszul if and only if there is a connected $R$-coring $C$ such that $(A,C)$ is Koszul. This result allows us to investigate the Hochschild (co)homology of Koszul rings. We apply our method to show that the twisted tensor product of two Koszul rings is Koszul. More examples and applications of Koszul pairs, including a generalization of Fr\"oberg Theorem, are discussed in the last part of the paper.