On Koszulity of Finite Posets

TL;DR

This paper unifies various characterizations of Koszul rings and applies these to finite graded posets, providing an algorithm to identify and generate new Koszul posets with concrete examples.

Contribution

It offers a unifying framework for Koszul ring characterizations and introduces an algorithm to find new Koszul posets based on incidence rings.

Findings

Multiple equivalent descriptions of Koszul rings established

Algorithm for constructing Koszul posets developed

Examples of Koszul posets provided

Abstract

We prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of $\Tor_n^A(R,R)$. As an application of these characterizations we investigate the Koszulity of the incidence rings for finite graded posets. Based on these results, we describe an algorithm to produce new classes of Koszul posets (i.e. graded posets whose incidence rings are Koszul). Specific examples of Koszul posets are included.