Homological freeness criterion for operadic modules and application to Cohen-Macalayness of posets

TL;DR

This paper introduces a new homological freeness criterion for operadic modules and applies it to demonstrate Cohen-Macaulay properties of certain posets, resolving open questions in the field.

Contribution

It develops a variation of the homological freeness criterion and applies it to decorated partition posets, establishing their Cohen-Macaulayness and computing their homology.

Findings

Operadic modules over Koszul operads can satisfy a new homological freeness criterion.

Decorated partition posets are shown to be Cohen-Macaulay.

Homology of these posets is explicitly computed.

Abstract

We show a variation of the usual homological freeness criterion for operadic modules over a Koszul operad. We then apply this result to decorated partition posets for some operads, showing that their augmentation is Cohen-Macaulay and computing its homology. This work answers several open questions asked by B\'er\'enice Delcroix-Oger and Cl\'ement Dupont in a recent article.