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

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 Bérénice Delcroix-Oger and Clément Dupont in a recent article.