Higher Cohen-Macaulay property of squarefree modules and simplicial posets

TL;DR

This paper extends the concept of higher Cohen-Macaulay properties from cell complexes to squarefree modules, toric face rings, and simplicial posets, establishing new relationships between their Cohen-Macaulay levels.

Contribution

It introduces new results linking the Cohen-Macaulay properties of simplicial posets and their skeletons, expanding the theoretical framework.

Findings

If a simplicial poset is l-Cohen-Macaulay, its codimension one skeleton is (l+1)-Cohen-Macaulay.

Extends higher Cohen-Macaulay property results to squarefree modules and toric face rings.

Provides partial generalizations of Floystad's results to broader algebraic structures.

Abstract

Recently, G. Floystad studied "higher Cohen-Macaulay property" of certain finite regular cell complexes. In this paper, we partially extend his results to squarefree modules, toric face rings, and simplicial posets. For example, we show that if (the corresponding cell complex of) a simplicial poset is $l$-Cohen-Macaulay then its codimension one skeleton is $(l+1)$-Cohen-Macaulay.