Cohen-Macaulay lexsegment complexes in arbitrary codimension

TL;DR

This paper characterizes Cohen-Macaulay lexsegment complexes in any codimension, establishing conditions based on purity, connectivity, and joins with simplices, and relates these properties to Buchsbaum complexes and Stanley-Reisner ideals.

Contribution

It provides a complete characterization of Cohen-Macaulay lexsegment complexes in arbitrary codimension, including special cases like flag complexes and those with non-quadratic ideals.

Findings

A lexsegment complex is Cohen-Macaulay iff it is pure with connected one-dimensional links.

A lexsegment flag complex is Cohen-Macaulay iff it is pure and connected.

Non-Cohen-Macaulay lexsegment complexes are Buchsbaum iff they are pure disconnected flag complexes.

Abstract

We characterize pure lexsegment complexes which are Cohen-Macaulay in arbitrary codimension. More precisely, we prove that any lexsegment complex is Cohen-Macaulay if and only if it is pure and its one dimensional links are connected, and, a lexsegment flag complex is Cohen-Macaulay if and only if it is pure and connected. We show that any non-Cohen-Macaulay lexsegment complex is a Buchsbaum complex if and only if it is a pure disconnected flag complex. For $t\ge 2$, a lexsegment complex is strictly Cohen-Macaulay in codimension $t$ if and only if it is the join of a lexsegment pure disconnected flag complex with a $(t-2)$-dimensional simplex. When the Stanley-Reisner ideal of a pure lexsegment complex is not quadratic, the complex is Cohen-Macaulay if and only if it is Cohen-Macaulay in some codimension. Our results are based on a characterization of Cohen-Macaulay and Buchsbaum lexsegment complexes by Bonanzinga, Sorrenti and Terai.