Cohen-Macaulay Lexsegment Complexes in Arbitrary Codimension

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 1-dimensional links are connected, and that 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 [Formula: see text], 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 [Formula: see text]-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.