Equivalent condition for approximately Cohen-Macaulay complexes

TL;DR

This paper establishes a precise criterion for when a simplicial complex is approximately Cohen-Macaulay, linking it to properties of its Alexander dual ideal, thus extending previous characterizations of Cohen-Macaulayness.

Contribution

It provides a necessary and sufficient condition for approximate Cohen-Macaulayness based on the componentwise linearity and degree generation of the Alexander dual ideal.

Findings

Characterizes approximately Cohen-Macaulay complexes via Alexander duality.

Completes the understanding of Cohen-Macaulay conditions in simplicial complexes.

Links algebraic properties of ideals to topological properties of complexes.

Abstract

We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen-Macaulay. Namely it is approximately Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and generated in two consecutive degrees. This completes the result of Herzog and Hibi who proved that a simplicial complex is sequentially Cohen-Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear.