Simplicial complexes which are minimal Cohen-Macaulay

TL;DR

This paper characterizes minimal Cohen-Macaulay simplicial complexes, proving that such complexes with certain properties have exactly twice as many vertices as their dimension plus one, and relates shellability to Cohen-Macaulay properties.

Contribution

It establishes a precise vertex count for minimal Cohen-Macaulay pure simplicial complexes and links shellability with Cohen-Macaulay properties of subcomplexes.

Findings

Minimal Cohen-Macaulay complexes have n=2d vertices.

Shellability of pure complexes is equivalent to Cohen-Macaulay properties of subcomplexes.

Provides a characterization connecting combinatorial and algebraic properties.

Abstract

Let $\D$ be a $(d-1)$-dimensional pure $f$-simplicial complex over vertex set $[n]$. In this paper, it is proved that $n=2d$ holds true if $\D$ is minimal Cohen-Macaulay. It is also indicated that the recent work of \cite{Dao2020} implies that shellable condition on a pure simplicial complex $\D$ is identical with CM properties of a full series of subcomplexes of $\D$.