Betti numbers of skeletons of thick trees

TL;DR

This paper investigates the Betti numbers and algebraic properties of skeletons of a specific class of simplicial complexes with 2-linear resolutions, providing formulas and conditions for Cohen-Macaulayness.

Contribution

It explicitly determines Betti numbers, projective dimension, depth, regularity, and Cohen-Macaulayness for skeletons of these complexes, and derives identities for their Hilbert series.

Findings

Betti numbers and algebraic invariants are explicitly computed.

Conditions for Cohen-Macaulayness are established.

Two methods for Hilbert series yield binomial coefficient identities.

Abstract

The starting point is the class of the following simplicial complexes $\Delta$ with 2-linear resolutions. The facets of $\Delta$ are $F_1,\ldots,F_n$, and we demand that for each $i$ $F_i\cap (F_1\cup \cdots\cup F_{i-1}\cup F_{i+1}\cdots\cup F_n)$ be a point. We will determine the Betti numbers, and thus the projective dimension, the depth, and the regularity of the Stanley-Reisner rings of all skeletons of such complexes. It follows that we know when these complexes are Cohen-Macaulay. Also, there are two ways to determine the Hilbert series of $\Delta$, giving sequences of identities for binomial coefficients.