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Betti numbers of skeletons of a class of squarefree monomial rings with l,inear resolution

Ralf Fröberg

Abstract

The starting point is the class of the following simplicial complexes $Δ$ with 2-linear resolutions. The facets of $Δ$ 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 $Δ$, giving sequences of identities for binomial coefficients.

Betti numbers of skeletons of a class of squarefree monomial rings with l,inear resolution

Abstract

The starting point is the class of the following simplicial complexes with 2-linear resolutions. The facets of are , and we demand that for each 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 , giving sequences of identities for binomial coefficients.
Paper Structure (5 sections, 12 theorems, 17 equations, 2 figures)

This paper contains 5 sections, 12 theorems, 17 equations, 2 figures.

Table of Contents

  1. Preliminaries
  2. Prerequisites
  3. Results
  4. Some identities for bimomial coefficients
  5. Competing interests

Key Result

Lemma 1.1

If $\Sigma$ is a simplicial complex with $f$-vector $(1,f_0,f_1,\ldots,f_d)$, then the Hilbert series of $k[\Sigma]$ is $1+f_0t/(1-t)+f_1t^2/(1-t)^2+\cdots+f_dt^{d+1}/(1-t)^{d+1}$.

Figures (2)

  • Figure 1: Pictorial representation of the simplicial complex $\{x_1,x_2,x_3\},\{x_3,x_4,x_5\}, \{x_{6}\}$.
  • Figure 2: The general Wollastonite graph $W(n_1,n_2,\dots,n_5).$

Theorems & Definitions (12)

  • Lemma 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Corollary 3.7
  • ...and 2 more