Betti numbers of fat forests and their Alexander dual

TL;DR

This paper computes the Betti numbers of fat forests, a class of simplicial complexes with 2-linear resolutions, and explores the Betti numbers of their Alexander duals, advancing understanding of their algebraic and combinatorial properties.

Contribution

It provides explicit calculations of Betti numbers for fat forests and their Alexander duals, a class previously characterized by 2-linear resolutions.

Findings

Betti numbers of fat forests are explicitly determined.

Betti numbers of Alexander duals of fat forests are analyzed.

Advances understanding of algebraic invariants of specific simplicial complexes.

Abstract

Let $k$ be a field and $R=k[x_1,\ldots,x_n]/I=S/I$ a graded ring. Then $R$ has a $t$-linear resolution if $I$ is generated by homogeneous elements of degree $t$, and all higher syzygies are linear. Thus $R$ has a $t$-linear resolution if ${\rm Tor}^S_{i,j}(S/I,k)=0$ if $j\ne i+t-1$. For a simplicial complex $\Delta$ on $[{\bf n}]=\{1,\ldots,n\}$ and a field $k$, the Stanley-Reisner ring $k[\Delta]$ is $k[x_1,\ldots,x_n]/I$, where $I$ is generated by those squarefree monomials $x_{i_1}\cdots x_{i_k}$ for which $\{ i_1,\ldots,i_k\}$ does not belong to $\Delta$. In \cite{Fr} the Stanley-Reisner rings with 2-linear resolution are determined. Their associated complexes has had different names in the literature. We call them fat forests here. In this article we determine the Betti numbers of fat forests. We also consider Betti numbers of Alexander duals of fat forests.