The Graded Betti Numbers of the Skeletons of Simplicial Complexes
Mohammed Rafiq Namiq
TL;DR
This work addresses how the graded Betti numbers of a simplicial complex $\Delta$ relate to those of its $i$-skeletons $\Delta^i$ by introducing the notion of a degree resolution and proving that $I_{\Delta^i}$ has a degree resolution with $\mathrm{reg}\mathbb{K}[\Delta^i]=\dim\mathbb{K}[\Delta^i]$ for all $-1\le i<\dim\Delta$. It provides an explicit Betti-number formula linking $\mathbb{K}[\Delta^k]$ to $\mathbb{K}[\Delta]$ through the $f$-vector and existing Betti numbers, and shows how skeletons can determine or preserve Betti data under certain reg/omega conditions. The results yield corollaries that allow computing or recovering Betti tables from skeletons and highlight the role of truncations and degree resolutions in simplifying the analysis of squarefree monomial ideals. Overall, the paper offers a concrete framework to transfer homological information between a complex and its skeletons, with practical implications for calculating Betti numbers in combinatorial settings.
Abstract
In this paper, we study a class $\mathcal{C}$ of squarefree monomial ideals $I\subseteq R=\mathbb{K}[x_1,\dots,x_n]$ over a field $\mathbb{K}$, defined by the condition that $\dim R/I$ equals the maximum degree of the minimal generators of $I$ minus one. We show that the Stanley-Reisner ideal of every $i$-skeleton of a simplicial complex $Δ$ belongs to $\mathcal{C}$ for all $-1\le i<\dimΔ$. To investigate their homological properties, we introduce the notion of a degree resolution and prove that every ideal in $\mathcal{C}$ possesses this property. Moreover, we show that every squarefree monomial ideal admits a truncation whose regularity coincides with that of the original ideal, thereby reducing the study of degree resolutions to that of linear resolutions. Finally, we provide an explicit formula describing the relationship between the graded Betti numbers of a simplicial complex and those of its skeletons.