The Graded Betti Numbers of the Skeletons of Simplicial Complexes

Abstract: In this paper, we study a class $C$ of squarefree monomial ideals $I$ in a polynomial ring $R=K[x_1,\dots,x_n]$ over a filed $K$ where $\dim R/I$ equals the maximum degree of the minimal generators of $I$ minus one. We show that the Stanley-Reisner ideal of any $i$-skeleton of a simplicial complex $\Delta$ in the class $C$ for $-1\le i<\dim\Delta$. Then, we introduce the notion of a degree resolution and prove that every ideal in the class $C$ possesses this property. Finally, we provides a formula to compute the graded Betti numbers of the $i$-skeletons of a simplicial complex in terms of the graded Betti numbers of the original complex. Conversely, we also present a way to express the graded Betti numbers of the original complex in terms of the graded Betti numbers of one of its skeletons.