The Graded Betti Numbers of the Skeletons of Simplicial Complexes

2025/10/11 by Namiq, Mohammed Rafiq · 1 citation
#13F55 #55U05 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 13D02 #Secondary 13A02

paper · doi:10.48550/arxiv.2510.09969

Abstract

In this paper, we study a class C of squarefree monomial ideals I⊆ R=\mathbbK[x₁,…,x_n] over a field \mathbbK, 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 C for all -1≤ i<dimΔ. To investigate their homological properties, we introduce the notion of a degree resolution and prove that every ideal in 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.

Citations

• The graded Betti numbers of truncation of ideals in polynomial rings
• Upper bounds for the regularity of powers of edge ideals of graphs
• Regularity of powers of forests and cycles
• The Regularity of Powers of Edge Ideals
• Monomial ideals whose powers have a linear resolution

Cited by

• Initially Cohen-Macaulay Modules

Related

• The graded Betti numbers of truncation of ideals in polynomial rings
• Betti numbers of skeletons
• Homological invariants of the Stanley-Reisner ring of a k-decomposable simplicial complex
• Dependence of Betti Numbers on Characteristic
• Graded Betti numbers of balanced simplicial complexes