Linear strand of edge ideals of comaximal graphs of commutative rings

Abstract

For an edge ideal I(G) of a simple graph $G,$ we study the $\mathbb{N}$-graded Betti numbers that appear in the linear strand of the minimal free resolution of $I(\Gamma ({\mathbb{Z}}_n)),$ where $\Gamma ({\mathbb{Z}}_n)$ is the comaximal graph of the integral modulo ring ${\mathbb{Z}}_n$. We show that the extremal Betti number of $I(\Gamma ({\mathbb{Z}}_n))$ is $\varphi (n),$ where $\varphi (n)$ is the Euler’s totient function and thereby we obtain a large class of edge ideals with even extremal Betti numbers. We find the regularity (Castelnuovo-Mumford) and the projective dimension of these ideals. Moreover, we exhibit explicit formulae that determine all the $\mathbb{N}$-graded Betti numbers in the linear strand of the minimal free resolution of $I(\Gamma ({\mathbb{Z}}_n))$ for certain values of $n.$

KEYWORDS:

• Betti numbers
• comaximal graphs
• edge ideals
• Euler’s totient function
• integers modulo ring
• linear strand
• projective dimension
• regularity

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D40
• 13D02
• 13F55
• 05C90
• 05C25
• 05E45
• 05E40

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There is no data associated with this article.

Disclosure statement

The authors declare that they have no competing interests.