Subadditivity, strand connectivity and multigraded Betti numbers of monomial ideals

TL;DR

This paper investigates subadditivity, strand connectivity, and multigraded Betti numbers of monomial ideals, providing new conditions and classes of graphs where these properties hold, advancing understanding in algebraic combinatorics.

Contribution

It establishes sufficient conditions for subadditivity, confirms it for complete intersection ideals, and explores strand connectivity and Betti number bounds for edge ideals.

Findings

Subadditivity holds for ideals generated by homogeneous complete intersections.

Identifies classes of graphs with subadditive edge ideals.

Provides upper bounds for multigraded Betti numbers of edge ideals.

Abstract

Let $R = \mathbb{K}[x_1, \ldots, x_n]$ and $I \subset R$ be a homogeneous ideal. In this article, we first obtain certain sufficient conditions for the subadditivity of $R/I$. As a consequence, we prove that if $I$ is generated by homogeneous complete intersection, then subadditivity holds for $R/I$. We then study a conjecture of Avramov, Conca and Iyengar on subadditivity, when $I$ is a monomial ideal with $R/I$ Koszul. We identify several classes of edge ideals of graphs $G$ such that the subadditivity holds for $R/I(G)$. We then study the strand connectivity of edge ideals and obtain several classes of graphs whose edge ideals are strand connected. Finally, we compute upper bounds for multigraded Betti numbers of several classes of edge ideals.