SUBADDITIVITY, STRAND CONNECTIVITY AND MULTIGRADED BETTI NUMBERS OF MONOMIAL IDEALS

Abstract

Let $R=\mathbb{𝕂}[x_1,\dots ,x_n]$ and $I\subset R$ be a homogeneous ideal. We first obtain certain sufficient conditions for the subadditivity condition of $R/I$. As a consequence, we prove that if $I$ is homogeneous complete intersection, then the subadditivity condition holds for $R/I$. We then study a conjecture of Avramov, Conca and Iyengar on the subadditivity condition, when $I$ is a monomial ideal with $R/I$ Koszul. We identify several classes of edge ideals of graphs $G$ such that the subadditivity condition 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.