Castelnuovo–Mumford regularity of monomial ideals arising from posets

Abstract

Let P be a finite poset and suppose I is the Stanley–Reisner ideal of the chain complex of P. In this paper, we prove the inequality $\text{reg}(I^{(s)})\le 2s+\text{rank}(P)$ for every integer s where $I^{(s)}$ denotes the sth symbolic power of I. As a consequence, we deduce that if I(G) is the edge ideal of a graph G such that its complementary graph G^c is bipartite, then the regularity of $I{(G)}^{(s)}$ is either 2s or $2s+1.$

Keywords:

• Castelnuovo–Mumford regularity
• chain complex
• edge ideal
• poset

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 13D02
• 05E40