Bounded powers of edge ideals: regularity and linear quotients

Abstract

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and let $I \subset S$ be a monomial ideal. For a vector $\mathfrak{c}\in\mathbb{N}^n$, we set $I_{\mathfrak{c}}$ to be the ideal generated by monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Also, let $δ_{\mathfrak{c}}(I)$ be the largest integer $k$ such that $(I^k)_{\mathfrak{c}}\neq 0$. It is shown that for every graph $G$ with edge ideal $I(G)$, the ideal $(I(G)^{δ_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Moreover, we show that for each integer $s=1, \ldots δ_{\mathfrak{c}}(I(G))$, the Castelnuovo--Mumford regularity of $(I(G)^s)_{\mathfrak{c}}$ is bounded above by $δ_{\mathfrak{c}}(I(G))+s$.

Theorems & Definitions (24)

• Definition 2.1
• Proposition 3.1
• proof
• Corollary 3.2
• Theorem 4.1
• proof
• Corollary 4.2
• Theorem 4.3
• proof
• Corollary 4.4