Componentwise linear monomial ideals

Abstract

Let $S=K[x_1,\dots ,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with $\mathrm{deg}{x}_1=\cdots =\mathrm{deg}{x}_n=1$ and $\mathbf{a}=(a_1,\dots ,a_n)\in {\mathbb{Z}}_{>0}^n$. Given a squarefree monomial $u=x_{i_1}\cdots x_{i_d}$ of $S$ with $1\le i_1<\cdots <i_d\le n$, we set $u^{[\mathbf{a}]}:=x_{i_1}^{a_{i_1}}\cdots x_{i_d}^{a_{i_d}}$. Let $I$ be a squarefree monomial ideal of $S$ and $G(I)$ its unique minimal set of monomial generators. We introduce the monomial ideal $I^{[\mathbf{a}]}$ with $G(I^{[\mathbf{a}]})=\{u^{[\mathbf{a}]}:u\in G(I)\}$. In the present paper, componentwise linearity of a squarefree monomial ideal $I$ and that of $I^{[\mathbf{a}]}$ is studied.