AI 中文总结
本文定义了由无平方因子单项式理想生成的分量线性单项式理想I^[a]，并研究I与I^[a]的分量线性性质。
AI 中文摘要
设S=K[x₁,…,xₙ]为域K上n个变量的多项式环，满足deg x₁=…=deg xₙ=1，且a=(a₁,…,aₙ)∈ℤ>0ⁿ。对S中无平方因子单项式u=xᵢ₁…xᵢd（1≤i₁<…<i_d≤n），定义u^[a]:=xᵢ₁^aᵢ₁…xᵢd^aᵢd。设I为S的无平方因子单项式理想，G(I)为其唯一极小单项式生成元集，引入单项式理想I^[a]，其极小生成元集G(I^[a])={u^[a]:u∈G(I)}。本文研究无平方因子单项式理想I与I^[a]的分量线性性质。
英文摘要
Let $S=K[x_1,\ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with ${\rm deg} x_1=\cdots ={\rm deg} x_n = 1$ and ${\bf a} = (a_1,\ldots,a_n) \in {\mathbb Z}_{>0}^n$. Given a squarefree monomial $u=x_{i_1} \cdots x_{i_d}$ of $S$ with $1 \leq i_1 < \cdots < i_d \leq n$, we set $u^{[{\bf a}]}:=x_{i_1}^{a_{i_1}}\cdots x_{i_d}^{a_{i_d}}$. Let $I$ be a squarefree monomial ideal of $S$ and $G(I)$ its unique minimal set of monomial generators. We introduce the monomial ideal $I^{[{\bf a}]}$ with $G(I^{[{\bf a}]})=\{u^{[{\bf a}]} : u \in G(I)\}$. In the present paper, componentwise linearity of a squarefree monomial ideal $I$ and that of $I^{[{\bf a}]}$ is studied.
