论文标题
顶点的可分解性和弱的多功能理想
Vertex decomposability and weakly polymatroidal ideals
论文作者
论文摘要
令$ k $为一个字段，$ r = k [x_1，\ ldots，x_n] $是$ n $ k $ $ n $变量的多项式环。令$δ$成为$ n $顶点的简单综合体，而$ i =i_Δ$为Stanley-Reisner理想。在本文中，我们表明，如果$ i $是矩形理想，那么以下条件是等效的：$（i）$ $δ$是依次是Cohen-Macaulay； $（ii）$ $δ$是可撒的； $（iii）$ $δ$是顶点分解的。另外，如果$ i $是由$ u_1，\ ldots，u_s $生成的，则$ s \ s \ leq 3 $或$ {\ rm supp}（u_i）\ cup {\ rm supp}（rm supp}（u_j）分解。此外，我们证明，如果$ i $是$ 2 $ $ 2 $的单一理想，那么$ i $在且仅当$ i $具有线性的时，且仅当$ i $才能分解时，当时$ i $是弱的。
Let $K$ be a field and $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$. Let $Δ$ be a simplicial complex on $n$ vertices and $I=I_Δ$ be its Stanley-Reisner ideal. In this paper, we show that if $I$ is a matroidal ideal then the following conditions are equivalent: $(i)$ $Δ$ is sequentially Cohen-Macaulay; $(ii)$ $Δ$ is shellable; $(iii)$ $Δ$ is vertex decomposable. Also, if $I$ is a minimally generated by $u_1,\ldots,u_s$ such that $s\leq 3$ or ${\rm supp}(u_i)\cup {\rm supp}(u_j)=\{x_1,\ldots,x_n\}$ for all $i\neq j$, then $Δ$ is vertex decomposable. Furthermore, we prove that if $I$ is a monomial ideal of degree $2$ then $I$ is weakly polymatroidal if and only if $I$ has linear quotients if and only if $I$ is vertex splittable.
