AI 中文总结
本文在Noether商环上证明幂零微分为零元素的余理想非包含定理，推广至孤立及非孤立临界轨迹，并构造反例否证积分闭包加强。
AI 中文摘要
设$k$为特征零的域，$R$为交换$k$-代数，$I$为$R$的真理想，并假设$A=R/I$为Noether环。我们证明：若$A$中$a=[f]$是幂零元且$d_{A/k}a=0$，则$I:f$不包含于$(I,f)$。若$R$为极大理想$\mathfrak n$的局部环，则$I:f$不包含于$(I,f)+\mathfrak n(I:f)$。作为应用，我们证明了任意维数下的孤立超曲面情形，并将结论推广到光滑簇上的非孤立临界轨迹。最后，我们构造一个五变量孤立奇点满足$J_f:f\subseteq\overline{J_f}$，从而否证了积分闭包加强版本。
英文摘要
Let $k$ be a field of characteristic zero, let $R$ be a commutative $k$-algebra, let $I$ be a proper ideal of $R$, and assume that $A=R/I$ is Noetherian. We prove that if $a=[f]$ in $A$ is nilpotent and $d_{A/k}a=0$, then $I:f$ is not contained in $(I,f)$. If $R$ is local with maximal ideal $\mathfrak n$, then $I:f$ is not contained in $(I,f)+\mathfrak n(I:f)$. As applications, we prove the isolated hypersurface case in arbitrary dimension and extend the conclusion to non-isolated critical loci on smooth varieties. Finally, we exhibit a five-variable isolated singularity satisfying $J_f:f\subseteq\overline{J_f}$, thereby disproving the integral-closure strengthening.
Comments11 pages, no figures
