AI 中文总结
本文证明余维至多为$c$的完全交性质不满足Nagata准则，构造反例并给出NC的刻画条件，恢复其轨迹开性，还证明相关子集在拟紧优概形上可构造。
AI 中文摘要
设$\boldsymbol{\textsf{CI}}_{\boldsymbol{\textsf{≤}} \boldsymbol{\textsf{c}}}$表示余维至多为$c$的完全交性质。尽管正则、完全交、Gorenstein及Cohen–Macaulay性质均满足Nagata准则（NC），但本文证明对任意$c \boldsymbol{\textsf{≥}} 1$，$\boldsymbol{\textsf{CI}}_{\boldsymbol{\textsf{≤}} \boldsymbol{\textsf{c}}}$不满足NC。我们先构造超曲面的反例，再通过平方零扩张得到任意$c$的反例。还为Noether局部环的性质引入三个条件，证明在关于局部化和合适正则序列的约化稳定时，NC可由法平幂零加厚上的提升性质刻画。不过，我们恢复了预期的开性结果：对每个满足Reg-Q0的Noether环，$\boldsymbol{\textsf{CI}}_{\boldsymbol{\textsf{≤}} \boldsymbol{\textsf{c}}}$的轨迹是开集，因此对每个拟优环也成立。最后，对每个拟紧优概形$X$，证明对任意$n \boldsymbol{\textsf{∈}} \boldsymbol{\textsf{N}}$，子集$\boldsymbol{\textsf{\big\bracevert}} \boldsymbol{\textsf{x}} \boldsymbol{\textsf{∈}} \boldsymbol{\textsf{X}} \boldsymbol{\textsf{\big\bracevert}} \boldsymbol{\textsf{edim}}(\boldsymbol{\textsf{O}}_{\boldsymbol{\textsf{X,x}}}) \boldsymbol{\textsf{−}} \boldsymbol{\textsf{dim}}(\boldsymbol{\textsf{O}}_{\boldsymbol{\textsf{X,x}}}) \boldsymbol{\textsf{≤}} \boldsymbol{\textsf{n}} \boldsymbol{\textsf{\big\bracevert}}$是可构造的，尽管函数$\boldsymbol{\textsf{x}} \boldsymbol{\textsf{↦}} \boldsymbol{\textsf{edim}}(\boldsymbol{\textsf{O}}_{\boldsymbol{\textsf{X,x}}}) \boldsymbol{\textsf{−}} \boldsymbol{\textsf{dim}}(\boldsymbol{\textsf{O}}_{\boldsymbol{\textsf{X,x}}})$一般不是上半连续的。
英文摘要
Let $\mathsf{CI}_{\leq c}$ denote the property of being a complete intersection of codimension at most $c$. Although the regular, complete intersection, Gorenstein, and Cohen--Macaulay properties satisfy the Nagata criterion (NC), we prove that $\mathsf{CI}_{\leq c}$ does not satisfy (NC) for any $c \geq 1$. We first construct a counterexample for hypersurfaces and then obtain counterexamples for arbitrary $c$ using square-zero extensions. We also introduce three conditions for a property of Noetherian local rings and show that, under stability with respect to localization and reduction by suitable regular sequences, (NC) is characterized by a lifting property across normally flat nilpotent thickenings. Nevertheless, we recover the expected openness result: the $\mathsf{CI}_{\leq c}$-locus is open for every Noetherian ring satisfying $\mathsf{Reg}$-Q0, and hence for every quasi-excellent ring. Finally, for every quasi-compact excellent scheme $X$, we prove that the subset $\left\{x \in X \mid \operatorname{edim}(\mathcal{O}_{X,x}) - \dim(\mathcal{O}_{X,x}) \leq n \right\}$ is constructible for every $n \in \mathbb{N}$, although the function $x \mapsto \operatorname{edim}(\mathcal{O}_{X,x}) - \dim(\mathcal{O}_{X,x})$ is not upper semicontinuous in general.
Comments22 pages, comments are very welcome
