Complete Intersections of Bounded Codimension: Failure of the Nagata Criterion and Recovery of Openness
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.