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

Let ${\mathsf{C}\mathsf{I}}_{\le c}$ 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 ${\mathsf{C}\mathsf{I}}_{\le c}$ does not satisfy (NC) for any $c\ge 1$. 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 ${\mathsf{C}\mathsf{I}}_{\le c}$-locus is open for every Noetherian ring satisfying $\mathsf{R}\mathsf{e}\mathsf{g}$-Q0, and hence for every quasi-excellent ring. Finally, for every quasi-compact excellent scheme $X$, we prove that the subset $\{x\in X\mid \mathrm{edim}({\mathcal{O}}_{X,x})-\dim ({\mathcal{O}}_{X,x})\le n\}$ is constructible for every $n\in \mathbb{N}$, although the function $x\mapsto \mathrm{edim}({\mathcal{O}}_{X,x})-\dim ({\mathcal{O}}_{X,x})$ is not upper semicontinuous in general.