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

Abstract: Let $\mathsf{CI}<em>{\leq c}$ 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 $\mathsf{CI}</em>{\leq c}$ 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 $\mathsf{CI}<em>{\leq c}$-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 $\left{x \in X \mid \operatorname{edim}(\mathcal{O}</em>{X,x}) - \dim(\mathcal{O}<em>{X,x}) \leq n \right}$ is constructible for every $n \in \mathbb{N}$, although the function $$x \mapsto \operatorname{edim}(\mathcal{O}</em>{X,x}) - \dim(\mathcal{O}_{X,x})$$ is not upper semicontinuous in general.

• The paper proves that ath]i_{leq c}]ath] fails the Nagata criterion for every ath]cath]eq 1ath], using square-zero extensions that raise codimension by one while preserving the spectrum.
• The paper identifies failure of lifting across normally flat nilpotent thickenings as the precise obstruction, while showing that localization and parameter-reduction properties alone are insufficient to guarantee the criterion.
• The paper recovers openness by proving that ath]i_{leq c}]ath] is open for every ath]m Reg]-Q0 ring, including all quasi-excellent rings, and that bounded codimension is constructible on quasi-compact excellent schemes.

Overview

This paper studies the property $\mathsf{CI}_{\leq c}$ of being a complete intersection of codimension at most $c$ for Noetherian local rings, where $c \in \mathbb{N}$. These classes interpolate between regularity ($\mathsf{CI}_{\leq 0} = \mathsf{Reg}$) and the full complete intersection property, forming the hierarchy

$$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$

The central question is whether $\mathsf{CI}_{\leq c}$ satisfies the Nagata criterion (NC): if $P(A/\mathfrak{p})$ contains a nonempty open subset of $\operatorname{Spec}(A/\mathfrak{p})$ for every prime $\mathfrak{p}$ of a Noetherian ring $A$, then $c$0 is open in $c$1. Nagata established (NC) for regularity [Nag59], Greco–Marinari for complete intersections and Gorenstein rings [GM78], and Massaza–Valabrega for Cohen–Macaulay rings [MV77]. Since $c$2 is stable under localization and polynomial extensions and behaves well under reduction by suitable regular sequences, one expects it to satisfy (NC). The paper proves this expectation fails in the strongest possible form, while simultaneously recovering the openness conclusion one would have derived from (NC).

Failure of the Nagata criterion

The first main result is categorical: for every $c$3, the property $c$4 does not satisfy (NC). This is a sharp contrast with all four classical singularity classes in the hierarchy above.

The construction proceeds in two steps. For $c$5, the author uses an example of Nishimura [Nis12]: a hypersurface local domain $c$6 containing a purely transcendental extension $c$7 of countably infinite transcendence degree whose regular locus contains no nonempty open subset of $c$8. Setting $c$9 with $c \in \mathbb{N}$0 yields a ring satisfying $c \in \mathbb{N}$1-Q0 — because every quotient $c \in \mathbb{N}$2 is either $c \in \mathbb{N}$3 itself or essentially of finite type over $c \in \mathbb{N}$4, hence excellent — yet the hypersurface locus $c \in \mathbb{N}$5 is not open: every neighborhood of $c \in \mathbb{N}$6 contains points $c \in \mathbb{N}$7 with $c \in \mathbb{N}$8 a hypersurface but $c \in \mathbb{N}$9 of codimension 2.

For arbitrary $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$0, the argument proceeds by induction using square-zero extensions $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$1. The key lemma shows that such extensions leave the spectrum unchanged as a topological space while increasing codimension by exactly one at every point:

$\mathsf{CI}_{\leq 0} = \mathsf{Reg}$2

Thus each counterexample for $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$3 produces one for $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$4. Notably, the resulting counterexample can be taken to be a three-dimensional Noetherian local Nagata domain, so the failure is not attributable to pathological non-Nagata behavior of the base ring alone.

A structural criterion for (NC)

To explain why (NC) fails, the paper introduces three conditions on a property $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$5 of Noetherian local rings:

• (A-1) stability under localization;
• (A-2) existence, at each $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$6, of a system of parameters cutting down to a ring still satisfying $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$7, together with a local lifting statement along that reduction;
• (A-3) lifting across normally flat nilpotent thickenings: if $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$8 satisfies $\mathsf{CI}_{\leq 0} = \mathsf{Reg}$9-Q0, $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$0, each $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$1 is free over $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$2, and $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$3 satisfies $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$4, then $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$5 lifts from $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$6 to $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$7 for $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$8 in a neighborhood of $$\mathsf{Reg} = \mathsf{CI}_{\leq 0} \Longrightarrow \mathsf{HS} = \mathsf{CI}_{\leq 1} \Longrightarrow \cdots \Longrightarrow \mathsf{CI}_{\leq c} \Longrightarrow \cdots \Longrightarrow \mathsf{CI} \Longrightarrow \mathsf{Gor} \Longrightarrow \mathsf{CM}.$$9.

The main structural result is that (A-1), (A-2), and (A-3) jointly imply (NC), while (NC) conversely implies (A-3). Consequently, failure of (A-3) accounts precisely for failure of (NC). The paper verifies that $\mathsf{CI}_{\leq c}$0, $\mathsf{CI}_{\leq c}$1, $\mathsf{CI}_{\leq c}$2, and $\mathsf{CI}_{\leq c}$3 all satisfy (A-3) — the case of $\mathsf{CI}_{\leq c}$4 via André–Quillen homology and generic freeness — whereas $\mathsf{CI}_{\leq c}$5 satisfies (A-1) and (A-2) but fails (A-3) for $\mathsf{CI}_{\leq c}$6. The obstruction is exactly the square-zero thickening in the counterexample above: the property cannot be lifted from the reduced base across the nilpotent direction $\mathsf{CI}_{\leq c}$7, because the thickening strictly increases codimension.

Two technical observations underpin the verification of (A-2) for $\mathsf{CI}_{\leq c}$8. First, for a regular sequence $\mathsf{CI}_{\leq c}$9 with images spanning an $P(A/\mathfrak{p})$0-dimensional subspace of $P(A/\mathfrak{p})$1,

$P(A/\mathfrak{p})$2

so codimension drops by $P(A/\mathfrak{p})$3; the converse implication requires linearly independent images. Second, every Cohen–Macaulay local ring admits a linearly independent system of parameters, which supplies the required sequences. The paper also records a clean numerical characterization of hypersurfaces: $P(A/\mathfrak{p})$4 is a hypersurface if and only if $P(A/\mathfrak{p})$5, proved via the Auslander–Buchsbaum formula.

Recovery of openness

Despite the failure of (NC), the expected openness result survives under a hypothesis considerably weaker than excellence. A Noetherian ring $P(A/\mathfrak{p})$6 satisfies $P(A/\mathfrak{p})$7-Q0 if $P(A/\mathfrak{p})$8 contains a nonempty open subset of $P(A/\mathfrak{p})$9 for every prime $\operatorname{Spec}(A/\mathfrak{p})$0. The second main theorem states:

Main Theorem B. If $\operatorname{Spec}(A/\mathfrak{p})$1 is a Noetherian ring satisfying $\operatorname{Spec}(A/\mathfrak{p})$2-Q0, then $\operatorname{Spec}(A/\mathfrak{p})$3 is open in $\operatorname{Spec}(A/\mathfrak{p})$4 for every $\operatorname{Spec}(A/\mathfrak{p})$5.

In particular, the locus is open for every quasi-excellent ring. The proof circumvents (A-3) by exploiting the fixed hypothesis $\operatorname{Spec}(A/\mathfrak{p})$6-Q0 directly: near a point $\operatorname{Spec}(A/\mathfrak{p})$7 with $\operatorname{Spec}(A/\mathfrak{p})$8 of codimension $\operatorname{Spec}(A/\mathfrak{p})$9, the normally flat structure forces $\mathfrak{p}$0; lifting a regular system of parameters from the regular quotient $\mathfrak{p}$1 through the nilpotent ideal (via Matsumura's exercise on regular sequences across normally flat thickenings) reduces to a zero-dimensional complete intersection $\mathfrak{p}$2 with $\mathfrak{p}$3, which deforms back to $\mathfrak{p}$4 being $\mathfrak{p}$5. The essential input is that both the regular locus of the quotient and the full $\mathfrak{p}$6-locus are open — the latter following from (NC) for $\mathfrak{p}$7 — so the bounded-codimension condition is the only part requiring new work.

An important consequence concerns the quotient condition (QC), the converse-type statement that $\mathfrak{p}$8 implies $\mathfrak{p}$9-Q0. For properties satisfying both (NC) and (QC), surjections from $A$0-rings preserve openness of the $A$1-locus. Since $A$2 fails (NC), this route is unavailable, and indeed the paper leaves open whether $A$3 satisfies (QC) for $A$4. It also recalls Nagata's example showing that even $A$5 fails (QC) in general, and notes that whether regularity satisfies (QC) in characteristic zero remains unresolved.

Constructibility of the codimension function

The final section studies the codimension function $A$6 on a locally Noetherian scheme $A$7. On the complete intersection locus, the condition $A$8 detects $A$9, so the topological behavior of $c$00 governs the geometry of these loci.

Upper semicontinuity fails in general, even for affine excellent schemes: for $c$01, the set $c$02 is not stable under generalization, hence not open. Nevertheless, the third main theorem establishes:

Main Theorem C. On a quasi-compact excellent scheme $c$03, the set $c$04 is constructible for every $c$05.

The proof combines two upper semicontinuous functions built from André–Quillen homology: the complete intersection defect $c$06 [ATR89] and the deviation $c$07 [Rag80]. Since $c$08 pointwise, and since quasi-compactness forces both $c$09 and $c$10 to take only finitely many values, $c$11 decomposes into a finite union of differences of open sets, hence is constructible.

Combining constructibility with the fact that $c$12 is stable under generalization, and applying the standard lemma that a constructible subset of a Noetherian sober space stable under generalization is open, yields a second, geometrically flavored proof that $c$13 is open for every excellent ring $c$14. This proof is independent of the Nagata criterion machinery and explains the openness phenomenon as a consequence of the constructible variation of embedding dimension minus dimension.

Limitations and open questions

Several questions remain open. Whether $c$15 satisfies the quotient condition (QC) for any $c$16 is unresolved; an affirmative answer would restore the transfer principle for surjective maps from $c$17-rings. The older question of Valabrega — whether regularity satisfies (QC) in characteristic zero — also remains open. Finally, the paper asks whether the newer singularity classes intervening between Gorenstein and Cohen–Macaulay, namely nearly Gorenstein, canonical trace radical, and Cohen–Macaulay with canonical module, satisfy (NC) or (QC); answering this requires controlling the behavior of canonical modules and their trace ideals along the relevant constructions, which the author defers to future work. The counterexamples depend essentially on Nishimura's examples over countable fields with infinite purely transcendental extensions, and the openness theorem requires the $c$18-Q0 hypothesis; the paper does not claim openness of $c$19-loci for arbitrary Noetherian rings.

Conclusion

This paper establishes that the Nagata criterion, valid for regularity, complete intersections, Gorenstein, and Cohen–Macaulay properties, fails for complete intersections of bounded positive codimension, and isolates the failure in the lifting condition (A-3) across normally flat nilpotent thickenings. The failure is nevertheless benign in practice: the $c$20-locus is open for every $c$21-Q0 ring, hence for every quasi-excellent ring, and the codimension function is constructible on quasi-compact excellent schemes despite not being upper semicontinuous. The results clarify precisely which structural features of a singularity class govern the Nagata criterion and demonstrate that openness of singularity loci can persist well beyond the reach of that criterion.