Local constructibility of the openness and universal-openness loci of a morphism (EGA IV_3, Remarques 14.1.3(iii) and 14.3.9(ii))

Grothendieck's Éléments de géométrie algébrique IV, partie 3 (1966) poses two questions about the topology of a morphism of schemes $f : X \to Y$ that have never been answered: is the set of points of $X$ at which $f$ is open, and the set of points at which $f$ is universally open (at a point $x$: every base change of $f$ is open at every point lying over $x$), a locally constructible subset of $X$? Grothendieck's own examples show that neither locus is open or closed in general, while the constructibility results of EGA IV show that almost every other pointwise property of morphisms locally of finite presentation (flatness, fiber dimension, geometric properties of fibers) has constructible locus, making these two loci a conspicuous gap in the theory. Later work has built a substantial pointwise and global theory around the questions — Chevalley-type criteria over geometrically unibranch bases, an effective fibred power criterion over normal bases (Adamus–Bierstone–Milman), Kollár's pointwise theory of universal openness, and the Stacks Project's lemmas — but none of it addresses the loci. The problem asks whether the openness locus and the universal-openness locus are locally constructible under the natural hypotheses (morphisms locally of finite type between locally noetherian schemes, or locally of finite presentation).

Background

In algebraic geometry one studies a morphism of schemes $f : X \to Y$ (schemes are the basic spaces of the field, carrying a sheaf of rings; a morphism is a continuous map together with a compatible ring map) not only through algebraic invariants but through the continuity behavior of the underlying map of topological spaces. A map is open at a point $x$ if it sends every neighborhood of $x$ to a neighborhood of the image point, and open if it is open at every point. Following EGA IV_3 (Grothendieck, 1966), $f$ is universally open at $x$ if every base change $X \times_Y Y' \to Y'$ (the pullback along an arbitrary morphism $Y' \to Y$) is open at every point over $x$; openness does not imply universal openness, because a base change can create new irreducible components in the fibers. A subset of a scheme is constructible if it is a finite Boolean combination of open subsets, and locally constructible if this holds locally on the ambient scheme; these are the tame subsets of the subject: Chevalley's theorem states that images of constructible sets under morphisms of finite presentation are constructible, and the constructibility results of EGA IV state that for $f$ locally of finite presentation the locus where the fiber satisfies essentially any reasonable algebraic condition — flatness (exactness of tensoring), smoothness, a bound on fiber dimension, geometric irreducibility of fibers — is constructible, and often open.

Two loci conspicuously resist this program. Let $U(f)$ be the set of points $x \in X$ at which $f$ is open, and $U^u(f)$ the set of points at which $f$ is universally open. EGA IV_3, Remarque 14.1.3(iii), states: "Nous ignorons si, lorsque $X$ et $Y$ sont localement noethériens et $f : X \to Y$ un morphisme localement de type fini, l'ensemble des points de $X$ où $f$ est ouvert est ou non localement constructible" (we do not know whether the openness locus is locally constructible; "locally noetherian" means the closed subsets satisfy the descending chain condition, and "locally of finite type" is the standard finiteness hypothesis for morphisms). Remarque 14.3.9(ii) adds: "Il serait intéressant de savoir si l'ensemble des points où un morphisme est universellement ouvert est localement constructible" (it would be interesting to know whether the universal-openness locus is locally constructible). Both questions come with examples showing that no simpler answer can hold: in EGA 14.1.3(i) — the projection of the plane curve $\mathrm{Spec}\,K[S,T]/(ST)$, the union of the two axes, onto one axis — the openness locus is an entire irreducible component, a closed and non-open subset, and EGA 14.1.3(ii) shows that the openness locus of a dominant morphism of finite type between integral schemes need not be closed; in EGA 14.3.9(i) — the normalization of the curve singularity $\mathrm{Spec}\,R[[U,V]]/(U^2+V^2)$, which is unibranch but not geometrically unibranch (EGA offers the normalization of an ordinary double point of a curve as an analogous example) — the universal-openness locus is the complement of a single closed point, the morphism being a local isomorphism at every other point, an open and non-closed subset. So each locus is in general neither open nor closed, and local constructibility is the natural remaining finiteness property.

Substantial theory describes these loci pointwise without settling their global shape. The Chevalley criterion (EGA IV_3, Corollaire 14.4.4) says that if $f$ is locally of finite type, equidimensional at $x$ (the fiber dimension is constant in the strong sense of EGA 13.3.2) and the base point $f(x)$ is geometrically unibranch (every étale extension — the algebraic analogue of a local homeomorphism — of the local ring $\mathcal{O}_{Y,f(x)}$ has irreducible spectrum; for instance the local ring is normal with algebraically closed residue field), then $f$ is universally open at $x$; in particular, over a geometrically unibranch base the equidimensional locus, which is open, is contained in $U^u(f)$. EGA 14.4.8 reduces, for a point maximal in its fiber, the pointwise question over a noetherian base to the normalized base, and EGA 14.3.7 gives a valuative criterion: for $f$ locally of finite type over a locally noetherian base, universal openness along a fiber is equivalent to every irreducible component of every pullback to the spectrum of a discrete valuation ring (a local ring whose nonzero ideals are totally ordered by inclusion, the one-dimensional analogue of a local ring of a curve) dominating the base. Later work sharpened the pointwise picture: Adamus, Bierstone and Milman proved that for $f$ locally of finite type over a normal base $Y$ of dimension $n$ that is of finite type over a field, $f$ is open if and only if the $n$-fold fibred power $X \times_Y \cdots \times_Y X$ has no vertical irreducible component (one mapped into a proper closed subset of $Y$) — an effective and computable, but global, criterion over normal bases. Kollár (preprint 2019, published 2021) developed the pointwise theory (an étale-local criterion, valuative criteria, an example separating openness from universal openness at a single point over a normal base, and a characterization of global universal openness via the fibred square), and the Stacks Project records basic criteria and a limit-descent lemma for locally quasi-finite morphisms in its section on universally open morphisms. None of these results addresses the constructibility of the loci.

Problem Statement

Let $f : X \to Y$ be a morphism of schemes. Define

$$U(f) = \{ x \in X : f \text{ is open at } x \}, \qquad U^u(f) = \{ x \in X : f \text{ is universally open at } x \},$$

where "open at $x$" means that $f$ maps every open neighborhood of $x$ to a neighborhood of $f(x)$, and "universally open at $x$" (EGA IV_3, Définition 14.3.3) means that for every morphism $Y' \to Y$ the base change $X \times_Y Y' \to Y'$ is open at every point lying over $x$. The problem is the pair of questions posed in EGA IV_3:

(1) (Remarque 14.3.9(ii)) If $f$ is locally of finite type or locally of finite presentation, is $U^u(f)$ a locally constructible subset of $X$?

(2) (Remarque 14.1.3(iii)) If $X$ and $Y$ are locally noetherian and $f$ is locally of finite type, is $U(f)$ a locally constructible subset of $X$?

The known constraints are: both loci can fail to be open and fail to be closed (EGA 14.1.3(i)–(ii) and 14.3.9(i)–(ii)); over a geometrically unibranch base the open equidimensional locus is contained in $U^u(f)$ (EGA 14.4.4(ii)); and $U(f)$ and $U^u(f)$ can differ at a single point even over normal bases (Kollár, arXiv:1906.11816, Example 25), so neither question reduces to the other.

References

1. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. Troisième partie, Publications Mathématiques de l'IHÉS 28 (1966), 5–255, DOI 10.1007/BF02684343, https://doi.org/10.1007/BF02684343
2. The Stacks Project Authors, Stacks Project (sections and tags: More on Morphisms, Section 37.74 "Universally open morphisms", tags 0F31–0F35; More on Flatness, Section 38.25, tag 0ASZ, Theorem 38.26.1, tag 05UA, Lemma 38.13.11, tag 05IQ; Algebra, Lemma 10.128.8, tag 00R7; Morphisms, Lemma 29.24.2, tag 01U1; each tag resolves at https://stacks.math.columbia.edu/tag/), https://stacks.math.columbia.edu
3. J. Kollár, Fundamental groups and path lifting for algebraic varieties, in: Singularities and Their Interaction with Geometry and Low Dimensional Topology (Trends in Mathematics), Springer, 2021, pp. 97–116, DOI 10.1007/978-3-030-61958-9_6, https://doi.org/10.1007/978-3-030-61958-9_6; preprint arXiv:1906.11816 (2019; theorem and example numbers follow the preprint), https://arxiv.org/abs/1906.11816
4. J. Adamus, E. Bierstone, P. D. Milman, Geometric Auslander criterion for openness of an algebraic morphism, Bulletin of the London Mathematical Society 45 (2013), 1060–1064, DOI 10.1112/blms/bdt028, arXiv:1006.1872, https://doi.org/10.1112/blms/bdt028
5. P. Poczobut, An algebraic variant of the Fischer-Grauert Theorem, preprint (2021), arXiv:2108.10041, https://arxiv.org/abs/2108.10041
6. M. Raynaud, L. Gruson, Critères de platitude et de projectivité. Techniques de «platification» d'un module, Inventiones mathematicae 13 (1971), 1–89, DOI 10.1007/BF01390094, https://doi.org/10.1007/BF01390094
7. M. Kaddar, Universally Equidimensional Morphisms and Weakly or Strongly rational Singularities, preprint (2025), arXiv:2507.03345, https://arxiv.org/abs/2507.03345