In EGA IV Grothendieck introduced notion of constructible topology. Is it only interesting gadget or can it be use for some practical purposes in algebraic geometry?

32$\begingroup$ There is nothing in EGA that has been introduced just because it's cute. $\endgroup$– AngeloJan 25, 2013 at 11:47

1$\begingroup$ I don't know if this question will survive, but I'll admit that, as a nonexpert who's never really had occasion to work with constructible sets, this is something I've been idly curious about on a handful of occasions. I'd like to hear the short version of why they're useful and maybe see a quick example or two. $\endgroup$– RamseyJan 25, 2013 at 15:01

2$\begingroup$ Are you wondering if it is introduced in EGA for more than putting pro and indconstructible sets into a systematic framework (which is does)? There are lots of important results in IV$_3$ concerning openness and constructibility of various loci, as well as pulling down results from "limit objects", and those proofs use pro and indconstructibility in very creative ways (e.g., via 1.9.11 and 1.9.12; also see 1.10). Any way to better understand fundamental concepts is always a good thing. Also see 1.9.16, now obsolete. More recently, these concepts are relevant to adic spaces...and so on. $\endgroup$– user30180Jan 25, 2013 at 15:02

5$\begingroup$ Very briefly, to prove openness results on the base one first proves constructibility results on the base (often by deducing it from constructibility on the source and applying Chevalley's theorem on images of constructible sets) and then uses specialization criteria for constructible sets to be open. But how to prove the constructibility? Sometimes it can be done "by hand", but Grothendieck never argues by hand when there is a more conceptual viewpoint to unify many results by a common technique. Hence the usefulness of 1.9.11 and 1.9.12. Read IV$_3$ sections 9 and 12. $\endgroup$– user30180Jan 25, 2013 at 15:06

1$\begingroup$ Together with its constructible topology, the spectrum of $k[X_1,...,X_n]$ is homeomorphic to the space of $n$types with parameters in $k$ (in the sense of model theory) for the firstorder theory ACF. (If $a\in K^n$, where $K$ is an overfield of~$a$, send its type $\mathrm{tp}(a)$ to the prime ideal of polynomials vanishing at $a$). Two apparently (but only apparently) distinct worlds in which the same object is defined... $\endgroup$– ACLMar 25, 2013 at 18:11
3 Answers
Julien Sebag tells me that the constructible topology is useful for the study of the Grothendieck ring of varieties. More precisely, it is relevant to the following question: "if $k$ is a field and $X$ is a $k$variety with a birational endomorphism $X\to X$inducing an isomorphism between open subsets $U$ and $V$, are $X\setminus U$ and $X\setminus V$ piecewise isomorphic? You may read about this in Liu and Sebag  The Grothendieck ring of varieties and piecewise isomorphisms.
Another place where the constructible topology is essential is in motivic integration, where constructible sets play the role of the measurable sets of usual integration theory.
The constructible topology is used in Bhatt and Scholze  The proétale topology for schemes, henceforth [BS], in an essential way (as far as I understand it).
The proétale topology is a generalization of the étale topology where $\overline{\mathbb{Q}}_\ell$étale cohomology is actually the sheaf cohomology (in the naive sense) of the constant sheaf $\overline{\mathbb{Q}}_l$ (rather than $\varprojlim_n H^i(X_{\text{et}}, \mathbb{Z} / \ell^n \mathbb{Z}) \otimes_{\mathbb{Z}_\ell} \mathbb{Q}_\ell$). The point is to enlarge the étale site to include "limits" of coverings, (so for example $\mathrm{Spec}(k^{\text{sep}})$ is always in the proétale site of $Spec(k)$, and $\mathrm{Spec}(k^{\text{sep}}) \to \mathrm{Spec}(k)$ is a proétale covering).
The Zariski version of this is a little more accessible. In this prosetup, there is an "initial" "Zariski" covering $X^Z \to X$, which is the "intersection" of all Zariski coverings. The closed points $(X^Z)^c$ of $X^Z$ are in bijection with the points of $X$, and $(X^Z)^c$ equipped with the subspace topology (via the canonical inclusion $(X^Z)^c \subseteq X^Z$) is homeomorphic to $X$ equipped with the constructible topology [BS, Lemma 2.1.10].
The fact that the constructible topology is quasicompact is used to produce a "contractible" covering of $(X^Z)^c$ in the site of profinite topological spaces [BS, Def.2.4.4, Thm.2.4.5, Exa.2.4.6, Exa.4.1.10], and this is one of the main steps in the construction of a "contractible" proétale covering of an arbitrary scheme [BS, Lemma 2.4.9, Prop.4.2.8].
The existence of "contractible" coverings is reason the proétale topology works [BS, Def.3.2.1, Prop.3.2.3, Def.3.1.1, Lem.3.3.2, Prop.3.3.3, Prop.5.5.4, Prop.5.6.2]. The fact that the constructible topology is quasicompact is one of the main ingredients for this existence of "contractible" coverings.
In Hochster  Prime Ideal Structure in Commutative Rings (MSN), the author uses it to characterize spectral spaces. This in turn is used in Huber's work on Adic spaces, cf. Huber  Étale cohomology of Rigid Analytic Varieties and Adic Spaces (MSN) and Scholze  Perfectoid spaces (MSN).

2$\begingroup$ If I remember correctly, Hochster wrote his thesis on spectral spaces (with Shimura as "advisor"...) solely out of idle curiosity about the nature of the lattice of prime ideals in a general ring (e.g., the fact that the underlying topological space of the scheme $\mathbf{P}^7_{\mathbf{Q}}$ is also the underlying topological space of the spectrum of a ring is not something for which he had a use in mind). He was very surprised to hear from time to time afterwards that others found these results to be useful, since he never used them in his later work. $\endgroup$ Jan 25, 2013 at 15:41

$\begingroup$ @pz Can you explain where spectral spaces are used in the theory of Adic spaces? I mean, I know that Huber proved that adic spaces are spectral (maybe with some conditions), but I always thought that he did this just to give an idea of how to visualize the topology on the adic spaces (in contrast with Berkovich spaces that have a "real" topology) and not to prove anything about adic spaces. $\endgroup$– RickyJan 25, 2013 at 17:36

2$\begingroup$ @Ricky I think you are right. Huber uses spectral spaces in some proofs, but I think he doesn't do anything with them that could not be done without them. They are afaik not crucial to the theory of adic spaces. $\endgroup$ Jan 25, 2013 at 18:21