# Why do we have to deal with constructible sets?

I'm a beginner in algebraic geometry. Recently, I learned about constructible sets and Chevalley's theorem. In a Noetherian space, constructible sets are some kind of finite union of open and closed sets, and Chevalley's theorem states that if $f : X \to Y$ is a morphism of finite type between Noetherian schemes, then the image of any constructible set $C \subset X$ is also constructible.

Is there any good application of the theorem? I can't understand why we have to consider constructible sets. It resembles Borel sets in measure theory (but the situation is completely different since topology is far from Hausdorff), so maybe we can think of constructible sets as some sort of topologically simple sets. But such intuition doesn't give any result.

• For an arbitrary scheme $X$, a subset $U \subset |X|$ is open iff it is constructible and stable under generization (see exercise II.3.18 c), in Hartshorne). Using this fact and Chevalley's theorem, you can show that any flat morphism $f : X \to Y$ (of finite type, and between noetherian schemes) is an open map — giving a nice intuition for flatness (see exercise III.9.1 in Hartshorne). – Watson May 31 '18 at 6:44
• @Watson Thank you! That's gives a good intuition. – Seewoo Lee May 31 '18 at 7:15

This is more of an extended comment, since I hope we get some more comprehensive answers.

First, you mean a finite union of locally closed sets, i.e., finite union and intersection of open and closed sets.

The first and maybe most important point is probably Chevalley's theorem, as you point out. IMO the remarkable property is that closed and open sets map to constructible sets under the hypothesis, which basically says that varieties (and their subvarieties) can at worst map to constructible sets, instead of just some arbitrary set. I believe this is reason enough to care about this kind of sets.

The other very cool observation that comes to mind is that the Nullstellensatz is in fact a corollary of Chevalley's theorem! Instead of reproducing the argument here here's the link to one version of Vakil's notes where he gives a one paragraph proof of it, see the proof 7.4.3 and the exercise 7.4.B: http://math.stanford.edu/~vakil/216blog/FOAGdec3014public.pdf

In fact, shortly after this you can find one or two more applications of the theorem.

PS. I know there's also the concept of constructible sheaf, of which I know nothing about. Perhaps someone can shed some light on what this is about.

• Very helpful answer, thanks! Also, definition of constructible looks similar to the definition of constructible sets in some sense, according to wiki. It seems that it is important in the theory of \'etale cohomology.. – Seewoo Lee May 31 '18 at 7:19

The first time I discovered the use of constructible sets was through linear algebraic groups. A linear algebraic group $G$ is an affine variety such that the composition map $m : G\times G \rightarrow G$ and inversion map $i : G\rightarrow G$ are regular maps.

Suppose we have a map of linear algebraic groups $\phi : G \rightarrow H$ (i.e. a regular map which is a group homomorphism). Since $\phi$ is continuous, the kernel will be a closed subgroup of $G$. What about its image? By Chevalley's theorem, $\phi(G)$ will be constructible and obviously it is a subgroup of $H$. Now it's not too hard to observe that a constructible subgroup of a linear algebraic group is closed (see Humphreys book below). So the image of $\phi$ is in fact a closed subgroup of $H$, i.e. a linear algebraic group!

Reference: Humphreys' book on linear algebraic groups, section 7.4.

• This is also a good application. Thanks! – Seewoo Lee Jun 5 '18 at 1:51