In classical algebraic geometry, one defines a subset of a variety over an algebraically closed field to be constructible if it is a union of locally closed subsets (in the Zariski topology). One of the basic results that one proves, which can be called “elimination theory” and is due to Chevalley, states that constructible sets are preserved under taking images: if is a regular map and is constructible, then so is . In general, this is the best one can say: even very nice subsets of (e.g. itself) need not have open or closed (or even locally closed) images.
In the theory of schemes, one can formulate a similar result. A morphism of finite type between noetherian schemes sends constructible sets into constructibles. One proves this result by making a sequence of reductions to considering the case of two integral affine schemes, and then using a general fact from commutative algebra. It turns out, however, that there is a more general form of the Chevalley theorem:
Theorem 1 Let be a finitely presented morphism of schemes. Then if is locally constructible, so is .
I will explain today how one deduces this more general fact from the specific case of noetherian schemes. This will highlight a useful fact: oftentimes, general facts in algebraic geometry can be reduced to the noetherian case since, for instance, every ring is an inductive limit of noetherian rings. This can be developed systematically, as is done in EGA IV-8, but I shall not do so here.
N.B. As a result, this post is written entirely for those whom Ravi Vakil would call “non-noetherian people.” I will simply assume as known the noetherian results (which can be found easily, e.g. in Hartshorne or EGA I) and explain how they can be generalized. Nonetheless, even noetherian readers have a very good reason to care. In fact, it is through such a “finite presentation” argument that Grothendieck proves the general quasi-finite form of Zariski’s main theorem; the finite presentation trick is a very ingenious strategy, about which I hope to say more soon, that can reduce many results not only to the noetherian case, but also to the local case.
Nonetheless, there are several caveats. The first is that the notion of constructibility needs to be modified for general (non-noetherian) schemes. Also, we have to explain what “locally constructible” means!
So, first, let us give the modified definition of constructibility.
Definition 2 If is a topological space and an open subset, then is called retrocompact if the inclusion is a quasi-compact morphism. That is, if is a quasicompact open set, then is quasicompact as well.
Classically, the algebra of constructible sets is defined to be that generated by the open sets. In general, we just work with retrocompact open sets.
Definition 3 The algebra of constructible sets is that generated by the retrocompact open sets. So a set is constructible if and only if it is expressible as
for the retrocompact.
Naturally, we can define what locally constructible means: a subset is locally constructible if there is an open covering by open sets such that is constructible in . We want to show that constructible sets behave well under morphisms of schemes. To do this, let us first show that always, they are closed under taking inverse images.
Proposition 4 If is a morphism of schemes, and is constructible, so is .
It is sufficient to show that retrocompact open sets pull back to retrocompact ones, because constructible sets are obtained from them by taking finite unions and differences. So let be retrocompact. This is equivalent to saying that the morphism of schemes is quasicompact. But the morphism is the pull-back of via , so it is also quasicompact. Thus is retrocompact in , and we have proved the claim.
The result now implies that the same is true for local constructibility:
Corollary 5 If is a morphism of schemes, and is locally constructible, so is .
We are interested in when the image of a constructible set is constructible. To start with, we will handle the case of a quasi-compact open immersion.
Proposition 6 Let be a quasi-compact open immersion. Then the image of a constructible is constructible in .
This will follow at once if we show that if is retrocompact open, then is constructible in . Note that any constructible set is a finite union of differences of sets of this form, so it is sufficient to handle the case of a retrocompact open. If is retrocompact, then is quasicompact. But by assumption is quasicompact, so the composite is quasicompact as well. This means that is constructible in .
Quasicompact and quasiseparated schemes are nice. In there, every quasicompact open set is retrocompact, and conversely. This is the definition of quasiseparatedness: that the intersection of two quasicompacts be quasicompact.
Proposition 7 If is a quasicompact, quasiseparated scheme and is locally constructible, it is globally constructible.
By definition, there is a finite cover of by small neighborhoods , which we may take affine (by the proposition), such that is constructible for each .
2. We can get any constructible set
So now we are interested in proving the Chevalley theorem. First, however, to make things clear, we prove a type of converse.
Proposition 8 If is a quasicompact and quasiseparated scheme and a constructible subset, there is an affine scheme and a morphism , quasicompact and locally of finite presentation, whose image is .
To do this, we start by covering by open affine subsets . Since is quasicompact, we only need finitely many of them. The morphism
is quasi-compact because is quasiseparated. As a result, by reducing to the case of the inverse image of in , we may suppose that itself is affine.
is constructible, so we can write as a finite union for quasicompact. By taking finite coproducts, we see that it is enough to handle the case of for retrocompact in (hence quasicompact, being affine). This means that each of is a finite union of basic open sets for each . By taking finite unions, we can reduce to the case
The claim is that arises as the image of an affine scheme, of finite presentation over . Suppose .
Then we can take the natural map whose image is ; moreover, it is immediate that this is of finite presentation, from the explicit form.
3. The main theorem
As I mentioned earlier, this is the generalization of the usual Chevalley theorem that I wanted to prove today:
Theorem 9 Let be a quasicompact morphism, locally of finite presentation. Then if is locally constructible, so is .
Interestingly, it turns out that Grothendieck does not (in EGA IV-1) state the theorem in maximal generality. Grothendieck assumes that is of “finite presentation,” which means that he assumes in addition that is quasiseparated. The more general form of the theorem is in the Stacks project. So let us prove this fact.
3.1. Step 1: Reduction to affine
The first step is to make some reductions. Since we are trying to prove an assertion which is local on , we may assume that is itself affine, and consequently that is quasi-compact. It follows that is a fintie union , where each is affine (and thus quasicompact and quasiseparated).
In particular, is not only locally constructible, but constructible. If we show (assuming affine, as usual) that is constructible for each , then we will be done. The upshot is that we can reduce to the case of affine and affine. In particular, we need to prove:
Lemma 10 Let be a morphism of affine schemes which is of finite presentation. If is constructible, so is .
But in fact we can make a further reduction. We know that there is an affine scheme and a morphism of finite presentation such that is the image of , by the above lemmata. So we are reduced to something even more concrete:
Lemma 11 Let be a morphism of affine schemes which is of finite presentation. Then is constructible.
3.2. Step 2: Reduction to the noetherian case
Finally, it is here that we introduce the “finite presentation” trick most clearly. Consider a map
coming from a finitely presented morphism of rings
The claim is that the image in is constructible. Now the key observation is that we know this, when is noetherian, because then it is the usual Chevalley theorem, which I don’t want to go into here. However, in general, is a filtered colimit of noetherian rings—namely, of its finitely generated -subalgebras. The claim is that we can reduce to this case by:
Lemma 12 Let be a finitely presented morphism of rings. Then there is a finitely generated subring and a morphism of finite presentation such that is a base-change of .
So visually, this looks like a cocartesian diagram
Let us assume this lemma for the moment and see how it implies the big theorem. Recall that our goal is to show that the image of is constructible. But we have a cartesian diagram of schemes:
Consequently, we know that the image of is the preimage of the image of , by basic properties of fibered products. But has constructible image by noetherianness, so the same is true for the top map, as inverse images preserve constructibility. We’re done, modulo the lemma!
3.3. Step 3: the finite presentation lemma
Finally, we are left with the lemma. This lemma is the crucial step in the above approach, and is one of the reasons that “finite presentation” results can often be reduced to noetherian ones. There are more results on this theme, which essentially boil down to the idea that in the category of rings, finitely presented objects are compact.
So we have a finitely presented morphism . This means that we can write , so is a quotient of a finitely generated polynomial ring over by a finitely generated ideal. However, the coefficients in the finitely many all lie in some finitely generated subring . This means that the ring
makes sense, and is a finitely presented -algebra. However, it is also easy to see that is a base-change of this. This proves the lemma.