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 {f: X \rightarrow Y} is a regular map and {C \subset X} is constructible, then so is {f(C)}. In general, this is the best one can say: even very nice subsets of {X} (e.g. {X} 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 {f: X \rightarrow Y} be a finitely presented morphism of schemes. Then if {C \subset X} is locally constructible, so is {f(C)}.

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.

1. Preliminaries

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 {X} is a topological space and {U \subset X} an open subset, then {U} is called retrocompact if the inclusion {U \hookrightarrow X} is a quasi-compact morphism. That is, if {V \subset X} is a quasicompact open set, then {U \cap V} 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\displaystyle \bigcup_{i=1}^n (U_i - V_i)

for the {U_i, V_i} retrocompact.

Naturally, we can define what locally constructible means: a subset {T } is locally constructible if there is an open covering by open sets {U} such that {T \cap U} is constructible in {U}. 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 {f: X \rightarrow Y} is a morphism of schemes, and {T \subset Y} is constructible, so is {f^{-1}(T)}.

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 {U \subset Y} be retrocompact. This is equivalent to saying that the morphism of schemes {U \rightarrow Y} is quasicompact. But the morphism {f^{-1}(U) \rightarrow X} is the pull-back of {U \rightarrow Y} via {X \rightarrow Y}, so it is also quasicompact. Thus {f^{-1}(U)} is retrocompact in {X}, and we have proved the claim.

The result now implies that the same is true for local constructibility:

Corollary 5 If {f: X \rightarrow Y} is a morphism of schemes, and {T \subset Y} is locally constructible, so is {f^{-1}(T)}.

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 {U \hookrightarrow X} be a quasi-compact open immersion. Then the image of a constructible {C \subset U} is constructible in {X}.

This will follow at once if we show that if {V \subset U} is retrocompact open, then {V} is constructible in {X}. 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 {V \subset U} is retrocompact, then {V \hookrightarrow U} is quasicompact. But by assumption {U \hookrightarrow X} is quasicompact, so the composite {V \hookrightarrow X} is quasicompact as well. This means that {V} is constructible in {X}.
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 {X} is a quasicompact, quasiseparated scheme and {T \subset X} is locally constructible, it is globally constructible.

By definition, there is a finite cover of {X} by small neighborhoods {U_i \subset X}, which we may take affine (by the proposition), such that {T \cap U_i \subset U_i} is constructible for each {i}.

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 {X} is a quasicompact and quasiseparated scheme and {T \subset X} a constructible subset, there is an affine scheme {X'} and a morphism {X' \rightarrow X}, quasicompact and locally of finite presentation, whose image is {T}.

To do this, we start by covering {X} by open affine subsets {X_i, 1 \leq i \leq n}. Since {X} is quasicompact, we only need finitely many of them. The morphism

\displaystyle \sqcup X_i \rightarrow X

is quasi-compact because {X} is quasiseparated. As a result, by reducing to the case of the inverse image of {T} in {\sqcup X_i}, we may suppose that {X} itself is affine.

{T} is constructible, so we can write {T } as a finite union {U_j - V_j} for {U_j, V_j \subset X} quasicompact. By taking finite coproducts, we see that it is enough to handle the case of {T = U-V} for {U,V} retrocompact in {X} (hence quasicompact, {X} being affine). This means that each of {U, V} is a finite union of basic open sets {X_{g_a}} for each {g_a \in \Gamma(X,\mathcal{O}_X)}. By taking finite unions, we can reduce to the case

\displaystyle U = X_g, \quad V = X_{f_1} \cup \dots \cup X_{f_r}.

The claim is that {U - V} arises as the image of an affine scheme, of finite presentation over {X}. Suppose {X = \mathrm{Spec} R}.

Then we can take the natural map {\mathrm{Spec} R_g/(f_1, \dots, f_r) \rightarrow \mathrm{Spec} R} whose image is {U-V}; 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 {f: X \rightarrow Y} be a quasicompact morphism, locally of finite presentation. Then if {C \subset X} is locally constructible, so is {f(C) \subset Y}.

Interestingly, it turns out that Grothendieck does not (in EGA IV-1) state the theorem in maximal generality. Grothendieck assumes that {f} is of “finite presentation,” which means that he assumes in addition that {f} 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 {X, Y} affine

The first step is to make some reductions. Since we are trying to prove an assertion which is local on {Y}, we may assume that {Y} is itself affine, and consequently that {X} is quasi-compact. It follows that {X} is a fintie union {\bigcup X_i}, where each {X_i} is affine (and thus quasicompact and quasiseparated).

In particular, {C \cap X_i} is not only locally constructible, but constructible. If we show (assuming {Y} affine, as usual) that {f(C \cap X_i)} is constructible for each {i}, then we will be done. The upshot is that we can reduce to the case of {Y} affine and {X} affine. In particular, we need to prove:

Lemma 10 Let {f: X \rightarrow Y} be a morphism of affine schemes which is of finite presentation. If { C \subset X} is constructible, so is {f(C)}.

But in fact we can make a further reduction. We know that there is an affine scheme {X'} and a morphism of finite presentation {X' \rightarrow X} such that {C} is the image of {X'}, by the above lemmata. So we are reduced to something even more concrete:

Lemma 11 Let {f: X \rightarrow Y} be a morphism of affine schemes which is of finite presentation. Then {f(X)} 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

\displaystyle \mathrm{Spec} S \rightarrow \mathrm{Spec} R

coming from a finitely presented morphism of rings

\displaystyle R \rightarrow S.

The claim is that the image {\mathrm{Spec} S} in {\mathrm{Spec} R} is constructible. Now the key observation is that we know this, when {R} is noetherian, because then it is the usual Chevalley theorem, which I don’t want to go into here. However, in general, {R} is a filtered colimit of noetherian rings—namely, of its finitely generated {\mathbb{Z}}-subalgebras. The claim is that we can reduce to this case by:

Lemma 12 Let {R \rightarrow S} be a finitely presented morphism of rings. Then there is a finitely generated subring {R' \subset R} and a morphism of finite presentation {R' \rightarrow S'} such that {R \rightarrow S} is a base-change of {R' \rightarrow S'}.

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 {\mathrm{Spec} S \rightarrow \mathrm{Spec} R} is constructible. But we have a cartesian diagram of schemes:

Consequently, we know that the image of {\mathrm{Spec} S \rightarrow \mathrm{Spec} R} is the preimage of the image of {\mathrm{Spec} S' \rightarrow \mathrm{Spec} R'}, by basic properties of fibered products. But {\mathrm{Spec} S' \rightarrow \mathrm{Spec} R'} 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 {R \rightarrow S}. This means that we can write {S = R[x_1, \dots, x_n]/(f_1, \dots, f_k)}, so {S} is a quotient of a finitely generated polynomial ring over {S} by a finitely generated ideal. However, the coefficients in the finitely many {f_i} all lie in some finitely generated subring {R'}. This means that the ring

\displaystyle S' = R'[x_1, \dots, x_n]/(f_1, \dots, f_k)

makes sense, and is a finitely presented {R'}-algebra. However, it is also easy to see that {R \rightarrow S} is a base-change of this. This proves the lemma.