This paper is a survey about recent progress on submersive morphisms of schemes combined with new results that we prove. They concern the class of quasicompact universally subtrusive morphisms that we introduced about 30 years ago. They are revisited in a recent paper by Rydh, with substantial complements and key results. We use them to show Artin-Tate-like results about the 14th problem of Hilbert, for a base scheme either Noetherian or the spectrum of a valuation domain. We look at faithfully flat morphisms and get “almost” Artin-Tate-like results by considering the Goldman (finite type) points of a scheme. Bjorn Poonen recently proved that universally closed morphisms are quasicompact. By introducing incomparable morphisms of schemes, we are able to characterize universally closed surjective morphisms that are either integral or finite. Next we consider pure morphisms of schemes introduced by Mesablishvili. In the quasicompact case, they are universally schematically dominant morphisms. This leads us to a characterization of universally subtrusive morphisms by purity. Some results on the schematically dominant property are given. The paper ends with properties of monomorphisms and topological immersions, a dual notion of submersions.

Our aim is to give a survey on recent progress on submersions and new results that commutative algebraists may find useful. We also recall results that are needed. The paper is written in the language of schemes because it is sometimes necessary to enlarge the category of commutative rings to get proofs, but the results can be easily translated.

Submersive morphisms of schemes (or submersions)

We recall the following facts.

The closed subsets of the

The Zariski topology induces a partial ordering on the underlying set of points of a scheme

Let

Every ordered pair

Then

Clearly, a subtrusive morphism is surjective.

We will mainly consider quasicompact morphisms of schemes. In that case, a morphism of schemes is subtrusive if and only if the above condition (1) holds [

Note that a quasicompact morphism of schemes

Let

Our main results are Artin-Tate-like results, about the descent of the finite type property of morphisms by universally subtrusive morphisms of finite presentation. Artin-Tate's result may be read as follows and exhibits a solution to the 14th problem of Hilbert.

Let

The affine version of this result is quite easy to establish, once the Eakin-Nagata theorem is known. Let

The results offered are consequences of a result of Rydh about the structure of universally subtrusive morphisms of finite presentation. Among a lot of nice results, Rydh proved the following.

Let

Theorem

Let

Thus even in the ring context, we cannot provide a ring theoretic proof of the main theorem of this paper and have to consider morphisms of schemes.

The following results [

Let

It follows that a universally subtrusive morphism

Then we have the following valuative criterion.

Let

If

Let

For every valuation ring

there is a valuation ring

The above condition (2) is the Nagata’s definition of a strongly submersive morphism [

In order to ease reading, we introduce the following definition.

Let

Now we give some comments about the terminology used in this paper. In the literature, a morphism of schemes

If

A ring morphism

Suppose that a morphism

Some results about immersions and monomorphisms are involved. They will be recalled when needed, especially in Section

Section

In Section

Section

In Section

Section

Considering Theorem

A faithfully flat morphism of schemes of finite presentation descends

We next derive some consequences of this result. Some authors say that a scheme (resp., a morphism) is

We need some considerations about monomorphisms of schemes. Note that monomorphisms between affine schemes correspond to epimorphisms of the category of commutative rings.

Let

Properties of monomorphisms are as follows.

Let

The following lemma is useful to reduce proofs to affine schemes.

Let

Each morphism

The canonical morphism

Let

In view of [

In the affine context, we can give a simpler proof. We introduce the following definition.

A universally injective ring morphism is called pure. A pure ring morphism is universally subtrusive [

Results about pure ring morphisms used in this paper come from the work of Olivier [

Let

Since each

The following result is well known. A proof may be found in the Columbia Stack project [

Let

Let

In case

Use the base change

In case

We recall that a scheme is called absolutely flat if each of its stalks is a field. Olivier and Hochster proved independently in [

Let

Let

Let

Let

Observe that

We intend to apply the above result in order to obtain “almost” Artin-Tate-like results.

Let

Note that a reduced ring

The above letter

Let

Similar to the proof of Remark

We can factorize

Let

In view of [

Note that if

We will prove an Artin-Tate-like result under Noetherian hypotheses. We need a result proved by Onoda [

Let

If

Let

In view of [

Another results proved by Fogarty will be useful [

Let

Let

Next result is decisive.

Let

It deserves to be compared with the following Alper's result.

Let

Then

We recall the following descent result by base changes.

Let

Let

For each

We can now state the main result of the section.

Let

First observe that

Let

Some known results are not a consequence of Theorem

In this section, we give a survey about new results on universally closed morphisms and add some commentaries.

We thank Bjorn Poonen for his kind authorization to reproduce his proof of the next result, published electronically in [

A universally closed morphism of schemes

Without loss of generality, we may assume that

Write

If

Rydh observed in the same item of [

Let

Let

In case

The following result generalizes [

Let

By the above proposition,

Let

We have defined a preorder

Olivier defined absolutely flat morphisms as flat morphisms, whose diagonal morphisms are flat [

Let

A proper, incomparable, and surjective morphism of schemes

Let

The equivalence (1)

Let

(1)

We recover the result: if

Consider two morphisms of schemes

Let

Use Theorem

Replace the integral hypothesis on

We end with descent results.

Let

The morphism

Note that the quasiseparated case needs only that

A universally closed morphism

Use [

Hashimoto proved the following result [

Let

We note here that

We are thus led to consider a class of scheme morphisms introduced by Mesablishvili, that is, a generalization to schemes of the class of pure ring morphisms [

Let

A schematically dominant morphism is dominant. If

Recall that a morphism of schemes is a

Let

We first show that (1) is equivalent to (2). A first step is done by considering [

We observe that a composite of quasicompact pure morphisms is pure by Proposition

The following result extends to schemes [

A quasicompact

Assume that

The converse is clear.

We give some examples of descent of the property

In particular, if

We conclude this section by a criterion for a flat morphism of schemes to be schematically dominant, as suggested in [

Let

Clearly, if

We prove the converse when

We end by recalling a result of Rydh, which is similar to Proposition

Let

It follows that a universally submersive morphism

The Nagata compactification theorem for schemes says that if

Recall the following result on monomorphisms, bearing in mind that a morphism of affine schemes is a monomorphism of schemes if and only if its associated ring morphism is an epimorphism of the category of rings. The case of a flat monomorphism is [

Let

Let

For instance, let

Olivier calls

Let

Let

Let

For each

For each ring morphism

An injective integral ring morphism whose target is an integral domain is essential. A flat epimorphism is essential by [

Let

Clearly, a topological immersion is topologically essential. We also note that the notion of topological immersion is dual to the notion of submersive morphism.

Let

If, in addition,

If

It is enough to rework the proof of [

An immersion of schemes

Let

Then

in [

A continuous map of topological spaces

Let

We show (1). Let

If

Let

We can replace the schematically dominant hypothesis by dominant. Then the result follows from Proposition

We address the following question. Is the reduced hypothesis necessary in the above proposition? By the way, we have the following result.

A quasicompact flat monomorphism

First observe that under the hypotheses on

We suspect that the above proposition has a more general version because flat epimorphisms of rings are essential on concentrated morphisms. To see this, suppose that

We end by some considerations on quasiaffine schemes. A scheme is called quasiaffine if it is isomorphic to a quasicompact open subscheme of an affine scheme. A morphism

The following may be useful.

Let

We observe that

We intend to apply the following result [

Let

Let