Recent Progress on Submersions : A Survey and New Properties

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.


Introduction
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)  :  →  are surjective morphisms inducing the quotient topology on ; that is,  ⊆  is an open (closed) subset if and only if  −1 () is open (closed).They are also called topological epimorphisms by some authors like Voevodsky who defines and uses the ℎ and ℎ- (Grothendieck) topologies [1].They appear naturally in many situations such as when studying quotients, homology, descent, and the fundamental group of schemes.A morphism of schemes  :  →  is called universally submersive if ×    →   is submersive for each morphism   → .The first proper treatment of submersive morphisms was settled by Grothendieck, with applications to the fundamental group of a scheme.We singled out a subclass of submersive morphisms in [2] and dubbed them subtrusive morphisms (or subtrusions).Submersive morphisms used in practice are subtrusive.Our study was established in the affine schemes context.But as Rydh showed, the theory can be extended to the arbitrary schemes context [3].Over a locally Noetherian scheme, a universally submersive morphism is universally subtrusive, showing again that the class of subtrusive morphisms is natural.In this section, we give information about our aims, notational conventions, and definitions, recalling results that are needed in the other sections.
We recall the following facts.
The Zariski topology induces a partial ordering on the underlying set of points of a scheme .We let  ≤   if  ∈ {  }, that is, if  is a specialization of   , or equivalently, if {} ⊆ {  }.A maximal point of  is the generic point of an irreducible component of .If  is quasicompact, for each  ∈ , there is some closed point , such that  ≤  because the set of all closed subsets is inductive for the relation ⊇ [4, 0.2.1.2].

Algebra
Definition 1 (see [3,Definition 2.2]).Let  :  →  be a morphism of schemes.Then  is called subtrusive if the following two conditions hold.
(1) Every ordered pair  ≤   of points in  lifts to an ordered pair of points  ≤   in .
Then  is called universally subtrusive if ×    →   is subtrusive for each morphism of schemes   → .
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 [3,Proposition 1.6].As quasicompactness is a universal property, a quasicompact morphism  is universally subtrusive if the condition (1) universally holds for .
Note that a quasicompact morphism of schemes  :  →  is closed if and only if  is specializing; that is, () is stable under specializations for each subset  ⊆ , stable under specializations.To see this use [4, I. Proposition 7.2.12(iv)] which tells us that if  is quasicompact, then  is closed for the patch topology and [4, I, Corollaire 7.3.2]which states that  = ∪[{};  ∈ ] for a proconstructible subset .
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.
Proposition 3 (see [5,Lemma 3.14.1]).Let  :  →  be a finite and surjective morphism of schemes over a locally Noetherian scheme .Then  is (locally) of finite type over  if and only if  is (locally) of finite type.
The affine version of this result is quite easy to establish, once the Eakin-Nagata theorem is known.Let  →  →  be a composite of ring morphisms, such that  →  is of finite type,  is Noetherian, and  → B is injective integral (equivalently, finite), then  →  is of finite type and  is Noetherian.To see this, let { 1 , . . .,   } be a system of generators of  over  and let  ⊆  be the -algebra generated by the coefficients of unitary polynomials   () ∈  [𝑋], such that   (  ) = 0. Then  →  is injective and finite, so that  is Noetherian by the Eakin-Nagata theorem.
It follows that  →  is finite and the proof is complete.In passing we note that the Eakin-Nagata theorem is not valid for integral extensions by [6, 2.3]: there exists a non-Noetherian domain  such that   is Noetherian for every prime ideal  of  and such that there exists an integral extension  →   , where   is a Noetherian domain.
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.
Theorem 4 (see [3,Theorem 3.10]).Let  be an affine or Noetherian scheme.Let  :  →  be a universally subtrusive morphism of finite presentation.Then there is a refinement   :   →  of  and a factorization of   into a faithfully flat morphism   →   of finite presentation followed by a proper surjective morphism   →  of finite presentation.If in addition  is universally open, then one may choose   such that   → ×    is a nil-immersion.Theorem 4 allows us to reduce our study to proper surjective morphisms.It may be asked whether there is a ring-theoretic version of the preceding result.The answer is no, as the following personal communication of Rydh shows.As a consequence, we cannot reduce the proof to finite morphisms, at least if we wish to use Rydh's above result.
Example 5. Let  be the affine plane and let  →  be the blow-up in a point , which is proper.Choose an affine covering  =  1 ∐ 2 of  and let  :  →  be the natural map.Then  is affine and universally subtrusive but does not admit a refinement of the form   :   →   →  where the first map is faithfully flat and the second is finite and surjective.Indeed, let  be the ideal sheaf defining the point .Since   factors as   →  →  → , the ideal sheaf    is principal.Since   →   is faithfully flat, this means that the ideal sheaf    is principal.But as   →  is finite, the inverse image of  has codimension 2 and we have a contradiction.
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 [2, Proposition 16] were extended to schemes by Rydh in [3, Proposition 2.7, Theorem 2.8].Proposition 6.Let  be a valuation ring and  :  → Spec() a morphism of schemes.The following statements are equivalent.
(4) Any chain of points in Spec() lifts to a chain of points in .
(5) There is a closed subscheme  ⊆  such that  | is faithfully flat.
It follows that a universally subtrusive morphism  :  →  lifts to  chains of points of  [3, Proposition 2.11], a generalization to schemes of [7,Theorem 3.26].Actually, the statement of [3,Proposition 2.11] is established for chains that have a lower bound in , a superfluous condition.To see this, let  be a subset of a scheme  such that each pair of elements of  has a lower bound in .Then the closure of  is irreducible, and whence has a generic element.
Then we have the following valuative criterion.
(2) For every valuation ring  and diagram of solid arrows there is a valuation ring   and morphisms such that the diagram becomes commutative and such that the left vertical morphism is surjective.
The above condition (2) is the Nagata's definition of a strongly submersive morphism [8,9].He proved our main theorem for a -algebra, where  is a Nagata ring (pseudogeometric for Nagata).
In order to ease reading, we introduce the following definition.
Definition 9. Let  :  →  be a morphism of schemes.We say that  descends the property A (resp., property P) if for each morphism  :  → , such that  ∘  is of finite type (resp., of finite presentation), then  is of finite type (resp., of finite presentation).We say that  belongs to the class A (resp., P) if  descends A and is of finite type (resp., descends P and is of finite presentation).Now we give some comments about the terminology used in this paper.In the literature, a morphism of schemes  →  is usually said to descend a property Q of morphisms of schemes if for each morphism  →  such that  ×   →  has Q, then  →  has Q.In order to avoid confusions in this paper, we say that such a morphism  →  D-descends the property Q (D for diagram).

Remark 10. (1)
In case  ∘  is of finite type and  is either quasiseparated or  is Noetherian, then  is of finite type (see [4,I,Proposition 6.3.4]).
(3) The classes A and P are stable under right division and composition.In particular, if  :  →  is a ring morphism,  → / descends A (P) for each ideal  ⊆ ker  of  if  does.
(4) We examine a converse.Suppose that / ker  →  descends A and that ker  is a nilpotent ideal.Then  →  descends A by [10, Lemma 4] Some results about immersions and monomorphisms are involved.They will be recalled when needed, especially in Section 2. Section 6 is concerned with topological immersions of schemes, a notion "dual" of submersions.They are considered because immersions of schemes are topological immersions.Moreover, the results we get have their own interest.A morphism of schemes  :  →  is called a topological immersion if  is injective and if the topology of  coincides with the inverse image topology on , with respect to .In case  is quasicompact, the topological immersion property can be characterized with ordered pair of points of , similarly to the definition of subtrusive morphisms.We also consider topological essentiality, which in the affine case is linked to essential morphisms of rings.
Section 2 deals with faithfully flat morphisms of schemes, of finite presentation, that are known to descend A and P. We derive some results from this case.In particular, we prove our main result for universally subtrusive -morphisms of schemes  → Spec(), where  is a valuation domain.

Algebra
In the affine context, the hypothesis that  is a valuation domain can be replaced by  is a Prüfer domain.As an easy consequence of the theorem of generic flatness for a surjective -morphism of schemes  → , where  is concentrated and reduced, we get that if  →  is of finite type (of finite presentation), there is a dense open subset  of , such that  →  →  is of finite type (of finite presentation).This will be useful in Section 3, where the Noetherian case is studied.We introduce absolutely flat schemes and recall their main properties, in order to get "essentially" Artin-Tate-like results.Absolutely flat schemes are also used in Section 4. For instance, consider ring morphisms  →  → , where  →  is of finite type and  →  is of finite type and spectrally surjective.Then for each finite type point (Goldman point)  of  (i.e.,  → () is of finite type),  →  → () is of finite type.In particular,  → / is of finite type for each maximal ideal  of .This is established in the scheme context.
In Section 3, we consider the Noetherian case and exhibit an Artin-Tate-like result.A result established by Onoda is crucial: if  is a Noetherian domain and  an overdomain of , then the ring morphism  →  is of finite type provided that   →   is of finite type for each  ∈ Spec() and there is a nonzero element  of  such that  →   is of finite type.The second condition is gotten as a consequence of the Theorem of generic flatness.The local condition is deduced by faithfully flat descent from a result by Hashimoto, when the base ring is excellent.
Section 4 examines properties of universally closed morphisms of schemes.We thank Bjorn Poonen for his kind authorization to reproduce his proof of the following result, electronically published in [12].A universally closed morphism is quasicompact.A version of the proof also appears in the Stacks Project [13,Lemma 39.9].Within the category of affine schemes, universally closed morphisms and integral morphisms coincide.Recall that a morphism of schemes  :  →  is called a Stein morphism if O  →  ⋆ O  is an isomorphism.We give Stein factorization results published in [13] for concentrated universally closed morphisms  :  → .In that case, there is a factorization  →   → , where the first morphism is Stein and the second is integral.We define incomparable morphism of schemes in the same way as in Commutative Algebra.If a universally closed (proper) separated morphism  is incomparable, then  is integral (finite).As a consequence, we give a short proof of a Raynaud's result: if  ∘  is integral,  surjective, and  separated, then  is integral.
In Section 5, we introduce pure morphisms of schemes defined and characterized by Mesablishvili, an extension to schemes of pure morphisms of rings.We consider only the quasicompact context, in which case pure morphisms are nothing but universally schematically dominant morphisms of schemes.We show that they are universally subtrusive when concentrated.Moreover, quasicompact universally subtrusive morphisms are shown to be quasicompact morphisms that become pure after each base change with respect to a valuation domain.We also establish a criterion for a flat morphism to be schematically dominant by using the set of all (weak Bourbaki) associated primes of a scheme.
Section 6 is concerned with monomorphisms of schemes and (topological) immersions.We are mainly interested in quasicompact flat monomorphisms.We look at relations with strict monomorphisms and quasiaffine morphisms.For a quasicompact morphism, the topological immersion property is equivalent to some property of pairs of comparable elements of schemes.We define topologically essential and schematically essential morphisms of schemes.We examine their properties linked to topologically minimal continuous maps.

Faithfully Flat Morphisms
Considering Theorem 4, we see that faithfully flat morphisms of finite presentation are involved.In this case the main theorem is already known.
We next derive some consequences of this result.Some authors say that a scheme (resp., a morphism) is concentrated if it is quasicompact and quasiseparated.Let  :  →  be a morphism of schemes, such that  is concentrated.Then  is concentrated if and only if  is concentrated (see [4, I, Section 6] for proofs).Noetherian schemes and affine schemes are concentrated.Note that a quasicompact  subset of a concentrated scheme  is such that  →  is quasicompact because  is quasiseparated.Actually, if  is concentrated,  →  is concentrated for each quasicompact open subset  of , and so is .For a concentrated morphism of schemes, the finite type (finite presentation) property is equivalent to the locally finite type (locally finite presentation) property by the very definition of these properties.
We need some considerations about monomorphisms of schemes.Note that monomorphisms between affine schemes correspond to epimorphisms of the category of commutative rings.
Proposition 13.Properties of monomorphisms are as follows.
(1) A monomorphism is separated because its diagonal morphism is an isomorphism.
(2) in [19, page 100], an immersion is a monomorphism.A quasicompact flat monomorphism  is an isomorphism if and only if  is surjective.A finite monomorphism is a closed immersion.
Let  be a scheme and  ∈ , then the natural map O , →  is a flat monomorphism with image g(), the set of all generalizations of .

Proposition 15.
Let  be a valuation domain,  →  a ring morphism,  a quasiseparated scheme, and  :  → Spec() a universally subtrusive morphism of schemes of finite type, such that  → Spec() → Spec() is of finite type (of finite presentation).Then  → V is of finite type (of finite presentation).
Proof.In view of [3, Proposition 2.7(vii)], there is a closed subscheme  of  such that  →  → Spec() is faithfully flat.Then  →  is of finite type and so is  → Spec().Moreover, a closed immersion is quasicompact and separated, whence  →  → Spec() is concentrated because  is concentrated.Since  is concentrated, we can suppose that  is affine by Lemma 14.We are now in position to apply [20, I, Corollaire 3.4.7]which tells us that a flat ring morphism of finite type, whose domain is an integral domain, is of finite presentation.Then  → Spec() is faithfully flat of finite presentation and therefore  →  is of finite type (resp., of finite presentation) by Theorem 11.
In the affine context, we can give a simpler proof.We introduce the following definition.Results about pure ring morphisms used in this paper come from the work of Olivier [11].Pure morphisms of schemes are introduced in Section 5.

Proposition 17.
Let  be a Prüfer domain,  →  a ring morphism, and  :  →  a universally subtrusive ring morphism of finite type, such that  →  is of finite type, then so is  → .
Proof.Since each   for  ∈ Spec() is a valuation domain, we get that   →   is pure by [2, Théorème 37(a)] and then  →  is pure.Let  be the quotient field of  and  := ker( →  ⊗  ) the torsion ideal of .Then by [11,  In case  = Spec() and  = Spec() are affine schemes where  is reduced, one gets that there is some regular element  ∈  such that  →   is of finite type (resp., of finite presentation), with   ⊆ Tot().
Proof.Use the base change  →  defined in Proposition 18, pick some nonempty affine open subset  ⊆  and observe that  →  is a quasicompact open immersion by Lemma 14, whence of finite presentation and so is   → .
In case  = Spec() and  = Spec() are affine schemes where  is reduced in Proposition 18, we see that  = D() is such that 0 :  = 0; that is,  is dense.
We recall that a scheme is called absolutely flat if each of its stalks is a field.Olivier and Hochster proved independently in [21,22] the existence of a universal absolutely flat scheme  cons (or ()) associated with an arbitrary scheme , a construction announced by Grothendieck in [4, I, 7.2.14].Actually, () is the final object of the category of absolutely flat -schemes.The structural morphism   : () →  is an affine surjective monomorphism, and the canonical map   : () →  is a homeomorphism when  is endowed with the constructible topology.In particular for a ring , there is a ring epimorphism  → () which gives for  = Spec(), the structural morphism () → .Since an affine morphism is quasicompact and separated, we see that   is concentrated for an arbitrary scheme .
(2)  is an affine scheme if and only if  is a concentrated scheme.If the preceding condition is verified, then  is an absolutely flat scheme.
Corollary 21.Let  be a concentrated scheme, then T() is an affine scheme.We intend to apply the above result in order to obtain "almost" Artin-Tate-like results.
Remark 24.Let  be a reduced -ring (that is,  → Tot() is of finite type or, equivalently Tot() =   for some regular  ∈ ) and suppose that Tot() is absolutely flat, then for a composite of ring morphisms which is of finite type  →  → , with  :  →  of finite presentation and such that   is surjective, then  →  → Tot() is of finite type because Tot() → Tot() ⊗   is faithfully flat and of finite presentation.In particular, if  is a -domain, we get that there is some nonzero  ∈  such that  →   is of finite type.
Note that a reduced ring  is such that Tot() is absolutely flat if and only if Min() is compact (Hausdorff) and  is a McCoy ring; that is, each finitely generated ideal contained in () has a nonzero annihilator.Such rings are called decent in [24, page 259].In particular a ring with few zero divisors is decent, like a Noetherian reduced ring.
The above letter  refers to some Goldman property of a ring.We can also define Goldman points of a scheme .A finite type point  of  is a point such that Spec(()) → Spec(O , ) →  is of finite type (see [13,Section 15]).In Commutative Algebra, such points are called Goldman points, a terminology we keep.A point  ∈  is Goldman if and only if {} is a locally closed subset of , that is, of the form {} ∩ , where  is an affine open subset.We denote by Gold() the set of all Goldman points of .Observe that Gold() is strongly dense in  by [13,Lemma 15.6] and [4, Section 0.2.6].It contains the set of all closed points and the set of all isolated points, the so-called -points.Proposition 25.Let  :  →  be a surjective morphism of schemes of finite type and  :  →  a morphism of schemes such that  ∘  is of finite type (resp., of finite presentation).Let  ∈ Gold(), then Spec(()) →  →  is of finite type (resp., of finite presentation).
Proof.Similar to the proof of Remark 24.
We can factorize Spec(()) →  as Spec(()) → Spec(O , ) →  if  ∈ , where the first morphism is a closed immersion and the second is a quasicompact flat monomorphism.In case Spec(O , ) is Artinian and the above hypotheses hold, Spec(O , ) →  →  is of finite type, by [13,

The Noetherian Context
We will prove an Artin-Tate-like result under Noetherian hypotheses.We need a result proved by Onoda [25, Lemma 2.14, Theorem 2.20] in a more general setting.Theorem 28.Let  be a Noetherian domain and  an overdomain of  such that   is finitely generated over  for some nonzero  ∈ .Then the following statements hold.
(1) If  is a multiplicative subset of  such that   →   verifies A, then there is some  ∈  such that  →   verifies A.
( It deserves to be compared with the following Alper's result. Proposition 32 (see [28,Proposition 1.3]).Let  be an excellent ring and  :  →  a morphism of -schemes.Suppose the following: (1)  is surjective.
(2)  is of finite type over .
Then  is of finite type over .
We recall the following descent result by base changes.
Proposition 33 (see [11,Proposition 5.3,page 22]).Let  :  →  be a ring morphism.Then  is pure if and only if  universally D-descends the property P.

Corollary 34.
Let  be a Noetherian ring and  :  →  a surjective proper morphism of -schemes, where  := Spec() is an affine Noetherian scheme.If  is of finite type over , so is   →   for each prime ideal  of .
Proof.For each  ∈ Spec(), consider the base change  →   → Ŝ , where Ŝ is the   -adic completion of the local ring   and is excellent.It follows by Proposition 31 and the descent of the finite type property by the faithfully flat morphism   → Ŝ that   →   is of finite type.
We can now state the main result of the section.Theorem 35.Let  :  →  be a universally subtrusive morphism of -schemes and suppose that  and  are Noetherian.If  →  is of finite type, so is  → .
Proof.First observe that  →  is of finite presentation because of finite type and  is Noetherian [4, I, 6.3].In view of [3,Theorem 3.10] there is a morphism of schemes   →  such that   :   →  can be factored into a faithfully flat morphism of finite presentation   →   followed by a proper surjective morphism   → .Clearly,   →  is of finite type and by Theorem 11,   →  is of finite type.Therefore, we can assume that  →  is a proper surjective morphism.We can also assume that  := Spec() and  := Spec() are affine schemes with a ring morphism  :  → .Using the base changes  →  red and  → / for each minimal prime ideal  of  for  → , we can assume that  is integral by [27, page 169] or by Lemma 30 (2).Furthermore, we can suppose that  is integral by changing  with / ker  because  →  → Spec(/ ker ) is of finite type.The proof is completed by combining Theorem 28, Corollaries 29 and 19.
Remark 36.Let  :  →  be a finite ring morphism of algebras such that the canonical map Spec() → Spec() is surjective and suppose that  →  verifies A. In view of [30, Corollary 1] there is some affine -subalgebra   of , such that   →  is a finite morphism and the module generators of  over  and   are the same.To see this it is enough to reduce the proof to an injective ring morphism  → .
Remark 37. Some known results are not a consequence of Theorem 35 and Proposition 17.They concern the so-called strongly affine pairs of rings  ⊆  (such that  is a subring of  and such that each -subalgebra of  is of finite type).For instance, if  is a field and  = [] an -algebra with a single generator,  ⊆  is a strongly affine pair [31].The reader will find much more examples in a paper by Papick [32].Note also that in case  ⊆  is a pair of rings sharing an ideal , then an -subalgebra  of  is of finite type if and only if so is / → /.

Universally Closed Morphisms
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 [12].A version of the solution appears also in the Stacks Project [13,Lemma 21.39.9].

Theorem 38. A universally closed morphism of schemes 𝑓 :
→  is quasicompact.If, in addition,  is also surjective, then it is universally subtrusive.
Proof.Without loss of generality, we may assume that  = Spec() for some ring  and that  is surjective.Suppose that  is not quasicompact.We need to show that  is not universally closed.
Write  = ∪ ∈   where the   are affine open subschemes of .Let  := Spec([{  :  ∈ }]), where the   are distinct indeterminates.Let   := D(  ) ⊆ .Let  be the closed set ( ×  ) \ ∪ ∈ (  ×    ).It suffices to prove that the image   () of  under   :  ×   →  is not closed.There exists a point  ∈ Spec() such that there is no neighborhood  of  in  such that   is quasicompact, since otherwise we could cover  with finitely many such  and prove that  itself was quasicompact.Fix such , and let  be its residue field.First we check that   (  ) ̸ =   .Let  ∈ () be the point such that   () = 1 for all .Then  ∈   for all , and the fiber of   →   above  is isomorphic to ( \ ∪ ∈   )  , which is empty.Thus,  ∈   \   (  ).If   () were closed in , there would exist a polynomial  ∈ [{  :  ∈ }] vanishing on   () but not at .Since () ̸ = 0, some coefficient of  would have nonzero image in , and hence be invertible on some neighborhood  of .Let  be the finite set of  ∈  Algebra such that   appears in .Since   is not quasicompact, we may choose a point  ∈  \ ∪ ∈   lying above some  ∈ .Since  has a coefficient that is invertible on , we can find a point  ∈  lying above  such that () ̸ = 0 and   () = 0 for all  ∉ .Then  ∉   for each  ∉ .A point  of  ×   mapping to  ∈  and to  ∈  then belongs to .But (  ()) = () ̸ = 0, so this contradicts the fact that  vanishes on   ().
If  is surjective and universally closed, then  is universally subtrusive by the first part of the proof and Example 2.
Rydh observed in the same item of [12] that Bjorn Poonen's argument can be somewhat simplified by using the simple (topological) fact that a closed morphism with quasicompact fibers is quasicompact (this is implicit in his argument).In the above proof we can thus assume that  is a point and that  is not quasicompact.Another comment by Rydh is Proposition 41 (1), which shows that a separated and universally closed morphism is "almost proper." We will need the following results.
Let  :  →  be a concentrated morphism, so that  (1)  has finite dimension.
(3)  is concentrated and the canonical map   → O  (  ) is an isomorphism, where each  −1 (D()) =   for  ∈  is open and closed.
Proof.(1) It is enough to use [36,Corollary 8.4] because a closed immersion is finite and the diagonal of  is a closed immersion.
(3) By [4, Corollaire 6.1.10], is quasiseparated.The proof of the isomorphism is [37,Lemma 14.2].Now the reduced ring of  is absolutely flat so that each element of  red is of the form  where  is an idempotent and  is a unit.It follows that   is open and closed because so is D() in Spec( red ).
We have defined a preorder ≤ on the underlying set of a scheme  by  ≤   ⇔  ∈ {  }.A morphism of schemes  :  →  is called incomparable if for any pair  ≤   of elements of , then () = (  ) ⇒  =   , or equivalently, each of the points of   is maximal for each  ∈ ; that is, each fiber is zero-dimensional.Clearly, an integral morphism of schemes is incomparable and so is an injective morphism.A composite of incomparable morphisms is incomparable.If  ∘  is incomparable and  is subtrusive, then  is incomparable.Note that the morphism  of Lemma 14 is incomparable.
Olivier defined absolutely flat morphisms as flat morphisms, whose diagonal morphisms are flat [38].We may find in [19, (1.4)] that a morphism of schemes  → , where  is the spectrum of a field , is absolutely flat if and only if   := O , is a field and  →   is a separable algebraic extension for each  ∈ .In this case,  is an absolutely flat scheme.It follows that an absolutely flat morphism of schemes is universally incomparable.Recall that a morphism is étale if and only if it is absolutely flat and locally of finite presentation.
Proposition 42 (see [39,Corollary 12.90, page 359]).Let  :  →  be a proper morphism of schemes and let  be the set of  ∈  such that  −1 () is a finite set.Then  is open in  and the restriction  −1 () →  of  is a finite morphism.Corollary 43.A proper, incomparable, and surjective morphism of schemes  :  →  is finite.
Proof.Let  be a point of .Then   :=  −1 () is nonempty, quasicompact, and Noetherian and each of its points is closed.By [4, I, Proposition 2.8.1]   is finite.It follows that  = .
(1) ⇔ (3).Clearly an integral morphism verifies (3).Let  :  →  be a separated, universally closed, and incomparable morphism.Let  be a fiber, then  is quasicompact because  is quasicompact in view of Theorem 38.It follows that  is concentrated.As in the preceding corollary, each point of  is closed.Set  :=  red and let  be a point of , then   is a zero-dimensional reduced local ring, whence a field.By Proposition 20,  is an affine scheme.Now  is also an affine scheme by [36,Corollary 8.2].The conclusion is a consequence of (1) ⇔ (2).
We recover the result: if  →  is universally closed and injective, then  →  is integral [14-17, Corollaire 18.12.10,page 183], because such a morphism is affine, whence separated.
Consider two morphisms of schemes  :  → ,  :  → , such that  ∘  is universally closed and  is surjective.Then  is clearly universally closed.Moreover, if  is separated, it is well known that  is universally closed, whence quasicompact.We deduce from these observations the following result of Raynaud, obtained after a long proof [18, Lemme 3.2].Proposition 45.Let  :  → ,  :  →  be morphisms of schemes, such that  ∘  is integral,  is surjective, and  is separated.Then  is integral.
Replace the integral hypothesis on  ∘  with  ∘  is finite in the setting of the above proposition and suppose in addition that  and  are Noetherian.As the integral and the finite type properties are equivalent to the finite property, we see that  is finite by Theorem 35.
We end with descent results.

Pure Morphisms
Hashimoto proved the following result [40,Theorem 1] on pure ring morphisms, that is, universally injective ring morphisms.This result is also a consequence of Theorem 35.
Proposition 48.Let  be a Noetherian ring, then a pure morphism  →  of -algebras and of finite (presentation) type descends the property A of -algebras.
Proof.We note here that  is Noetherian and so is  because  = ∩ for each ideal  of .It follows that all the involved algebras are of finite presentation.
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 [41,42].We refer the reader to [41,42] for a definition of arbitrary pure morphisms of schemes.We will only consider quasicompact morphisms.
Schematically dominant morphisms of schemes  :  →  are involved.They are defined by O  →  ⋆ O  is injective [4, I, Section 5.4] and generalize the injective property of ring morphisms.A Stein morphism  :  →  (such that O  →  ⋆ O  is an isomorphism) is schematically dominant.
Let  :  →  be a concentrated morphism (quasicompact and quasiseparated).For such a morphism, it is well known that O  →  ⋆ O  defines a quasicoherent algebra.In view of [4, I, 9.1.21],there is an -morphism of schemes   :  →   , where   := Spec( ⋆ O  ) is the relative spectrum of  and   →  is an affine morphism.We can add that   is Stein, whence schematically dominant.
A schematically dominant morphism is dominant.If  is reduced, then  is schematically dominant if and only if  is dominant [4,I,Proposition 5.4.3].The main problem is that the schematically dominant property does not need to be universal.Actually, universally dominant morphisms define pure morphisms, at least for quasicompact morphisms.
Recall that a morphism of schemes is a regular epimorphism if  is a co-equalizer for the projections  1 ,  2 :  ×   → .
The converse is clear.
We give some examples of descent of the property A, by special pure morphisms that do not involve Noetherian properties Example 51. (1) We proved that if the -algebra  is a retract of the -algebra  then  →  descends P [43, Lemma 2.3].Note that in that case  →  is a pure ring morphism.
(2) Let  :  →  be a pure finite morphism of finite presentation.Then  →  descends A. To prove this, first observe that the trace ideal   () of the -module  is  by [44,Lemme 5.5].Then it is enough to apply [30, Corollary 2].
In particular, if  is a normal ring containing Q, then a finite morphism  →  of finite presentation descends A by [44,Proposition 5.7].The same result is valid if either  is Prüfer or  is an integrally closed domain [44, (2), (3) page 307].
(4) Let  →  be a universally subtrusive ring morphism, where  is a quasi-Prüfer domain; that is, its integral closure   is Prüfer.By using Theorem 50, we get that   →  ⊗    is pure and then a composite  →   →  ⊗    of  →  →  ⊗    , where the first morphism is integral injective and the second is pure.This result is similar to Theorem 4.
We conclude this section by a criterion for a flat morphism of schemes to be schematically dominant, as suggested in [ We prove the converse when  is concentrated.For  ∈ Ass(), consider the flat base change  := Spec(O , ) → .Then  ×   →  is concentrated, flat, and schematically dense.We thus can assume that  and Y are affine by using Lemma 14, in which case the result follows from [45, Chapter II, Proposition 3.3].
We end by recalling a result of Rydh, which is similar to Proposition 49.Note also that schematically dominant morphisms of -schemes are epimorphisms in the subcategory of separated -schemes, with a converse for concentrated morphisms (see [4, I, 5.4.6]).
Proposition 53 (see [3,Proposition 7.2]).Let  :  →  be a schematically dominant universally submersive morphism of schemes.Then  is an epimorphism in the category of schemes.
It follows that a universally submersive morphism  :  → , with  reduced is an epimorphism by [4, I, Proposition 5.4.3].

Monomorphisms of Schemes
The Nagata compactification theorem for schemes says that if  is a concentrated scheme (e.g., any Noetherian scheme) and if  :  →  is a separated morphism of finite type of schemes, then  fits into a factorization  =  ∘ , where  :  →  is an open immersion and  is proper (see [46]).Then  is called an -compactification of .Hence, surjective morphisms as above can be factored into an open immersion followed by a universally subtrusive morphism.This justifies that we look at immersions and monomorphisms beyond the fact that they are involved in proofs.
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 [18,Proposition 1.1].
Lemma 54.Let  :  →  be a morphism of schemes.Then  is a monomorphism of schemes (resp., a flat monomorphism) if and only if  is injective and the ring morphism O ,() → O , is a ring epimorphism (resp., isomorphism) for each  ∈ .
Let  :  →  be a morphism of schemes and denote by Λ  the class of all pairs of morphisms  1 ,  2 :  →  equalizing .Then  is called a strict monomorphism if any morphism  :  →  such that Λ  ⊆ Λ  can be uniquely factored  =  ∘   .Note that the monomorphism property and the strict monomorphism property are universal.
For instance, let  :  →  be an open immersion of schemes , then  is a strict monomorphism; actually  is an equalizer (a kernel) of the double arrow  →  ∪  , where  ∪   is the standard gluing of  and  along  [47,Example 3.1].
Olivier calls immersive a morphism of schemes which is a composite of strict monomorphisms [48].We have the following results.
Definition 57.Let  :  →  be a ring morphism.Then  is called essential if one of the following equivalent conditions is verified.
(3) For each ring morphism  →  such that  →  is injective, then  →  is injective.
An injective integral ring morphism whose target is an integral domain is essential.A flat epimorphism is essential by [45, IV, Proposition 2.1].
Definition 58.Let  :  →  be a morphism of schemes.We say the following.
(1)  is topologically essential if for each nonempty open subset  of , there is an open subset  of  such that 0 ̸ =  −1 () ⊆ .
(2)  is a topological immersion if  is injective and if the inverse image topology on  with respect to  coincide with the topology of , or equivalently,  defines a homeomorphism  → ().
Clearly, a topological immersion is topologically essential.We also note that the notion of topological immersion is dual to the notion of submersive morphism.
Proposition 59.Let  :  →  be a morphism of schemes.
The following statements are equivalent.
If, in addition,  is quasicompact, then (1) is equivalent to Remark 60.Let  : Spec() → Spec() be a morphism of affine schemes, corresponding to a ring morphism  :  → .
(1) Then  is topologically essential if and only if for each  ∈  \ Nil(), there is some element  ∈  \  −1 (Nil()) such that () ∈ .Assume that  is topologically essential, then so is  red and we see that

Definition 16 .
A universally injective ring morphism is called pure.A pure ring morphism is universally subtrusive [2, page 535].A faithfully flat ring morphism is pure.

The following result generalizes [ 34 ,Corollary 40 .
Proposition 3.18].Let  :  →  be a universally closed and quasiseparated morphism of schemes, where  = Spec() is an affine scheme.Then  :  → O  () is an integral ring morphism.Proof.By the above proposition,  can be factored  ∘   , where  is integral and   is a Stein morphism.In view of[35, Proposition 6.1.4], → O   (  ) is integral and then the proof is complete since O  () = O   (  ).Proposition 41.Let  :  →  be a separated and universally closed morphism of schemes.Let  :=   be the fiber of  for some  ∈ ,  := O  () and  :  → Spec() the canonical morphism.

( 1 )
If  is a quasicompact flat monomorphism of schemes, then  is a strict monomorphism, which is quasi-finite and of finite presentation.(2) If  is (affine) quasi-local with closed point  such that  −1 () ̸ = 0, then the following statements are equivalent: (a)  is immersive; (b)  is affine and O  () → O  () is surjective; (c)  is a strict monomorphism of schemes.
(1) If  is locally Noetherian, then it is enough to consider discrete valuation rings in(1), and f is universally subtrusive if and only if f is universally submersive.Corollary 8. Let  :  →  be a quasicompact morphism of schemes.Then the following statements are equivalent.
1) Each morphism   →  is concentrated and a flat monomorphism of finite presentation.(2) The canonical morphism  :  →  is concentrated, of finite presentation, faithfully flat and  is an affine scheme.Proof.(1) We can use the above remark on concentrated schemes.We give a direct proof.Each   →  is an open immersion, whence a flat monomorphism locally of finite presentation by Proposition 13.Moreover,   is proconstructible, whence retrocompact [4, I, Proposition 7.2.3(ix),(v)].It follows that   →  is quasicompact and therefore of finite presentation.(2) Clearly,  is faithfully flat and of finite presentation by [4, I, Proposition 6.3.11].Now  is concentrated because of [4, I, Proposition 6.1.15].
[20,llaire 1.5],  → / is pure and flat because  is Prüfer and  → / is torsion-free.Therefore,  → / is faithfully flat and  → / of finite type.It follows from[20, I, Corollaire 3.4.7]that→/ is of finite presentation and  → / of finite type.Therefore,  →  is of finite type.The following result is well known.A proof may be found in the Columbia Stack project [13, Proposition 25.2] included in a stronger result.Proposition 18 (Theorem of generic flatness).Let  :  →  be a morphism of schemes of finite type, such that  is reduced.Then there exists a dense open subscheme  of  such that   :=  ×   →  is flat and of finite presentation.Let  :  →  be an -morphism of schemes of finite type, surjective, and such that  is reduced and concentrated.If  →  is of finite type (resp., of finite presentation), there is a nonempty affine open subset  of  such that  →  →  is of finite type (resp., of finite presentation).
[23,osition22(see[23, Lemme 8.6]).Let  :  →  be a quasicompact monomorphism of schemes.If  is absolutely flat, then  is a flat closed immersion.Let  →  →  be scheme morphisms and   →  a scheme morphism of finite type, where   is absolutely flat.If  →  is surjective and of finite presentation and  →  of finite type, then   →  →  is of finite type.Proof.Observe that  ×    →   is faithfully flat of finite presentation and that  ×    →  is of finite type.
Lemma 15.2].Let  :  → Spec() be a surjective morphism of schemes of finite type and  :  →  a ring morphism, such that  → Spec() is of finite type (resp., of finite presentation).Then  →  → / is of finite type (resp., of finite presentation) for each maximal ideal  of .In view of[13,Lemma 15.2], if  →  → / is of finite type and (, ) is Artinian local, then  →  is of finite type.Remark 27.Note that if  is a subring of  and [ 1 , . . .,   ] = [ 1 , . . .,   ], then  →  is of finite presentation because  → [ 1 , . . .,   ] is faithfully flat of finite presentation.This is an application to the nonunicity of the coefficient ring of a polynomial ring.
Let Spec() = ∪  Spec   be the irreducible decomposition.Then  is finitely generated over  if and only if each   is finitely generated over .
)  →  verifies A if and only if   →   verifies A for each maximal ideal (resp., prime ideal )  of .Corollary 29.Let  be a Noetherian domain and  an subalgebra of a finitely generated overdomain of .Then the statements (1) and (2) of Theorem 28 hold.(1)  is finitely generated over  if and only if  red is finitely generated over .
algebra.We consider below its relative spectrum Spec  (   is the normalization of  in , (6) if  →  is locally of finite type, then   →  is finite.In case  is proper then   is proper with geometrically connected fibers.
(3),first step is done by considering [41, Proposition 3.12], which tells us Algebra that an arbitrary  is pure if and only if there is an affine open cover (  ) ∈ of  such that  −1 (  ) →   is pure.As  is quasicompact, so are each  −1 (  ) →   and then the result is a consequence of [41, Theorem 5.12 (xi) ⇔ (xii)].The rest is[42, Theorem 6.5].We observe that a composite of quasicompact pure morphisms is pure by Proposition 49(3).If a composite  ∘  of quasicompact morphisms of schemes is pure, so is  [42, Corollary 6.2, Theorem 6.5].A quasicompact faithfully flat morphism of schemes is pure[41, Remark 3.13].A quasicompact pure morphism of schemes  :  →  is surjective.To see this, use a base change () →  for  ∈  and Proposition 49(3).The following result extends to schemes [2, Théorème 37] and add a result to Example 2.Theorem 50.A quasicompact  :  →  morphism of schemes is universally subtrusive if and only if  ×  Spec() → Spec() is a pure morphism of schemes, for each morphism of schemes Spec() → , where  is a valuation domain.In particular, a quasicompact pure morphism of schemes is universally subtrusive.
45, Chapter II, Remarques 3.4].It generalizes the fact that a quasicompact faithfully flat morphism is schematically dominant.We say that a point  of a scheme  is associated with  if the maximal ideal m  of O , is the radical of some annihilator 0 : , where  ∈ O , .This means that m  is a weak Bourbaki associated prime ideal of O , (see for instance, [45, Chapter II]), or equivalently, O , is an autoassociated quasi-local ring.We denote by Ass() the set of all associated elements of .Let  :  →  be a flat morphism of schemes.Then  is schematically dominant if Ass () ⊆ ().The converse holds if in addition  is concentrated.Proof.Clearly, if  is an open subscheme of , then Ass() =  ∩ Ass().As the ring morphism F() → ∏ ∈ F  is injective for every open subset  of a space  and sheaf F on , we claim that O  () → ∏ ∈Ass() O , is injective.Indeed, for an -module  over a ring , we have that  → ∏ ∈Ass()   is injective.It follows then that a flat morphism of schemes  :  →  is schematically dominant if Ass() ⊆ (), because O ,() → O , is faithfully flat.
red →  red is essential.The converse holds if  is dominant.(2) in [2, Proposition 20, page 545],  is called universally essential if and only if  can be factored  ∘ , where  is a flat epimorphism and  is surjective, or also if and only if   →   is surjective for each  ∈ Spec(), such that  −1 () ̸ = 0, or also, if and only if  is immersive.A continuous map of topological spaces  :  →  is called minimal if () ⊂ () for each closed subset  ⊂ .This definition is a weakening of injectiveness.