Research Article | Open Access
Gabriel Picavet, "Recent Progress on Submersions: A Survey and New Properties", Algebra, vol. 2013, Article ID 128064, 14 pages, 2013. https://doi.org/10.1155/2013/128064
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.
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 is open (closed). They are also called topological epimorphisms by some authors like Voevodsky who defines and uses the and -(Grothendieck) topologies . 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  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 . 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].
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 . (2) is submersive in the constructible topology.
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 (iv)] which tells us that if is quasicompact, then is closed for the patch topology and [4, I, Corollaire ] which states that for a proconstructible subset .
Example 2 (see [3, Remark 2.5]). Let be a quasicompact surjective morphism of schemes. Then is universally subtrusive in the following cases.(1) is universally specializing (closed). (2) is proper. (3) is integral. (4) is universally generalizing. (5) is faithfully flat. (6) is universally open.
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 ]). 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 is injective integral (equivalently, finite), then is of finite type and is Noetherian. To see this, let be a system of generators of over and let be the -algebra generated by the coefficients of unitary polynomials , such that . 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 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 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.
Proposition 6. Let be a valuation ring and a morphism of schemes. The following statements are equivalent.(1)is universally subtrusive. (2)is subtrusive. (3)The pair in lifts to in . (4)Any chain of points in 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.
Theorem 7. Let be a quasicompact morphism. (1) is universally subtrusive (resp., submersive) if and only if, for any valuation ring and morphism with , the pull-back is subtrusive (resp., submersive). (2)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) is universally subtrusive.(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.
In order to ease reading, we introduce the following definition.
Definition 9. Let be a morphism of schemes. We say that descends the property (resp., property ) 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 (resp., ) if descends and is of finite type (resp., descends 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 of morphisms of schemes if for each morphism such that has , then has . In order to avoid confusions in this paper, we say that such a morphism D-descends the property (D for diagram).
Remark 10. In case is of finite type and is either quasiseparated or is Noetherian, then is of finite type (see [4, I, Proposition ]).
If descends , is of finite presentation and is quasiseparated, then is of finite presentation (see [4, I, Proposition ]).
The classes and are stable under right division and composition. In particular, if is a ring morphism, descends () for each ideal of if does.
We examine a converse. Suppose that descends and that is a nilpotent ideal. Then descends by [10, Lemma 4]. The nilpotent condition on is verified in two cases.(a)If is of finite presentation, surjective on the spectra and descends the property , then is a nilpotent ideal, because and is an ideal of of finite type by [4, I, Remarque 220.127.116.11], since is of finite presentation.(b)A ring morphism is called a strong Nakayama morphism if for each nonzero -module . A strong Nakayama morphism D-descends the finite type and finite properties of ring morphisms [11, Proposition 2.6, page 30]. Morphisms that D-descend the finite presentation property of ring morphisms are Nakayama morphisms such that is a Noetherian -module contained in , by the two lemmas of [11, page 31] and since D-descends the finite presentation property.(c)Suppose that a morphism of schemes universally descends the property and is of finite type, then universally D-descends the property . To see this consider the pull-back defined by the maps and . An example is given by a faithfully flat morphism of finite presentation by Theorem 11.
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 and . We derive some results from this case. In particular, we prove our main result for universally subtrusive -morphisms of schemes , where is a valuation domain. 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 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 . 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 is an isomorphism. We give Stein factorization results published in  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.
2. 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 12 (see [18, Proposition 1.1]). Let be a scheme morphism. The following statements are equivalent.(1)is a flat monomorphism.(2) is injective and is an isomorphism for each .
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. (3) in [ 14, 15, 16, 17, Proposition ], an open immersion is a flat monomorphism locally of finite presentation. In case is quasicompact, then is of finite presentation.
Let be a scheme and , then the natural map is a flat monomorphism with image , the set of all generalizations of .
The following lemma is useful to reduce proofs to affine schemes.
Lemma 14. Let be a concentrated scheme and be an affine open covering of . Set , the following statements hold.(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. 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 (ix),(v)]. It follows that is quasicompact and therefore of finite presentation.
Clearly, is faithfully flat and of finite presentation by [4, I, Proposition ]. Now is concentrated because of [4, I, Proposition ].
Proposition 15. Let be a valuation domain, a ring morphism, a quasiseparated scheme, and a universally subtrusive morphism of schemes of finite type, such that is of finite type (of finite presentation). Then is of finite type (of finite presentation).
Proof. In view of [3, Proposition 2.7(vii)], there is a closed subscheme of such that is faithfully flat. Then is of finite type and so is . Moreover, a closed immersion is quasicompact and separated, whence 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 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.
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.
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 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 the torsion ideal of . Then by [11, Corollaire 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.
Corollary 19. 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).
In case and 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 ⊆.
In case and are affine schemes where is reduced in Proposition 18, we see that is such that ; 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 (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 , the structural morphism . Since an affine morphism is quasicompact and separated, we see that is concentrated for an arbitrary scheme .
Proposition 20 (see [23, Lemma 8.4, Proposition 8.5]). Let be a ringed space, such that is a field for each . (1)is absolutely flat. (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 is an affine scheme.
Proposition 22 (see [23, Lemme 8.6]). Let be a quasicompact monomorphism of schemes. If is absolutely flat, then is a flat closed immersion.
Proposition 23. 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.
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, is of finite type or, equivalently for some regular ) and suppose that 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 is of finite type because 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 is absolutely flat if and only if 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 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 the set of all Goldman points of . Observe that is strongly dense in by [13, Lemma 15.6] and [4, Section ]. 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 , then is of finite type (resp., of finite presentation).
Proof. Similar to the proof of Remark 24.
We can factorize as if , where the first morphism is a closed immersion and the second is a quasicompact flat monomorphism. In case is Artinian and the above hypotheses hold, is of finite type, by [13, Lemma 15.2].
Corollary 26. Let be a surjective morphism of schemes of finite type and a ring morphism, such that 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 , then is of finite presentation because is faithfully flat of finite presentation. This is an application to the nonunicity of the coefficient ring of a polynomial ring.
3. 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 , then there is some such that verifies .(2) verifies if and only if verifies 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 and of Theorem 28 hold.
Proof. In view of [26, Proposition ], there is some such that is of finite type.
Lemma 30. Let be a ring morphism between Noetherian rings.(1) is finitely generated over if and only if is finitely generated over .(2)Let be the irreducible decomposition. Then is finitely generated over if and only if each is finitely generated over .
Next result is decisive.
Proposition 31 (see [29, Theorem 4.2]). Let be an excellent ring, and a surjective proper morphism of -schemes. If is of finite type over and is a Noetherian affine scheme, then is of finite type over .
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 .(3)Each irreducible component of dominates .(4) is normal and Noetherian.
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 .
Corollary 34. Let be a Noetherian ring and a surjective proper morphism of -schemes, where is an affine Noetherian scheme. If is of finite type over , so is for each prime ideal of .
Proof. For each , consider the base change , where 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 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 and are affine schemes with a ring morphism . Using the base changes 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 because 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 is surjective and suppose that verifies . 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 . The reader will find much more examples in a paper by Papick . 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 .
4. 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 . A version of the solution appears also in the Stacks Project [13, Lemma ].
Theorem 38. A universally closed morphism of schemes is quasicompact. If, in addition, is also surjective, then it is universally subtrusive.
Proof. Without loss of generality, we may assume that 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 , where the are distinct indeterminates. Let . Let be the closed set . It suffices to prove that the image of under is not closed. There exists a point 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 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 , some coefficient of would have nonzero image in , and hence be invertible on some neighborhood of . Let be the finite set of 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 and for all . Then for each . A point of mapping to and to then belongs to . But , 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  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, 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 is a quasicoherent -algebra. We consider below its relative spectrum over (see [4, I, Section 9]).
Proposition 39 (Stein Factorization [33, Lemmas 29.1 and 29.5]). Let be a scheme. Let be a universally closed (whence quasicompact) and quasiseparated morphism of schemes. There exists a factorization of where and with the following properties: (1)the morphism is universally closed, quasicompact, quasiseparated, and surjective, (2)the morphism is integral whence affine, (3)one has ; that is, is a Stein morphism, (4)one has , (5) 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.
The following result generalizes [34, Proposition 3.18].
Corollary 40. Let be a universally closed and quasiseparated morphism of schemes, where is an affine scheme. Then 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 ], is integral and then the proof is complete since .
Proposition 41. Let be a separated and universally closed morphism of schemes. Let be the fiber of for some , and the canonical morphism. (1) has finite dimension. (2) is integral over , whence is zero-dimensional.(3) is concentrated and the canonical map is an isomorphism, where each for is open and closed.
Proof. It is enough to use [36, Corollary 8.4] because a closed immersion is finite and the diagonal of is a closed immersion.
Use Corollary 40.
By [4, Corollaire ], 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 is of the form where is an idempotent and is a unit. It follows that is open and closed because so is in .
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 . 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 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 is a finite set. Then is open in and the restriction of is a finite morphism.
Corollary 43. A proper, incomparable, and surjective morphism of schemes is finite.
Proof. Let be a point of . Then is nonempty, quasicompact, and Noetherian and each of its points is closed. By [4, I, Proposition ] is finite. It follows that .
Theorem 44. Let be a separated morphism of schemes. The following statements are equivalent: (1) is integral, (2) is universally closed and has affine fibers [36, Theorem 8.5], (3) is universally closed and incomparable.
Proof. (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 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).
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.
Proof. Use Theorem 44(1) (3) and the fact that is quasicompact and then universally subtrusive.
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.
Proposition 46. Let be a universally closed surjective morphism of -schemes. If is quasiseparated (separated), then so is .
Proof. The morphism is universally closed by [4, I, Proposition ], whence quasicompact. From the equation , where the and are the diagonal morphisms of and , we deduce easily that the quasicompactness of implies the quasicompactness of because is surjective. So the quasiseparated case is proved. The proof of the separated case is similar, since is separated if and only if is closed.
Note that the quasiseparated case needs only that be surjective and quasicompact.
Corollary 47. A universally closed morphism is concentrated if is quasiseparated.
Proof. Use [4, I, Corollaire 6].
5. Pure Morphisms
Proposition 48. Let be a Noetherian ring, then a pure morphism of -algebras and of finite (presentation) type descends the property 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 is injective [4, I, Section 5.4] and generalize the injective property of ring morphisms. A Stein morphism (such that is an isomorphism) is schematically dominant.
Let be a concentrated morphism (quasicompact and quasiseparated). For such a morphism, it is well known that defines a quasicoherent algebra. In view of [4, I, 9.1.21], there is an -morphism of schemes , where 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 ]. 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 .
Proposition 49. Let be a quasicompact morphism of schemes. The following statements are equivalent: (1) is pure; (2)there is an open affine covering of such that is a pure ring morphism for each ; (3)is universally schematically dominant; (4)is universally a regular epimorphism.
Proof. We first show that (1) is equivalent to (2). A first step is done by considering [41, Proposition 3.12], which tells us that an arbitrary is pure if and only if there is an affine open cover of such that is pure. As is quasicompact, so are each 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 is a pure morphism of schemes, for each morphism of schemes , where is a valuation domain. In particular, a quasicompact pure morphism of schemes is universally subtrusive.
Proof. Assume that is universally subtrusive. By Corollary 8, for each morphism , where is a valuation domain, there is a surjective affine morphism of schemes inducing , where is a valuation domain. We deduce from [2, Proposition 16((1) (3))] that is pure and so is that factorizes .
The converse is clear.
We give some examples of descent of the property , by special pure morphisms that do not involve Noetherian properties
Example 51. We proved that if the -algebra is a retract of the -algebra then descends [43, Lemma 2.3]. Note that in that case is a pure ring morphism.
Let be a pure finite morphism of finite presentation. Then descends . 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 , then a finite morphism of finite presentation descends 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].
Let be a pure ring morphism and a morphism of -algebras. Then descends , () if descends , (). The proof is a consequence of [11, Chapitre I, Proposition 5.3] and a remark of [11, page 22].
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 [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 of is the radical of some annihilator , where . This means that is a weak Bourbaki associated prime ideal of (see for instance, [45, Chapter II]), or equivalently, is an auto-associated quasi-local ring. We denote by the set of all associated elements of .
Proposition 52. Let be a flat morphism of schemes. Then is schematically dominant if . The converse holds if in addition is concentrated.
Proof. Clearly, if is an open subscheme of , then . As the ring morphism is injective for every open subset of a space and sheaf on , we claim that is injective. Indeed, for an -module over a ring , we have that is injective. It follows then that a flat morphism of schemes is schematically dominant if , because is faithfully flat.
We prove the converse when is concentrated. For , consider the flat base change . Then is concentrated, flat, and schematically dense. We thus can assume that and 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 ].
6. 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 ). 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 is a ring epimorphism (resp., isomorphism) for each .
Let be a morphism of schemes and denote by the class of all pairs of morphisms equalizing . Then is called a strict monomorphism if any morphism such that 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 . We have the following results.
Proposition 55 (see [48, Lemme, page 2, Lemme, page 4]). Let be a morphism of schemes. (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 , then the following statements are equivalent:(a)is immersive;(b)is affine and is surjective;(c)is a strict monomorphism of schemes.
Lemma 56 (see [45, Lemme 1.6]). Let and be two morphisms of schemes such that is separated, is a quasicompact flat monomorphism of schemes and is schematically dominant, then is a quasicompact flat monomorphism.
Definition 57. Let be a ring morphism. Then is called essential if one of the following equivalent conditions is verified. (1) For each in , there is some such that with . (2), for each ideal of . (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 . (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. (1)is a topological immersion. (2)is injective and for each subset of .
If, in addition, is quasicompact, then is equivalent to (3)if , then .
If is a topological immersion then identifies to for each .
Proof. It is enough to rework the proof of [7, Proposition 3.20], taking into account that is proconstructible whenever is a proconstructible subset of and is quasicompact. The end of the proof follows from [4, 0, 3.4.5].
An immersion of schemes is a topological immersion by [4, Section I, 4.2], with locally closed, open, or closed according to whether is an immersion, an open immersion, or a closed immersion. A quasicompact flat monomorphism is also a topological immersion by [3, Proposition 1.2].
Remark 60. Let be a morphism of affine schemes, corresponding to a ring morphism . (1) Then is topologically essential if and only if for each , there is some element such that . Assume that is topologically essential, then so is and we see that 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 , such that , 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.
Proposition 61. Let be a dominant morphism of schemes. (1)If is topologically essential, then is a minimal map. (2)If is closed and minimal, then is topologically essential.
Proof. We show (1). Let be a closed subset of , with . Then is a nonempty open subset. Since is topologically essential, let be an open subset of such that . We get that with , so that . This shows that is minimal. Next we prove (2). Let be an open subset of , then is a closed subset that we can suppose different from . It follows that is a closed subset of , different from . There exists an open subset of such that , so that . As is surjective, shows that .
If is schematically dominant, then is clearly schematically dominant. We define a schematically essential morphism of schemes as a morphism of schemes such that for each morphism of schemes with schematically dominant, then is schematically dominant. Recall that a morphism is dominant if maximal points of can be lifted up to . The converse holds if is quasicompact.
Proposition 62. Let be a quasicompact morphism of reduced schemes which is a topological immersion, then is schematically essential.
Proof. We can replace the schematically dominant hypothesis by dominant. Then the result follows from Proposition 59(3). To see this, consider a morphism of schemes , such that is dominant. Let be a maximal point and let be a maximal point such that . There is some such that , so that . Since is a topological immersion, we get that and finally, .
We address the following question. Is the reduced hypothesis necessary in the above proposition? By the way, we have the following result.
Proposition 63. A quasicompact flat monomorphism is schematically essential on concentrated flat morphism and on closed immersions.
Proof. First observe that under the hypotheses on , the quasi-local ring is auto-associated if is auto-associated. Indeed, by Lemma 54, and are isomorphic. To conclude, use the fact that a flat concentrated morphism is schematically dominant if and only if by Proposition 52. The case of closed immersions is [19, Lemme 1.5].
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 is schematically dominant, with concentrated and a flat monomorphism of affine schemes. We can reduce to the affine case by using a Stein factorization of , with affine. In that case the result holds [45, IV, Proposition 2.1].
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 is called quasiaffine if there exists a covering of by affine open subsets such that is quasiaffine [35, Section 5]. If is a concentrated scheme with ring of global sections , then is quasiaffine if and only if is an open immersion. In that case, is dominant and quasicompact, whence a flat monomorphism of finite presentation.
The following may be useful.
Remark 64. Let be a morphism between quasiaffine schemes and set , . Then is of finite type if is a ring morphism of finite type. Indeed, consider the commutative diagram:
We observe that is of finite type, is quasiseparated, and is of finite type. Then is of finite type. To see this, consider the composite in the above diagram.
Theorem 65. Let be a morphism of schemes locally of finite presentation. If is quasi-finite and separated, then is quasiaffine. In particular, a quasicompact flat monomorphism is quasiaffine.
Proposition 66. Let be a morphism of schemes. (1)If is a monomorphism of finite presentation, then is quasiaffine.(2)If is a morphism of -schemes where is locally of finite type over and quasiseparated, then the graph morphism is a quasiaffine monomorphism.(3)If is locally of finite type and quasiseparated, then the diagonal morphism is quasiaffine.
Proof. It is enough to observe that a monomorphism is separated and use the above result.
is a monomorphism since it factors . Then is locally of finite type and quasiseparated. In view of [4, Proposition (v)], we get that is of finite presentation because . Hence, is a quasiaffine monomorphism by (1). And (3) follows from (2).
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