On relative birational geometry and Nagata’s compactification

On relative birational geometry and Nagata’s compactification for algebraic spaces

Michael Temkin, Ilya Tyomkin Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Giv’at Ram, Jerusalem, 91904, Israel temkin@math.huji.ac.il Department of Mathematics, Ben-Gurion University of the Negev, P.O.Box 653, Be’er Sheva, 84105, Israel tyomkin@math.bgu.ac.il
Abstract.

In [Tem11], the first author introduced (relative) Riemann-Zariski spaces corresponding to a morphism of schemes and established their basic properties. In this paper we clarify that theory and extend it to morphisms between algebraic spaces. As an application, a new proof of Nagata’s compactification theorem for algebraic spaces is obtained.

Key words and phrases:
Riemann-Zariski spaces, algebraic spaces, Nagata compactification.
Both authors were supported by the Israel Science Foundation (grant No. 1018/11). The second author was also partially supported by the European FP7 IRG grant 248826.

1. Introduction

In [Tem11], the first author introduced relative Riemann-Zariski spaces associated to a separated morphism of schemes, established their basic properties, and obtained several applications such as a strong version of stable modification theorem for relative curves, and a theorem about factorization of separated morphisms, which generalizes Nagata’s compactification theorem [Nag63]. The aim of the current paper is to extend the methods and the results of [Tem11] to algebraic spaces, and thereby, prepare the ground for further generalizations, e.g., the equivariant case and the case of stacks.

In this paper, we introduce the language of models, which makes the method more intuitive. Its choice is motivated by relations between classical birational geometry, Raynaud’s theory, and, what we call, relative birational geometry. The results in this paper include (i) a new proof of Nagata’s compactification theorem for algebraic spaces and of a stronger result on factorization of separated morphisms of algebraic spaces, (ii) a valuative description of Riemann-Zariski spaces associated to a separated morphism of algebraic spaces, (iii) a valuative criterion for schematization, and (iv) an application of our theory to the study of Prüfer morphisms and Prüfer pairs of qcqs algebraic spaces.

1.1. Relative birational geometry

Relative birational geometry is the central topic of the current paper. In this section, we explain the concept of relative birational geometry, the motivation for its study, and the relations with the classical birational and non-archimedean geometries.

1.1.1. The absolute case

Classical birational geometry studies algebraic varieties up-to birational equivalence, and, in particular, addresses questions about (i) birational invariants, and (ii) existence of “good” representatives of birational classes. More precisely, birational geometry studies the localization procedure of the category of integral schemes with dominant morphisms with respect to proper birational morphisms, and the birational problems can be roughly divided into two classes: the study of the localized category, including (i), and the study of the fibers of the localization, including (ii), weak factorization problem, Chow’s Lemma, etc.

1.1.2. Birational geometry of pairs

Often, one is only interested in modifications of a scheme that preserve a certain (pro-)open subscheme. For example, in the desingularization problems, one usually wants to preserve the regular locus of . In this case, it is more natural to consider the category of pairs , where is pro-open, quasi-compact, and schematically dense; and to study the localization functor that inverts -modifications, i.e., morphisms such that is proper and . We call the latter study birational geometry of pairs, and the classical birational geometry is obtained when is a point.

1.1.3. The general relative case

It makes sense to study birational geometry of morphisms that are not necessarily pro-open immersions. For this aim, we introduce the language of models (see §1.1.6 for some analogies explaining the terminology). A model is a triple such that and are quasi-compact and quasi-separated (qcqs) algebraic spaces, and is a separated schematically dominant morphism. If is an open immersion (resp. affine morphism) then we say that is good (resp. affine). In particular, a good model can be viewed as a pair . A morphism of models is a pair of morphisms and such that . If is an isomorphism and is proper then we say that is a modification. The relative birational geometry studies localization of the category of models with respect to the family of modifications.

1.1.4. Factorization and Nagata’s compactification theorems

Nagata’s compactification theorem is a classical result first proved by Nagata [Nag63] in 1963. Since then few other proofs were found, see [Con07, Lüt93, Tem11]. Recently, it has been generalized to the case of algebraic spaces by Conrad, Lieblich, and Olsson following some ideas of Gabber [CLO12]. Note also that in a work in progress [Ryd], Rydh extends the compactification theorem to tame morphisms between Deligne-Mumford stacks.

In our study of models up to modifications, one of the main results is Theorem 4.3.4 asserting, in particular, that any model possesses a modification such that is affine. This is equivalent to the Factorization theorem asserting that is a composition of an affine morphism and a proper morphism . Since any affine morphism of finite type is quasi-projective, if is of finite type then above can be chosen to be good. In particular, is an -compactification of , and we obtain a new proof of Nagata’s compactification theorem for algebraic spaces. This proves a posteriori that the birational geometry of pairs contains the birational geometry of finite type models - a deep fact equivalent to Nagata’s compactification.

1.1.5. Formal birational geometry

Our discussion of birational geometry would not be complete without the formal case. Let be a complete height-one valued field with ring of integers . To any admissible (i.e., finitely presented and flat) formal -scheme one can associate a non-archimedean analytic space over called the generic fiber. There are three theories of such analytic spaces: Tate’s rigid spaces, Berkovich’s -analytic spaces, and Huber’s adic spaces, and can be defined in each of them. Any admissible blow up, i.e., a formal blow up along an open ideal, induces an isomorphism of generic fibers. Raynaud proved that the category of qcqs rigid spaces over is the localization of the category of admissible formal -schemes with respect to the family of all admissible blow ups (see [BL93]). A part of this result is the following version of Chow’s lemma: admissible blow ups of are cofinal among all admissible modifications, i.e., proper morphisms inducing an isomorphism of the generic fibers. In particular, Raynaud’s theory provides a way to treat analytic geometry over as the birational geometry of admissible formal -scheme. This approach had major contributions to the non-archimedean geometry.

1.1.6. Analogies

In classical birational geometry, a (proper) -model of a finitely generated -field is a representative of the birational equivalence class of . Similarly, in formal geometry one says that is a formal model of the non-archimedean space . In the same way, one can view a model as a representative of the relative birational equivalence class defined by . The analogy with the formal case is especially fruitful, and some of our arguments are motivated by their analogues in Raynaud’s theory (see also [Tem11]). Note, however, that the analogy becomes tight only when addressing good models.

1.2. Birational spaces

In order to study the fibers of the localization functor, it is useful to have nice invariants associated to the fibers. In this paper we consider several relative birational spaces - topological spaces associated to a model in a functorial way that depend on the model only up to modifications. In the absolute case the construction of such spaces goes back to Zariski.

1.2.1. The classical case

For a finitely generated field extension Zariski defined the Riemann manifold to be the set of all valuation rings such that , see [Zar40, A.II.5]. Zariski equipped with a natural quasi-compact topology, and proved that it is homeomorphic to the filtered limit of all projective -models of .

1.2.2. The formal case

A well known fact that appeared first in a letter of Deligne says that if is an admissible formal scheme then the filtered limit of all its admissible modifications (or blow ups) is homeomorphic to the adic generic fiber (and other generic fibers and are its natural subsets). In particular, any finite type adic space over is homeomorphic to the filtered limit of all its formal models.

1.2.3. The absolute Riemann-Zariski spaces

Zariski’s construction easily extends from projective varieties to arbitrary integral schemes, see [Tem10, §3.2]. The Riemann-Zariski space of such a scheme is the filtered limit of the family of all modifications of in the category of locally ringed topological spaces. In particular, is a birational space, i.e., it only depends on up to modifications. Such spaces play an important role in desingularization problems, in some proofs of Nagata’s compactification theorem (including Nagata’s original proof), and in other problems of birational geometry.

Similarly to Zariski’s definition, there exists a valuative description of points of . Let be the set of pairs consisting of a valuation ring of and a morphism extending the isomorphism of the generic points. It admits a natural structure of a locally ringed space. The valuative criterion of properness gives rise to a natural map , which turns out to be an isomorphism.

1.2.4. The relative Riemann-Zariski spaces

The above absolute construction was extended in [Tem10, §3.3] and [Tem11] to morphisms of schemes, i.e., to models in the category of schemes, and in this paper we generalize it further to models of algebraic spaces. We define to be the filtered limit of the topological spaces over the family of all modifications . Unlike the scheme case in [Tem11], we do not equip with any further structure, see Remark 2.2.9. Obviously, is a relative birational space, as it depends on only up to a modification.

The situation with valuative interpretation is subtler. We introduced in [TT13b, §5] certain valuative diagrams in the category of algebraic spaces that generalize Zariski’s valuations , and used them to refine the usual valuative criterion of properness for algebraic spaces. Such diagrams are called semivaluations in this paper (see §3.1.3), and the set of all semivaluations of is denoted by . It admits two natural topologies, called Zariski topology and constructible topology. The space is easy to work with, but, unfortunately, it is much bigger than , and hence only plays a technical role. However, there exists a class of “nice” semivaluations, called adic (see §3.1.5), and we denote the subspace of adic semivaluations by . Although, the space is more difficult to approach, it turns out to be the “right” space to consider.

It follows from the valuative criteria of [TT13b] that both and depend on only up to modifications. The spaces play central role in our study of birational geometry of models. In particular, we will often work locally around points of . As in the classical case, there exist natural restriction maps and , and we prove in Theorem 5.2.4 that is a homeomorphism.

1.2.5. Categorifications

As a future project, it would be desirable to provide relative birational spaces with enough structure in order to obtain a geometric realization of birational geometry. This should lead to an independent definition of the category of birational spaces, equivalent to the category of models localized by the modifications. For comparison we note that in the formal case, this program is fully realized by (one of) the definitions of non-archimedean spaces and Raynaud’s theory, in the classical case, this is done by the definition of the category in [Tem00, §1] and [Tem00, Prop. 1.4], and a geometric realization of birational geometry of schemes is constructed by U. Brezner in [Bre13].

1.3. Overview of the paper

1.3.1. Models

In Section 2, we introduce the category of models of algebraic spaces, and study basic properties of models. In particular, we define pseudo-modifications, quasi-modifications, modifications, and blow ups of models; and prove that they are preserved by compositions and base changes, and form filtered families. Here we only note that is a pseudo-modification if is an open immersion and is separated and of finite type, and it is a quasi-modification if it is also adic, i.e., the immersion is closed. Quasi-modifications provide the right tool to localize modifications, and our main task will be to construct large enough families of pseudo-modifications, quasi-modifications, and modifications.

1.3.2. Relative birational spaces

In Section 3, we associate to a model the birational spaces , , and . Then we show that these spaces are quasi-compact, , and construct natural maps whose continuity, excluding , is easily established:

Note that the topology of is generated by the subsets for compactifiable quasi-modifications , i.e., is an open submodel of a modification of , while the topology of is generated by the subsets for pseudo-modifications . In particular, the proof of the property in Theorem 3.3.5 reduces to a construction of a pseudo-modification that separates semivaluations. The construction involves two pushouts (affine and Ferrand) and an approximation argument. The reason that we have to use the affine pushout is quite technical: in general there is no localization of an algebraic space at a point.

At this stage we provide with the topology induced from . In fact, the natural topology on is generated by quasi-modifications because only when is a quasi-modification of (Lemma 3.2.2), but this topology is useless at this stage as we do not even know yet that there are enough quasi-modifications to distinguish points of . To begin with, we only need to use quasi-compactness of , so it suffices to work with the (a priori) stronger pseudo-modification topology.

1.3.3. The family of modifications

Theorem 4.3.4 is our main result about the family of modifications of a model. It asserts that affine models are cofinal among all modifications of , and if is good, then the family of blow ups of is cofinal among all its modifications. As was noted in §1.1.4, this implies Factorization and Nagata’s compactification theorems. Our proof of Theorem 4.3.4 consists of the following steps: First, we attack the problem locally, and prove in Theorem 4.1.1 that if is of finite type then any adic semivaluation factors through a good quasi-modification . The construction of is the most technical argument in this paper. It is analogous to the construction of a pseudo-modification in Theorem 3.3.5 but more involved because we have to work with -adic models throughout the construction. In addition, it only applies to uniformizable semivaluations, thus forcing us to assume that is of finite type. Then we prove in Lemma 4.2.2 that good quasi-modifications can be glued to a modification of after blowing them up. These two results together with the quasi-compactness of imply Nagata’s compactification theorem for algebraic spaces, see Theorem 4.2.4. Finally, we apply the approximation theory of Rydh [Ryd13] to remove unnecessary finite type assumptions, and complete the proof of Theorem 4.3.4.

1.3.4. Main results on Riemann-Zariski spaces

The quasi-modifications constructed in Theorem 4.1.1 are strict, i.e., satisfy . In Section 5 we study the whole family of quasi-modifications along an adic semivaluation, and prove that there exist “enough” (not necessarily strict) quasi-modifications, see Theorem 5.1.5 for precise formulation. As an easy corollary, we complete our investigation of the topological spaces , , and . Namely, we show in Theorem 5.2.2 that any open quasi-compact subset in is the image of for a quasi-modification , and conclude that the map is a homeomorphism (Theorem 5.2.4), and is a topological quotient (Corollary 5.2.5). In particular, all three topologies on coincide: the topologies generated by (1) pseudo-modifications, (2) quasi-modifications, and (3) compactifiable quasi-modifications.

We conclude the paper with two other applications of our theory. Theorem 5.3.4 establishes a valuative criterion for schematization of algebraic spaces, and Theorem 5.3.6 describes Prüfer morphisms of algebraic spaces, i.e., morphisms that admit no non-trivial modifications.

2. Models

2.1. Definition and basic properties

2.1.1. The category of models

A model is a triple such that and are quasi-compact and quasi-separated (qcqs) algebraic spaces, and is a separated schematically dominant morphism. The spaces and are called the model space and the generic space of respectively. We always denote the model, the model space, and the generic space by the same letter, but for model we use bold font, for the generic space calligraphic font, and for the model space the regular font. If both and are schemes then the model is called schematic.

A morphism of models is a pair of morphisms and such that . The maps and are called the generic and the model components of . Once again, we use the same letter but different fonts to denote the morphisms of models and their components. We say that a morphism is separated if is separated. In this case, is separated, and hence is also separated.

2.1.2. Good models

As we have mentioned in the introduction, the class of models for which is an open immersion plays an important role in the sequel. We call such models good.

Remark 2.1.3.

(i) Any good model can be equivalently described by its model space and the closed subset . Since is quasi-compact, for a finitely generated ideal that we call an ideal of definition of . Any other ideal of definition satisfies for .

(ii) We noted in the introduction that there is a large similarity between good models and formal models of non-archimedean spaces . In particular, corresponds to , corresponds to the generic fiber of , corresponds to the closed fiber , and ideals of definition of correspond to ideals of definition of .

2.1.4. Basic properties of models

The proof of the following result is almost obvious, so we omit it.

Proposition 2.1.5.

(1) The functor is a full embedding of the category of qcqs algebraic spaces into the category of models. By abuse of notation we shall not distinguish between and .

(2) Any model admits two canonical morphisms .

(3) The category of models admits fiber products. Furthermore, if and are morphisms of models then the generic space of is , the model space of is the schematic image of , and is the natural map.

2.1.6. Classes of models

Our next aim is to introduce basic classes of models and their morphisms that will be used in the sequel.

Definition 2.1.7.

(1) Let be one of the following properties of morphisms: affine, of finite type, of finite presentation. We say that a model satisfies if and only if satisfies .

(2) A model is called affinoid if both and are affine.

(3) The diagonal of a morphism is the natural morphism

2.1.8. Adic morphisms

A morphism is called adic (resp. semi-cartesian) if its diagonal is proper (resp. a clopen immersion).

Remark 2.1.9.

To illustrate importance of adic morphisms and to explain the terminology, let us consider the case of a morphism of good models. Set and . Then it is easy to see that is adic if and only if , or, equivalently, the pullback of an ideal of definition is an ideal of definition. Using the analogy with formal schemes from Remark 2.1.3 we see that adic morphisms of good models correspond to adic morphisms of formal schemes.

Lemma 2.1.10.

(1) Let be a morphism of models. If is proper and is separated then is adic.

(2) Composition of adic morphisms is adic.

(3) Adic morphisms are preserved by base changes.

(4) Let and be morphisms of models. If is adic and is separated then is adic.

Proof.

(1) The base change of is separated and the composition is proper, hence is proper.

(2) Let and be two adic morphisms. Then and are proper. Hence the base change of is proper, and the composition is proper too. Thus, is adic.

(4) The composition is proper, and is separated since so are and the diagonal . Thus, is proper.We leave the proof of (3) to the reader. ∎

2.1.11. Immersions

An adic morphism is called an immersion (resp. open immersion, resp. closed immersion) if both its components are so. In particular, by an open submodel of we mean a model together with an open immersion . If , are open submodels then the union and the intersection are defined componentwise. We say that is an open covering of if .

Remark 2.1.12.

It is easy to see that any open immersion is semi-cartesian: Indeed, the morphism and the composition are open immersions, hence so is the diagonal . Since is adic, is proper and hence is semi-cartesian. However, a closed immersion need not be semi-cartesian, for example, take , , , and is the closed immersion of the origin of .

2.2. Modifications of models

Definition 2.2.1.

Let be a morphism of models.

(1) is called a pseudo-modification if is an open immersion and is a separated morphism of finite type. A pseudo-modification is called strict if is an isomorphism.

(2) is called a quasi-modification if it is an adic pseudo-modification.

(3) is called a modification if it is a strict pseudo-modification and is proper.

By abuse of language, we will often refer to as a modification or quasi-modification or pseudo-modification of . In particular, we will say that a modification is good (resp. affine good) if is so. The following easy observation will be useful:

Proposition 2.2.2.

Any strict pseudo-modification is a quasi-modification. In particular, any modification is a quasi-modification.

Example 2.2.3.

(i) If is a modification then any open submodel is a quasi-modification of . In Theorem 4.3.4 (a strong version of Nagata’s compactification), we will prove that a partial converse is true. Namely, for any quasi-modification there exist modifications and such that is an open submodel, thereby justifying the terminology.

(ii) A simple example of a pseudo-modification, which is not a quasi-modification is the following: , , and is the natural map.

Remark 2.2.4.

Although, the notion of quasi-modification seems to be new, it is tightly related to ideas used in various proofs of Nagata’s compactification theorem for schemes. Assume that is a schematic model.

(i) If is a good model then quasi-modifications of are closely related to quasi-dominations that were introduced by Nagata and played an important role in the proofs of Nagata’s compactification theorem by Nagata, Deligne, and Conrad. See, for example [Con07, §2].

(ii) Modifications appeared in [Tem11] under the name of -modifications. Namely, is a modification if and only if is an isomorphism and is an -modification of . Quasi-modifications were not introduced terminologically in [Tem11], but they were used in the paper (e.g., see [Tem11, Prop. 3.3.1]).

Next, we establish basic properties of pseudo-modifications.

Lemma 2.2.5.

Let be a morphism of models. If the generic component is an immersion then so is the diagonal . In particular, if is a pseudo-modification then is an immersion.

Proof.

Obvious. ∎

Lemma 2.2.6.

(1) Modifications, quasi-modifications, and pseudo-modifications, are preserved by compositions and base changes. In particular, if and are modifications (resp. quasi-modifications, resp. pseudo-modifications) of models then so is .

(2) For any model , the families of modifications, quasi-modifications, and pseudo-modifications of are filtered.

Proof.

The second claim follows from the first one. The latter is obvious for modifications and pseudo-modifications, and the case of quasi-modifications follows from Lemma 2.1.10. ∎

For good models, one can say more.

Lemma 2.2.7.

Let be a pseudo-modification. If is good then,

(1) is good and the diagonal is an open immersion.

(2) is a quasi-modification if and only if it is semi-cartesian.

Proof.

(1) The open immersion factors through , hence is an immersion by [Sta, Tag:07RK]. Since is schematically dominant, it is, in fact, an open immersion.

(2) By definition, a pseudomodification is quasi-modification if and only if the open immersion is proper if and only if is semi-cartesian. ∎

2.2.8. Riemann-Zariski spaces of models

By the Riemann-Zariski or RZ space of a model we mean the topological space , where , , is the family of all modifications of . We claim that is a functor. Indeed, if is a morphism of models then for any modification the base change is a modification by Lemma 2.2.6, and therefore there exists a natural continuous map .

Remark 2.2.9.

One may wonder whether the limit of ’s exists in a finer category. It is easy to see that the limit does not have to be representable in the category of algebraic spaces, so it is natural to seek for an enlargement of this category. For example, in the case of schematic models, a meaningful limit exists in the category of locally ringed spaces (see [Tem11]). It seems that in our situation the most natural framework is the category of strictly henselian topoi as introduced by Lurie (intuitively, this is a ringed topos such that the stalks of the structure sheaf are strictly henselian rings). However, even the foundations of such theory are only being elaborated; for example, the fact that algebraic spaces embed fully faithfully into the bicategory of such topoi was checked very recently by B. Conrad, see [Con, Th. 3.1.3]). For this reason, we prefer not to develop this direction in the paper.

2.3. Blow ups

Definition 2.3.1.

Let be a model, an ideal of finite type such that , and the natural map. Then the model is called the blow up of along . An open submodel of a blow up is called a quasi-blow up.

Any blow up (resp. quasi-blow up) is a modification (resp. quasi-modification). We will only consider blow ups of good models which can be identified with usual -admissible blow ups of by the following remark.

Remark 2.3.2.

Let be a blow up. If is good then , is an -admissible blow up, and is good. Conversely, for an -admissible blow up , the model is good and is a blow up of models.

2.3.3. Extension of blow ups

An important advantage of blow ups over arbitrary modifications is that they can be easily extended from open submodels. This property is very useful in Raynaud’s theory, and it will be crucial in §4.2.1 while gluing good quasi-modifications.

Proposition 2.3.4.

Let be a good model, an open submodel, and a blow up. Then there exists a blow up such that .

Proof.

The spaces , , and are open subspaces of , and by Remark 2.1.12. By definition, for a finite type ideal such that the subspace defined by is disjoint from . Then is also disjoint from , and hence so is the schematic closure of in , that we denote by . By [Ryd13, Theorem A], the ideal defining is the filtered colimit of ideals of finite type that extend to . By quasi-compactness of , there exists such that is disjoint from the subspace defined by . Then is as needed. ∎

Proposition 2.3.5.

Let be a good model. If and are blow ups (resp. quasi-blow ups) then so is the composition .

Proof.

For schematic models the case of blow ups follows from the fact that composition of -admissible blow ups of the algebraic space is an -admissible blow up by [RG71, Lem. 5.1.4]. Note that the proof in [RG71] is incomplete, and the complete proof (including an additional argument due to Raynaud) can be found in [Con07, Lemma 1.2]. In general, the proof is almost identical to the proof in [Con07], and the only difference is that one must replace Zariski-local arguments with étale-local, and use the approximation theory of Rydh [Ryd13, Corollary 4.12] instead of [Gro67, I, 9.4.7; , 1.7.7].

Assume, now, that the morphisms are quasi-blow ups. Then and for blow ups and . The latter blow up can be extended to a blow up by Proposition 2.3.4. So, the composition is a blow up, and we obtain that is a quasi-blow up of . ∎

3. Relative birational spaces

3.1. Semivaluations of models

3.1.1. Valuation models

We refer to [TT13b, §5.1.1] for the definition of valuation algebraic spaces. By a valuation model we mean a model such that is a valuation algebraic space and is its generic point.

3.1.2. Valuative diagrams

A valuative diagram as defined in [TT13b, §5.2.1] can be interpreted as a morphism from a valuation model to a model such that is separated.

3.1.3. Semivaluations

By a semivaluation of a model we mean a valuative diagram such that is a Zariski point. It follows from [TT13b, Lemma 5.2.3] that any valuative diagram factors uniquely through a semivaluation such that is surjective. As a corollary, it is shown in [TT13b, Proposition 5.2.9] that semivaluations can be used to test properness of .

3.1.4. The sets

The set of all semivaluations of is denoted by . For any separated morphism of models and a semivaluation , the composition is a valuative diagram, hence induces a semivaluation of by [TT13b, Lemma 5.2.3]. This defines a map in a functorial way, and hence is a functor from the category of models with separated morphisms to the category of sets.

3.1.5. Adic semivaluations

We say that a semivaluation is adic if it is adic as a morphism of models. It is proved in the refined valuative criterion [TT13b, Theorem 5.2.14] that adic semivaluations suffice to test properness of .

Remark 3.1.6.

(i) If is schematic then semivaluations of are exactly the -valuations of as defined in [Tem11, §3.1].

(ii) For the sake of comparison, we note that in adic geometry of R. Huber, one defines affinoid spaces to be the sets of all continuous semivaluations of the morphism , equipped with a natural topology and a structure sheaf. The continuity condition is empty if the topology on is discrete, and is equivalent to adicity of the semivaluation otherwise.

Lemma 3.1.7.

Assume that is a semivaluation and is a separated morphism of valuation models such that is surjective. If the composed valuative diagram is adic then is adic.

Proof.

By [TT13b, Lemma 5.2.12], if is not adic then the morphism extends to a pro-open subspace strictly larger than . Therefore, extends to the preimage of . Since is strictly larger than by the surjectivity of , we obtain that is not adic, which is a contradiction. ∎

3.1.8. The sets

We denote by the set of all adic semivaluations of a model . It is a subset of , and we denote the embedding map by . If is a separated adic morphism of models then for an adic semivaluation the composition is adic by Lemma 2.1.10, and hence takes to by Lemma 3.1.7. Set making to a functor from the subcategory of models with separated adic morphisms to the category of sets.

3.1.9. The retraction

Let be a semivaluation. By [TT13b, Lemma 5.2.12(i)], there exists a maximal pro-open subspace such that the -morphism extends to . Moreover, is quasi-compact and has a closed point . Set to be the closure of in , , and . Then is an adic semivaluation by [TT13b, Lemma 5.2.12(ii)].

3.1.10. Full functoriality of

It is easy to see that if is an adic morphism then the retractions and are compatible with the maps and , i.e., is a natural transformation of these functors.

Furthermore, we can use the retractions to extend the functor to the whole category of models with separated morphisms. Indeed, for a separated morphism we simply define to be the composition .

3.1.11. The reduction maps

By the center of a semivaluation we mean the image of the closed point of . The maps and that associate to a semivaluation its center are called the reduction maps. Plainly, , and the reduction maps and commute with and for any separated morphism .

Proposition 3.1.12.

The reduction maps and are surjective.

Proof.

It is sufficient to prove the surjectivity of since . Without loss of generality we may assume that is affinoid since commutes with for separated morphisms . Let be a point. By schematic dominance of , there exists a point such that is a generalization of . Indeed, for affine schemes this follows from [Sta, Tag:00FK], and the general case follows by passing to an affine presentation. Take any valuation ring of such that extends to a morphism that takes the closed point to , and let be any valuation ring of that extends . Set . Then the natural semivaluation belongs to . ∎

Remark 3.1.13.

The term “reduction map” is grabbed from the theory of formal models of non-archimedean spaces. Another reasonable name would be a “specialization map”.

3.2. Topologies

3.2.1. Featured subsets of and

Lemma 3.2.2.

Assume that is a pseudo-modification. Then,

(1) The map is injective.

(2) If is a quasi-modification then is injective.

(3) If is an adic semivaluation that lifts to then is also adic.

(4) The following are equivalent: (a) is a modification, (b) is a bijection, (c) takes to and induces a bijection between them.

Proof.

Assertion (1) follows from [TT13b, Proposition 5.2.6] applied to the morphism . Assertion (2) follows from (1) by the definition of (cf. § 3.1.8). (3) follows from Lemma 2.1.10 (4). It remains to prove (4).

Note that for any point there exists a trivial semivaluation , which is easily seen to be adic. Thus, if either or is bijective then is strict. Hence we may assume that is strict.

If either (b) or (c) holds then is proper by the strong valuative criterion of properness [TT13b, Theorem 5.2.14], and hence (a) holds. Vice versa, if (a) holds then is injective by (1) and is surjective by [TT13b, Proposition 5.2.9], and hence (b) holds. Furthermore, takes to (cf. § 3.1.8), and hence (c) holds by (3). ∎

Thanks to Lemma 3.2.2, for a pseudo-modification , we will freely identify with the subset of all semivaluations of that lift to . If is a quasi-modification then we will use similar identification for the sets .

Definition 3.2.3.

Let be a semivaluation. We say that a pseudo-modification is along (or, by abuse of language, along ) if .

Lemma 3.2.4.

Let be a semivaluation. Then the families of all pseudo-modifications and quasi-modifications of along are filtered.

Proof.

Follows from Lemma 2.2.6. ∎

Lemma 3.2.5.

Assume is a model with a semivaluation , and let be the associated adic semivaluation. Then,

(1) Any pseudo-modification along is also along .

(2) A quasi-modification is along if and only if it is along .

Proof.

Let and denote the morphisms of and , respectively. Recall, that is the closed point of the maximal pro-open subspace for which the -morphism extends to , and is the closure of in . Since is open and , the morphism factors through . By [TT13b, Proposition 4.3.11], is the Ferrand pushout. Thus, the compatible morphisms and define a morphism , and hence . This proves (1).

To prove (2) we assume that is along . Identify with a subset of . Since is adic, we have that , so . ∎

3.2.6. Maps to

By Lemma 3.2.2 (4), for any modification we can identify with . Therefore, the reduction maps induce a map , and in the same manner one obtains a map . Since the center maps are compatible with the retraction , we obtain the following diagram, in which both triangles are commutative.

(1)

Plainly, such diagrams are functorial with respect to separated morphisms of models.

3.2.7. Topologies on

If and are two pseudo-modifications of then . Therefore, the collection of sets for all pseudo-modifications , forms a base of a topology on , which we call Zariski topology. Since pseudo-modifications are preserved by base changes, the maps are continuous. This, enriches to a functor whose target is the category of topological spaces.

In addition, we consider the boolean algebra generated by the sets for pseudo-modifications , and call its elements constructible subsets of . They form a base of a topology, which we call constructible topology. Clearly, the maps are continuous with respect to the constructible topologies as well.

Remark 3.2.8.

We will show in Proposition 3.3.3 that the sets are quasi-compact, and so our ad hoc definition of the constructible topology coincides with what one usually takes for the definition, i.e., the topology associated to the boolean algebra generated by open quasi-compact sets. However, we will use both topologies in the proof, so it is convenient to start with the ad hoc definition.

3.2.9. The topology on

We define Zariski topology on to be the induced topology from the Zariski topology on , i.e., the sets for pseudo-modifications form a base of the Zariski topology. Note that by Lemma 3.2.2 (3), .

Remark 3.2.10.

(i) The functoriality of (the enriched) with respect to adic morphisms is obvious, while the question about general functoriality is much more subtle. In fact, our main results about the topology of are the following: (1) is a homeomorphism (Theorem 5.2.4), and (2) the retraction map is open and continuous (Corollary 5.2.5). Since the construction of is functorial in a natural way, we will obtain an interpretation of the functoriality of that does not involve the retractions . Moreover, this will extend the functoriality of to all morphisms, and will imply the continuity of the maps for an arbitrary morphism .

(ii) One could define quasi-modification topology on by using only quasi-modifications in the definition. It is a priori weaker than the Zariski topology, but a posteriori the two topologies coincide (Theorem 5.2.2).

(iii) As one might expect, the quasi-modification topology is not so natural on . In fact, it does not distinguish points in the fibers of by Lemma 3.2.5, while the pseudo-modification topology does so by Theorem 3.3.5.

3.3. Topological properties

In this section we establish relatively simple topological properties, whose proof does not involve a deep study of quasi-modifications.

Proposition 3.3.1.

The maps , , , , and are continuous in Zariski topology.

Proof.

The continuity of follows immediately from the definitions. For the continuity of and , let be an open subset. Set . Then and are open subsets. Finally, since is the limit of , where , , is the family of all modifications of , the continuity of and follows from the continuity of and . ∎

We conclude Section 3 by showing that and are quasi-compact. For the proof we need the following observation.

Lemma 3.3.2.

For any , the retraction is a specialization of in Zariski topology.

Proof.

Follows from Lemma 3.2.5(1). ∎

Proposition 3.3.3.

Let be a model, and a pro-constructible subset. Then the sets and are quasi-compact in the Zariski topology. In particular, and are quasi-compact. Furthermore, is compact in the constructible topology.

Proof.

First, we claim that if is quasi-compact in Zariski topology then so is . Indeed, any open covering of in is the restriction of an open covering of in . By Lemma 3.3.2, for any the retraction is a specialization of . Note that since . Thus, is also an open covering of in . If is quasi-compact then its covering admits a finite refinement, and hence admits a finite refinement too.

Assume, first, that is affinoid. This case is essentially due to Huber (see [Hub93] and [Tem11] for details). If and then, as explained in [Tem11, §3.1], the topological space is nothing but Huber’s adic space , where and are viewed as discrete topological rings. In [Hub93], it is proved that is compact in the constructible topology. Since is continuous, the set is pro-constructible. Thus, it is an intersection of constructible sets, which are closed, and hence compact in the constructible topology. We conclude that is compact.

For an arbitrary , pick an étale affine covering , and set . Now, pick an étale affine covering , and set and . We obtain an affinoid model and morphisms , whose components are étale and surjective. Consider the induced map . Note that is compact by the already established affinoid case, hence we should only prove that is surjective.

Let be a semivaluation. Set , and note that is an SLP space because it is étale over . Applying [TT13b, Proposition 5.1.9(i)] to we obtain that is a scheme. Let be a preimage of the closed point of and . Set . Then is a semivaluation, whose image in is . Indeed, is a surjective separated morphism of valuative spaces inducing an isomorphism of the generic points. Hence, by [TT13b, Lemma 5.2.3(ii)], it is an isomorphism. Let now be the generic point of . We identify it with a point of , and choose any preimage . Choose any valuation ring of whose intersection with coincides with . Then defines a valuation model and the morphisms and define a semivaluation of that lifts . ∎

Corollary 3.3.4.

The maps and are surjective.

Proof.

It is sufficient to show that is surjective. Pick a point , and consider the corresponding compatible family of points , where are the modifications of . The subsets are compact in the constructible topology since are continuous maps between compact spaces and are compact. Furthermore, for each by Proposition 3.1.12. Finally, since the family of modifications of is filtered by Lemma 2.2.6 (2), for any finite set of indices , there exists a modification that factors through all . Thus, . Hence , and by compactness, . ∎

Theorem 3.3.5.

For any model the spaces and are .

Proof.

Since the topology on is induced from , it suffices to prove the assertion for .

Let be two distinct points. We should construct a pseudo-modification such that belongs to the open set , but does not, or vice-versa. If as points of then without loss of generality we may assume that does not belong to the closure of . Thus, there exists an open containing and not containing . Set to be the schematic image of , and . Then is a pseudo-modification, belongs to , but does not.

Assume now that . Without loss of generality we may assume that does not factor through , and it suffices to construct a pseudo-modification along such that does not factor through . This will be done in few steps as shown in the diagram

where is a closed immersion and is an open immersion.

Pick any open affine in the closure of in . We may assume that is -affine: Indeed, let be an étale affine presentation. Set . Let be an open affine subset. Then there exists such that factors through . Thus, after replacing with we may assume that and are affine, and hence so is by descent.

Let be the -affine pushout, i.e., the pushout in the category of -affine algebraic spaces. Note that . Then is separated and schematically dominant. Furthermore, for the family of principal localizations of . We claim that , where . To check this we can replace with an affine étale covering and all the -spaces with their base changes, so assume that is affine. Then -affine pushouts are nothing but affine pushouts, and by the very definition, as a subring of . The equality follows, and as a consequence we obtain that the morphism does not factor through for large enough. After replacing with , we may assume that does not factor through .

Let be an open subspace such that is a closed immersion. Let be the Ferrand pushout, see [TT13a, §3.1]. Then is a closed immersion by [TT13a, Theorem 4.4.2(iii)]. Since the morphism takes the generic point of to and does not factor through , it does not factor through too. Finally, by approximation [TT13b, Lemma 2.1.10], we can choose separated of finite type such that factors through , is affine and schematically dominant, and does not factor through . Thus is as needed. ∎

We shall mention that a similar, but much more involved, argument will be used to prove Theorem 4.1.1 below.

4. Main results on modifications of models

4.1. Approximating adic semivaluations with quasi-modifications

The following theorem is the key result towards Nagata’s theorem. Its proof occupies whole §4.1.

Theorem 4.1.1.

Let be a finite type model, and an adic semivaluation. Then there exists a strict quasi-modification along such that is good and affine.

4.1.2. Plan of the proof

The proof is relatively heavy, so we first provide a general outline. To ease the notation we identify with the corresponding point . Let be the closure of .

Step 1. Choose a suitable open -affine subscheme and set to be the -affine pushout of and . Ensure that is a good affine model satisfying certain conditions; the main one being adicity over .

Step 2. Define to be the gluing of and along in the Zariski topology, and show that is a good affine model separated over .

Step 3. Set to be the Ferrand pushout (see [TT13a, §3.1]) and show that is a good affine model separated over .

Step 4. Construct an affine strict quasi-modification along by choosing to be a finite type approximation of .

Step 5. Modify so that it becomes good.

(2)

4.1.3. Comparison with the schematic case from [Tem11]

Before proving the theorem, let us compare this plan with the parallel proof in [Tem11]. That proof runs as follows. If is schematic then is also schematic by [TT13b, Proposition 5.1.9]. Consider the semivaluation ring of (see [Tem11, §3.1]), then is the affine Ferrand pushout (see [Tem11, §2.3]). Approximate by an affine -scheme of finite presentation, and note that for a small enough neighborhood of the morphism lifts to a morphism by approximation. Let be the schematic image of in . Then is of finite type. Set . Finally, care to choose and such that is adic; this is the most subtle part.

Assume, now, that is general. If the localization