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RELATIVE RIEMANN-ZARISKI SPACES M. TEMKIN Max Planck Institut für Matematik, Vivatgasse 7, Bonn 53111, Germany (e-mail: [email protected]) 1. Introduction Let k be an algebraically closed field and K be a finitely generated k-field. In the first half of the 20-th century, Zariski defined a Riemann variety RZK associated to K as the projective limit of all projective k-models of K. Zariski showed that this topological space, which is now called a RiemannZariski (or Zariski-Riemann) space, possesses the following set-theoretic description: to give a point x ∈ RZK is the same as to give a valuation ring Ox with fraction field K, such that k ⊂ Ox . The Riemann-Zariski space possesses a sheaf of rings O whose stalks are valuation rings of K as above. Zariski made extensive use of these spaces in his desingularization works. Let S be a scheme and U be a subset closed under generalizations, for example U = Sreg is the regular locus of S, or U = η is a generic point of S. In many birational problems, one wants to consider only U -modifications S 0 → S, i.e. modifications which do not modify U . Then it is natural to consider the projective limit S = RZU (S) of all U -modifications of S. It was remarked in [Tem2], that working with such generalized Riemann-Zariski spaces, one can extend the P -extension results of [Tem2] (see loc.cit. 1.3, 1.7.5 and 1.8.4) to the case of general U and S. This plan is realized in chapter 2. In section 2.1, we prove that any point x ∈ S is uniquely determined by a point x ∈ U and a valuation ring R of k(x), which is centered on the Zariski closure of x in S and satisfies some minimality condition. In the next section, we use this description of S to prove the stable modification theorem, which generalizes the main result of [Tem2]. The definition of relative Riemann-Zariski spaces may be generalized without efforts to the case of an arbitrary separated morphism f : Y → X. Let us say that f is decomposable if it factors into a composition of an affine morphism Y → Z and a proper morphism Z → X. Actually, in section 2.1, we study the case of a general decomposable morphism. We define a set ValY (X) whose points are certain X-valuations of Y , and construct a surjection ψ : ValY (X) → RZY (X) and a section RZY (X) → ValY (X). Later, we prove that these maps are actually bijections, see 3.2.5. It is natural to ask if the decomposition assumption is essential. The answer is negative because the assumption is actually empty. The main result of chapter 3 is 1 2 M. TEMKIN a decomposition theorem which states that a morphism of quasi-compact quasi-separated schemes is decomposable if and only if it is separated. Let us describe briefly other parts of the paper. In section 1.1 below, we recall Nagata compactification and Thomason’s noetherian approximation theorems, see [Nag] and [TT], C.9, and prove a slight generalization of the second theorem. Then we notice that the decomposition theorem is essentially equivalent to the union of Nagata’s and Thomason’s theorems. Thus, one could use Nagata’s theorem to prove that ψ : ValY (X)f →RZY (X) in general, but it is also possible to work in opposite direction. In section 3.1, we establish an interesting connection between Riemann-Zariski spaces and adic spaces of R. Huber; in particular, we obtain an intrinsic topology on ValY (X). Then we study Y -blow ups of X, which are analogs of formal blow ups from Raynaud’s theory, see [BL], and prove that in the decomposable case ψ is a homeomorphism. We give a local description of spaces ValY (X) in the next section; surprisingly, it is finer than the local structure of related adic spaces SpaY (X). Finally, we prove the decomposition theorem in section 3.4. In particular, we prove that ψ is a homeomorphism in general and obtain a new proof of Nagata’s theorem, see [Con] and [Lüt] for other proofs. Nevertheless, it seems that our proof has certain similarities with Deligne’s proof presented in [Con]. I want to mention that I was motivated by Raynaud’s theory in studying Riemann-Zariski spaces in the decomposable case, and the basic ideas are taken from [BL]. I give a simple illustration of those ideas in the proof of the generalized Thomason’s theorem. I want to express my gratitude to R. Huber for a useful discussion and to B. Conrad for pointing out various gaps and mistakes in an earlier version of the article. This article was written during my stay at the Max Planck Institute for Mathematics at Bonn, and I want to thank it for hospitality. 1.1. On noetherian approximation and Nagata compactification. For shortness, a filtered projective family of schemes with affine transition morphisms will be called affine filtered family. Also, we abbreviate the words ”quasi-compact and quasi-separated” by the single ”word” qcqs. In [TT], C.9, Thomason proved a very useful noetherian approximation theorem, which states that any qcqs scheme Y over a ring Λ is isomorphic to a scheme proj lim Yα , where {Yα }α is an affine filtered family of Λ-schemes of finite presentation. Due to the following lemma, this theorem may be reformulated in a more laconic way as follows: Y is affine over a Λ-scheme Y0 of finite presentation. Lemma 1.1.1. A morphism of qcqs schemes f : Y → X is affine if and only if Y f → proj lim Yα , where {Yα }α is a filtered family of X-affine finitely presented X-schemes. → proj lim Yα is as above, then Yα = Spec(Eα ) and Y = Spec(E), Proof. If Y f where E = inj lim Eα . Conversely, suppose that f is affine. By [EGA I], 6.9.16 RELATIVE RIEMANN-ZARISKI SPACES 3 (iii), f∗ (OY )f → inj lim Eα , where {Eα } is a direct filtered family of finitely presented OX -algebras. Hence Y = proj lim Spec(Eα ). ¤ We generalize Thomason’s theorem below. As a by-product, we obtain a simplified proof of the original theorem. Theorem 1.1.2. Let f : Y → X be a (separated) morphism of qcqs schemes. Then f may be factored into a composition of an affine morphism Y → Z and a (separated) morphism Z → X of finite presentation. Proof. If Y is X-separated, then Z may be chosen X-separated by [TT], .... Step 1. Preliminary work. If Y is affine and f (Y ) is contained in an open affine subscheme X 0 ⊂ X, then the claim is obvious. Hence Y admits a finite covering by open qcqs subschemes Y1 , . . . , Yn , such that the induced morphisms Yi → X satisfy the conclusion of the theorem. By induction, we may assume that n = 2. Then the schemes U = Y1 and V = Y2 , may be represented as U = proj lim Uβ and V = proj lim Vγ , where the limits are taken over affine filtered families of X-schemes of finite presentation. If f is separated, then we also require that Vγ ’s and Uβ ’s are X-separated. Step 2. Affine domination. By [EGA IV], 8.2.11, for β ≥ β0 and γ ≥ γ0 , the schemes Uβ and Vγ contain open subschemes Uβ0 and Vγ0 , whose preimages in U and V coincide with W = U ∩ V . By [EGA IV], 8.13.1, the morphism W → Uβ0 0 factors through Vγ0 for sufficiently large γ. Replace γ0 by γ. By the same reason, the morphism W → Vγ00 factors through some Uβ0 and the morphism W → Uβ0 factors through some Vγ0 . Let us denote the corresponding morphisms as fγ,β : Vγ0 → Uβ0 , fβ,γ0 and fγ0 ,β0 . Now comes an obvious but critical argument: fβ,γ0 is separated, because the composition fγ0 ,β0 ◦ fβ,γ0 : Uβ0 → Uβ0 0 is separated; fγ,β is affine because its composition with a separated morphism fβ,γ0 is affine. We gather the already defined objects in the left diagram below. Notice that everything is defined over X, the horizontal arrows are open immersions, the vertical arrows are affine morphisms and the indexed schemes are of finite X-presentation. V o ? _W ² ² Vγ o /U φ0 ? _ Vγ0 ² Uβ0 h ² / Uβ V o ? _W ² ² ? _V 0 Vγ o φ0 γ ² Uβ0 /U ² φ / Uγ ² / Uβ Step 3. Affine extension. The main task of this step is in producing the right diagram from the left one. It follows from the previous stage that Vγ0 = Spec(E 0 ), where E 0 is a finitely presented OUβ0 -algebra. The morphism φ0 : W → Vγ0 to a Uβ0 -affine scheme corresponds to a homomorphism ϕ0 : E 0 → h0∗ (OW ), where h0 : W → Uβ0 is the projection. Obviously 4 M. TEMKIN h∗ (OU )|Uβ0 f →h0∗ (OW ), where h : U → Uβ is the projection. Hence we can apply [EGA I], 6.9.10.1, to find a finitely presented OUβ -algebra E and a homomorphism ϕ : E → h∗ (OU ), such that E|Uβ0 f →E 0 and the restriction of ϕ to Uβ0 is ϕ0 . Set Uγ = Spec(E), then ϕ induces a morphism φ : U → Uγ . Finally, we glue Uγ and Vγ along Vγ0 obtaining a finitely presented X-scheme Z, and notice that the affine morphisms U → Uγ and V → Vγ glue to an affine morphism Y → Z over X. ¤ Our proof is a simple analog of Raynaud’s theory. Thomason used the first two steps (induction argument in the proof of C.9 and lemma C.6). Our simplification of his proof is due to the third step. The same arguments are used in Raynaud’s theory, see [BL], the end of the proof of 4.1, statement (d) from that proof and lemma 2.6 (a). In our paper, they also appear in the proofs of 1.1.3, 3.2.1 (i) and 3.2.3. Next, we recall the Nagata compactification theorem, see [Nag]. A scheme theoretic proof may be found in [Con] or [Lüt]. Recall that a morphism f : Y → X is called compactifiable if it may be factored as a composition of an open immersion g : Y → Z and a proper morphism h : Z → X. Nagata proved that a finite type morphism f : Y → X of qcqs schemes is compactifiable iff it is separated. Actually, Nagata considered noetherian schemes. The general case is due to [Con]. Let I ⊂ OZ be an ideal with support Z \ Y , and Z 0 the blow up of Z along I. We can choose a finitely generated I, because the morphism Y ,→ Z is quasi-compact. The open immersion g 0 : Y → Z 0 is affine, because Z 0 \ Y is a locally principal divisor. It follows that g is a composition of an affine morphism g 0 of finite type and a proper morphism Z 0 → X. Conversely, assume that g : Y → Z is affine of finite type and Z → X is proper. Then Y is quasi-projective over Z, hence there is an open immersion of finite type Y ,→ Y with Zprojective and, therefore, X-proper Y . Thus, Nagata’s theorem may be reformulated as follows: a finite type morphism is separated iff it may be represented as a composition of an affine morphism of finite type and a proper morphism. Now, one sees that a weak form of theorem 1.1.2 (f is separated and Z → X is of finite type) and Nagata’s theorem are together equivalent to the following decomposition theorem, which will be proven in section 3.4. Theorem 1.1.3. A morphism f : Y → X of quasi-compact quasi-separated schemes is separated if and only if it may be factored as a composition of an affine morphism Y → Z and a proper morphism Z → X. 2. Preliminary description of Riemann-Zariski spaces and applications Throughout this chapter, f : Y → X denotes a separated morphism of qcqs schemes. We are going to recall some notions introduced in [Tem2]. RELATIVE RIEMANN-ZARISKI SPACES 5 Consider the family of all factorizations of f into a composition of a schematically dominant morphism fi : Y → Xi and a proper morphism gi : Xi → X, it has an initial object corresponding to the schematic image of Y in X. We call the pair (fi , gi ) a Y -modification of X, usually it will be denoted simply as Xi . Given two Y -modifications of X, we say that Xj dominates Xi , if there exists an X-morphism gji : Xj → Xi compatible with fi and fj . A standard argument shows that if gji exists, then it is unique (one uses only, that fj is schematically dominant and Xi is X-separated). The family of Y -modifications of X is filtered, because any two Y -modifications Xi , Xj are dominated by the scheme-theoretic image of Y in Xi ×X Xj . A relative Riemann-Zariski space X = RZY (X) is defined as the projective limit of the underlying topological spaces of Y -modifications of X. If X is integral and Y is its generic point, then one recovers the classical RiemannZariski spaces. A slightly more general case, when Y is a dominant point, was considered in [Tem1], Sect. 1. We have the obvious projections πi : X → Xi and a map η : Y → X, and we provide X with the sheaf MX = η∗ (OY ) of meromorphic functions. We will immediately see that the sheaf πi−1 (OXi ) embeds into MX , and we define a subsheaf OX ⊂ MX of regular functions by OX = inj lim πi−1 (OXi ) = ∪πi−1 (OXi ). Lemma 2.0.4. The natural homomorphism πi−1 (OXi ) → MX is monomorphism. Proof. ¤ By schematical dominance of fi , 2.1. Riemann-Zariski space of an affine morphism. Let ValY (X) be the set of triples y = (y, R, φ), where y ∈ Y is a point, R is a valuation ring of k(y) (in particular Q(R) = k(y)) and φ : S = Spec(R) → X is a morphism compatible with y = Spec(k(y)) → Y and such that the induced morphism y → S ×X Y is a closed immersion. Let Oy denote the preimage of R in OY,y . For any Y -modification Xi → X, φ uniquely factors through Xi by the valuative criterion of properness. Let xi ∈ Xi be the center of R on Xi , i.e. the image of the closed point of S, then the family of points (xi ) defines a point x ∈ X. Thus, we obtain a map ψ : ValY (X) → X. For any i, the image of OXi ,xi in k(y) lies in R. Therefore, the image of OXi ,xi in OY,y lies in Oy , and we obtain a homomorphism OX,x = inj lim OXi ,xi → Oy . Proposition 2.1.1. Suppose that f is decomposable. Then for any point x ∈ X, there exists a unique point y = λ(x) ∈ ValY (X), such that ψ(y) = x and Ox f →Oy . In particular, λ is a section of ψ. Proof. Factor f into a composition of an affine morphism Y → Z and a proper morphism Z → X. After replacing X by the scheme-theoretic image of Y in Z, we may assume that f is affine. Let x and xi ’s be the images of x in X and the Xi ’s. The schemes Ui = Spec(OXi ,xi ) ×Xi Y and U = Spec(OX,x ) ×X Y are affine, because f and, therefore, fi ’s are affine. 6 M. TEMKIN Each Ui consists of points y ∈ Y , such that xi is a specialization of fi (y), the morphisms Ui → Y are topological embeddings and OY |Ui f →OUi . Clearly, U and x satisfy similar properties. Notice that the schemes Ui = Spec(Bi ) form a filtered family and U∞ := proj lim Ui = Spec(B∞ ), where B∞ = inj lim Bi . By [EGA IV], ch. 8, U∞ = ∩Ui set-theoretically. Recall that schematical density is preserved under base changes with respect to schematically dense morphisms, therefore the morphism Ui → Spec(OXi ,xi ) is schematically dominant. Thus, OXi ,xi ,→ Bi and we obtain an embedding of direct limits OX,x = inj lim OXi ,xi ,→ B∞ . Lemma 2.1.2. Suppose that elements g, h ∈ B∞ do not have common zeros on U∞ , then either g ∈ hOx or h ∈ gOx . Proof. Find i such that g and h are defined and have not common zeros on Ui , and replace X by Xi . Notice that U∞ = ∩f −1 (Vj ), where Vj runs over affine neighborhoods of x. Hence we can choose a neighborhood X 0 = Spec(A) of x, such that g, h ∈ B and gB + hB = 1, where Y 0 = Spec(B) is the preimage of X 0 in Y . Now, the pair (g, h) induces a morphism λ0 : Y 0 → P1A = Proj(A[Tg , Th ]), whose scheme-theoretic image Xi0 is a Y 0 -modification of X 0 . It would suffice to extend the Y 0 -modification Xi0 → X 0 to a Y -modification fi : Xi → X, where Xi0 is an open subscheme of Xi . Indeed, either Tg ∈ Th Ox0 or Th ∈ Tg Ox0 , where x0 is the center of x on Xi0 . If fi exists, then we deduce that either g ∈ hO0 or h ∈ gO0 , where O0 denotes the image of Ox0 in B∞ . Probably such an extension fi does not exist in general, but fortunately the following trick works. Given elements a, b ∈ B with (a, b) = 1, the morphism Y 0 → Proj(A[Tag , Tah , Tbg , Tbh ]) factors through Proj(A[Tg , Th ]) × Proj(A[Ta , Tb ]). Therefore, Xj0 is dominated by the scheme-theoretic image Z of Y 0 in Proj(A[Tag , Tah , Tbg , Tbh ]). If we choose a, b such that ag + bh = 1, then Z is isomorphic to the scheme-theoretic image of Y 0 in P 0 = Proj(A[T1 , Tag , Tah , Tbg , Tbh ]). Finally, the morphism Y 0 → P 0 may be extended as follows. By [EGA I], 6.9.7, the A-submodule E ⊂ B generated by ag, ah, bg, bh may be extended to a finitely generated OX -submodule E ⊂ F = f∗ (OY ). Replacing E by E + OX , we can achieve that the morphism E ⊗F → F is onto and still E(X 0 ) = E. Since f is affine, we obtain an epimorphism f ∗ (E) → OY . Applying [EGA II], 4.2.3, we obtain a morphism Y → PX (E) = Proj(S(E)), which obviously extends Y 0 → P 0 . ¤ Let us prove that B∞ is a local ring. Suppose conversely that the affine U∞ contains two different closed points y and z, so there exist g, h ∈ B∞ such that g(y) = h(z) = 0 and g and h have no common zeros on U∞ . Now, the lemma implies that either g|h or h|g, contradicting our assumptions. We see that U∞ possesses a unique closed point y. In particular, it follows immediately that U∞ = Spec(OY,y ) and OX,x ⊂ OY,y . Moreover, the lemma implies that my ⊂ OX,x and the ring R = OX,x /my is a valuation ring of RELATIVE RIEMANN-ZARISKI SPACES 7 k(y). To prove the first claim we pick up an element g ∈ my and apply the lemma to g and h = 1. Since 1 ∈ / my ⊃ gOX,x , we obtain that h ∈ OX,x . To prove the second claim we pick up a non-zero element ge ∈ k(y) and apply the lemma to any its lifting g ∈ B∞ and h = 1. It follows that either ge or ge−1 is contained in R, thus proving the second claim. Define φ : S = Spec(R) → X as the composition of the closed immersion S → Spec(OX,x ) and the natural morphism Spec(OX,x ) → X. For any i, Ui = Spec(Oxi )×Xi Y is a closed subscheme of Spec(Oxi )×X Y , hence the closure of the image of y in Spec(Oxi ) ×X Y coincides with the closure of y in Ui . It follows that the closure of the image of y in Spec(Ox )×X Y coincides with the closure of y in U∞ = ∩Ui . Since y is closed in U∞ , we obtain that the morphism y → Spec(Ox ) ×X Y is a closed immersion. Hence the morphism y → S ×X Y is a closed immersion too, and we obtain that y = (y, R, φ) is the required element of ValY (X). ¤ In the sequel, by a semi-valuation ring we mean a ring O with a valuation | | such that for any pair g, h ∈ O with |g| ≤ |h| 6= 0, one has h|g. Notice that the prime ideal m = Ker(| |) is (O \ m)-divisible, so O embeds into A = Om , mA = m, and O is composed from the local ring A and the valuation ring R = O/m, i.e. O is the preimage of R ⊂ A/m in A. We say that A is a quotient ring of O. Conversely, any ring composed from a local ring and a valuation ring is easily seen to be a semi-valuation ring. In the situation of the above proposition, Ox is a semi-valuation ring composed from Oy and R. Semi-valuation rings play the same role in the theory of relative RZ spaces as valuation rings play in the theory of usual RZ spaces. 2.2. Applications. A preliminary description of relative Riemann-Zariski spaces, obtained in the previous section, suffices for some applications. In this section we fix a qcqs scheme S with a schematically dense quasi-compact subset U closed with respect to generalizations. An S-scheme X is called U -admissible if the preimage of U in X is schematically dense. By a U etale covering we mean a separated morphism h : S 0 → S, such that h is etale over U , S 0 is U -admissible and for any valuation ring R, any morphism Spec(R) → S lifts to S 0 . Note that in [BLR] one considers a more restrictive class of U -etale coverings S 0 → S, which split to a composition of a surjective flat U -etale morphism and a U -modification. However, it follows from the flattening theorem of Raynaud-Gruson, see [RG], 5.2.2, that the coverings of the latter type form a cofinal subclass. Let us consider the following situation: (A, m) is a local ring, U = Spec(A), K = A/m, R ⊂ K is such that Q(R) = K, S = Spec(O) and O is composed from A and R, i.e. O is the preimage of R in A. Notice that S is covered by U and T = Spec(R), and T ∩ U = η = Spec(K). Lemma 2.2.1. Suppose that we are given an η-admissible T -scheme XT , a U -scheme XU and an isomorphism XT ×T ηf →XU ×U η, then: 8 M. TEMKIN (i) there exists a unique way to glue XT and XU to a U -admissible Sscheme X; (ii) X is of finite type (resp. finite presentation) iff XU and XT are so. Proof. Let us prove first that an affine U -admissible S-scheme X = Spec(C) is uniquely determined by XU = Spec(CU ) and XT = Spec(CT ). By admissibility, C ⊂ C ⊗O A = CU and C/mC = CT ⊂ CU ⊗O K, hence C is the preimage of CT in CU . Moreover, if Y is another affine U -admissible S-scheme, then the same argument shows that any morphism Y → X is uniquely defined by its restrictions YU → XU and YT → XT , i.e. MorS (X, Y )f →MorU (XU , YU ) ×Morη (Xη ,Yη ) MorT (XT , YT ). The remaining part is very standard, so we only outline it. Using the above bijection, one proves that it holds for any pair of qcqs U -admissible Sschemes X and Y . Then (i) is easily deduced, because the required scheme X is glued from affine pieces in an essentially unique way. The second part of the lemma is exactly the Step 3 of proposition 2.4.4 from [Tem2]. ¤ Now we prove a stable modification theorem, which generalizes its analog from [Tem2]. We refer to loc.cit., 1.6 and 1.7, for terminology. Theorem 2.2.2. Let C be an S-multipointed curve with semi-stable U fibers, then there exists a U -etale covering S 0 → S, such that the curve C ×S S 0 admits a stable U -modification Cst . Proof. Step 1. Reduction to the case of S = Spec(O) and U = Spec(A), where O is a semi-valuation ring and A is its quotient ring. Since C is semi-stable over an open subscheme of S, we can enlarge U to an open schematically dense qcqs subscheme. Then by noetherian approximation, we can assume that S is of finite Z-type. Set S = RZU (S), it is a qcqs topological space by [Tem2], B.2.3. For any point x = (y, R, φ) ∈ S, set Sx = Spec(OS,x ), Ux = Spec(OY,y ) and Cx = C×S Sx . If the case mentioned in the formulation of the step is known, then there exists a Ux -etale covering Sx0 → Sx , such that the Sx0 -multipointed curve Cx ×Sx Sx0 admits a stable Ux -modification. Let xi be the centers of x on U -modifications Si of S, then ∪OSi ,xi = OS,x . By approximation, there exists i = i(x) and a U -etale morphism hx : Si0 → Si , such that xi lies in its image and C ×S Si0 admits a stable U -modification. Furthermore, by théorème de platification par éclatements, see [RG], 5.2.2, we can assume that hx is flat: replace Si by some its U -modification and replace Si0 by its strict transform with respect to that modification. Then the preimage of hx (Si0 ) in S is an open subset. Since S is quasi-compact, we need only finitely many points x1 , . . . , xn to cover the whole S. Then 0 tnj=1 Si(x is a required U -etale covering of S. j) Step 2. The theorem holds over a semi-valuation ring O. We assume that O is composed from a local ring (A, m) and a valuation ring R ⊂ K = A/m, S = Spec(O) and U = Spec(A). Set also T = Spec(R) and η = Spec(K). By RELATIVE RIEMANN-ZARISKI SPACES 9 [Tem2], 1.7.5, the theorem is known in the case of a valuation ring, i.e. the case when m = 0. Find a finite separable extension K 0 /K with a valuation ring R0 lying over R, such that C ×S T 0 admits a stable modification, where T 0 = Spec(R0 ). Let A0 be an A-etale local ring, such that A0 /mf →K 0 , and 0 0 0 let O be the semi-valuation ring composed from A and R . Replacing O by O0 , we can assume that already C ×S T admits a stable modification CT0 . By lemma 2.2.1, we can glue CU and CT0 to a U -modification C 0 → C. It remains to show, that C 0 is an S-curve, then it is clear that C 0 is a stable U modification of C. By 2.2.1, C 0 is of finite presentation over S. Notice, that S has no non-trivial U -modifications, because T is the only modification of itself. Applying the flattening theorem, see [RG], 5.2.2, we obtain that C 0 is S-flat. ¤ A scheme version of the reduced fiber theorem of Bosch-LütkebohmertRaynaud, see [BLR], 2.1’, may be proven absolutely analogously. Theorem 2.2.3. Let X → S be a schematically dominant finitely presented morphism, whose U -fibers are geometrically reduced. Then there exists a U -etale covering S 0 → S, and a finite U -modification X 0 → X ×S S 0 , such that X 0 is flat, finitely presented and has reduced geometric fibers over S 0 . Proof. If S is the spectrum of a valuation ring and U is its generic point, then the theorem follows from [Tem2], 1.8.4 (actually it was the content of Steps 1–4 of the loc.cit.). Acting as in Step 2 of the previous proof, we deduce the case when S is the spectrum of a semi-valuation ring and U is the corresponding local scheme. Then it remains to repeat the argument of Step 1. ¤ 3. Nagata compactification 3.1. Connection to adic spaces. Let A be a ring and B be an A-algebra. Following [Hub1], we define a valuation on B as a commutative ordered group Γ with a multiplicative map | | : B → Γ ∪ {0}, which satisfies the strong triangle inequality and sends 1 to 1. Recall, that if B is a field, then B ◦ = {x ∈ B| |x| ≤ 1} is a valuation ring, and the correspondence B 7→ B ◦ gives a one-to-one correspondence between the equivalence classes of valuations on B and valuation rings B ◦ with fraction field B. In general, a valuation is defined up to an equivalence by its kernel p, which is a prime ideal, and by the induced valuation on the residue field Q(B/p). By slight abuse of language, we will often identify equivalent valuations. In loc.cit., the set Spv(B) of all equivalence classes of valuations of B is provided with the weakest topology such that the sets of the form {| | ∈ Spv(B)| |a| ≤ |b| 6= 0} are open, where a, b ∈ B. Huber gives an elegant proof that the resulting topological space is quasi-compact, see [Hub1], 2.2. Furthermore, he considers the quasi-compact subspace Spa(B, A) ⊂ Spv(B) consisting of valuations of B such that |A| ≤ 1, see the definition on p. 467 (here A and B are considered as topological rings with discrete topology; note also that 10 M. TEMKIN Huber actually considers the case when A is an integrally closed subring of B, but it does not really restrict the generality, because replacing A by the integral closure of its image in B does not change the space Spa(B, A)). Actually, the topological space Spa(B, A) has a much finer structure of an adic space, but we do not need it. Let us generalize the above paragraph to schemes. If Y is a scheme, then by a valuation on Y we mean a pair y = (y, R), where y ∈ Y is a point and R is a valuation ring in k(y). One can define y by giving a valuation | |y : Oy → Γy , whose kernel is my . By Oy we denote the subring of Oy given by the condition | |y ≤ 1, it is the preimage of R in Oy . Furthermore, if f : Y → X is a morphism of schemes, then by an X-valuation on Y me mean a valuation y = (y, R) provided with a morphism φ : S = Spec(R) → X which is compatible with the natural morphism η = Spec(k(y)) → X. Recall that in the valuative criteria of properness/separatedness, one considers commutative diagrams of the form η i /Y ² φ ² S (1) f /X where S = Spec(R) is the spectrum of a valuation ring and η = Spec(K) is its generic point, and studies liftings of S to Y . We will prove in 3.1.6 that it suffices to consider the only case when k(i(η))f →K. The diagrams of the latter particular type are exactly the diagrams which correspond to X-valuations of Y . Note also, that a valuation y = (y, R, φ) gives rise to the following finer diagram η / Yy /Y ² ² /T ² /X S (2) where Yy = Spec(Oy ) and T = Spec(Oy ). e = Spa(Y, X) denote the set of all equivalence classes of X-valuations Let X of Y . Given a morphism f 0 : Y 0 → X 0 and a morphism g : f 0 → f consisting of a compatible pair of morphisms gY : Y 0 → Y and gX : X 0 → X, there is a natural map Spa(f ) : Spa(Y 0 , X 0 ) → X which to a point (y 0 , R0 , φ0 ) associates a point (y, R, φ), where y = gY (y 0 ),R = R0 ∩ k(y) and φ is defined as follows. The morphism gX ◦ φ0 : Spec(R0 ) → X factors through Spec(OX,x ), where x is the image of the closed point of the source, hence we obtain a homomorphism α : OX,x → R0 . Since the morphism Spec(k(y 0 )) → X factors uniquely through Spec(k(y)), the image of α lies R = R0 ∩ k(y). So, gX ◦ φ0 factors uniquely through a morphism φ : Spec(R) → X. We proved that Spa(Y, X) depends functorially on f . Also, it is easy to check that Spa respects fibred products, i.e. Spa(Y 0 ×Y Y 0 , X 0 ×X X 00 )f →Spa(Y 0 , X 0 ) ×X Spa(Y 00 , X 00 ). RELATIVE RIEMANN-ZARISKI SPACES 11 If Y 0 is a subscheme of Y and gX is separated, then φ is injective. Indeed, e has a non-empty preimage in X e 0 = Spa(Y 0 , X 0 ), if a point y = (y, R, φ) ∈ X then y ∈ Y 0 and any preimage of y is given by a lifting of φ : Spec(R) → X to X 0 , which is unique by the valuative criterion of separatedness. If Y 0 is e 0 is open and affine, and X 0 is affine and of finite X-type, then we say that X e If X1 , X2 ⊂ X are affine subsets, then X12 = X1 ∩ X2 is an affine subset of X. a finite union of affine subsets (use that X12 f →X1 ×X X2 ), and X12 is affine if e X and Y are separated. We provide X with the weakest topology in which all affine sets are open; obviously, affine sets form a basis of this topology. The two following statements follows easily: if Y 0 ,→ Y is an open immersion and X 0 → X is separated and of finite type, then Spa(Y 0 , X 0 ) ,→ Spa(Y, X) is an open embedding, and the maps Spa(g) are continuous. Lemma 3.1.1. If X = Spec(A) and Y = Spec(B) are affine, then the canonical bijection X = Spa(Y, X) → Spa(B, A) is a homeomorphism. Proof. It follows from the definitions of the topologies, that the map is e 0 = Spa(Spec(C), Spec(A0 )) be an affine subset, where continuous. Let X Spec(C) is an open subscheme of Spec(B) and A0 is a finitely generated e0 ⊆ X e is a neighborhood of each point A-algebra. It suffices to prove that X 0 z ∈ Spa(B, A) lying in it. Replacing A by its image in C, we can assume that it is an A-subalgebra of C generated by h1 , . . . , hn ∈ C. Notice that if {Ui } is an open covering of Spec(C), then the sets Spa(Ui , Spec(A0 )) cover X. Therefore, shrinking Spec(C), we may assume that C = Bb for an e 0 consists element b ∈ B. Then hi = bi /bm , where bi ∈ B, m ∈ N, and X e 0 is open in of all valuations of B with |bi | ≤ |bm | = 6 0 for any i. Thus, X Spa(B, A), and we are done. ¤ Corollary 3.1.2. If X and Y are qcqs schemes, then the space Spa(Y, X) is qcqs. Let B be a ring and | | : B → Γ ∪ {0} be a valuation. By y ∈ Spec(B) we denote the point corresponding to the kernel of | |. We say that a convex subgroup H ⊂ Γ bounds B, if for any element b ∈ B, there exists an element h ∈ H with |b| < h. For any such subgroup, we can define a valuation | |H : B → H by the rule |x|H = |x| if |x| ∈ H and |x|H = 0 otherwise. Recall that | |H is called a primary specialization of | |, see [Hub1], 2.3. It follows from H being convex, that the order on Γ induces an order on the group Γ/H, and | | induces a valuation | |/H : B → Γ/H. Notice that |B|/H ≤ 1 because H bounds B. Therefore the image of B in k(y) lies in a valuation ring R0 corresponding | |/H, and the morphism y → Spec(B) extends to a morphism Spec(R0 ) → Spec(B). Let, more generally, (y, R) be a valuation on a scheme Y . Notice that any ring R ⊆ R0 ⊆ k(y) is a valuation ring. Assume that the morphism Spec(k(y)) → Y extends to a morphism S 0 = Spec(R0 ) → Y and let y be the image of the closed point s ∈ S 0 . The image of R in k(s) is a valuation 12 M. TEMKIN ring whose intersection with k(y) will be denoted R. The pair (y, R) is a valuation on Y , which is called a primary specialization of (y, R). In general, by a primary specialization of an X-valuation y = (y, R, φ), we mean an Xvaluation y = (y, R, φ), such that (y, R) is a primary specialization of (y, R) and the image of φ in X is contained in the image of φ in X. Primary specialization is a particular case of a specialization relation in Spa(Y, X), i.e. any neighborhood of y contains y. A valuation (y, R, φ) is called minimal if it has no non-trivial primary specializations. In the sequel we assume that f : Y → X is a separated morphism of qcqs schemes, unless the contrary is stated explicitly. Lemma 3.1.3. Keep the above notations and let y = (y, R, φ) be an Xvaluation on Y , then: i) the set of primary specializations of y is non-empty, totally ordered and contains a minimal element; (ii) y is minimal if and only if the morphism h : Spec(k(y)) → Y ×X Spec(R) is a closed immersion. Proof. Let X 0 be a separated neighborhood of the image of φ and Y 0 be its preimage. Replacing X and Y by X 0 and Y 0 , we can achieve that Y is separated. It follows from the valuative criterion of separatedness, that for any valuation ring R ⊆ R0 ⊆ k(y), there is at most one morphism Spec(R0 ) → Y compatible with y → Y . Since the set of valuation rings R0 as above is totally ordered with respect to inclusion, we obtain a total order on the set {(yi , Ri }i∈I of primary specializations of y. Provide I with the corresponding order, then I possesses a maximal element corresponding to the trivial specialization (y, R). Choose elements i ≤ j in I, and let R ⊆ Ri0 ⊆ Rj0 ⊆ k(y) be the corresponding valuation rings. Then the morphism Spec(Rj0 ) → Y is a composition of Spec(Rj0 ) → Spec(Ri0 ) and Spec(Ri0 ) → Y , in particular yi specializes yj . By quasi-compactness of Y , there exists a point y ∈ Y which specializes all yi ’s. Each (yi , Ri ) induces a valuation | |i on Oy , and | |i is a primary specialization of | |j for i ≤ j. Then | |i = (| |0 )Hi , for some convex groups Hi bounding Oy . Hence H = ∩Hi bounds Oy and (| |0 )H defines a primary specialization of y, which specializes all valuations (yi , Ri ). Let us prove (ii). Notice that y is not minimal iff there exist open affines X 0 ⊂ X and Y 0 ⊂ Y , such that f (Y 0 ) ⊂ X 0 , y ∈ Y 0 and y is not minimal as an X 0 -valuation on Y 0 . Analogously, h is not a closed immersion iff for some choice of X 0 and Y 0 as above, the morphism Spec(k(y)) → Y 0 ×X 0 Spec(R) is not a closed immersion. Thus, we may assume that our schemes are affine, say X = Spec(A) and Y = Spec(B). The morphism from the lemma is not a closed immersion iff the homomorphism B ⊗A R → k(y) is not onto. The latter happens iff B is bounded by an element r ∈ |k(y)|, where we provide k(y) with a valuation corresponding to R. Finally, such r exists iff the valuation is not minimal. ¤ RELATIVE RIEMANN-ZARISKI SPACES 13 It follows from the second part of the lemma that the set X = ValY (X) e consisting of minimal defined in section 2.1 coincides with the subset of X X-valuations. Note that in affine situation, such subsets were considered by Huber, see [Hub1], 2.6 and 2.7. We provide X with the induced topology. The following lemma follows easily from the valuative criterion of properness, so we omit the proof. Lemma 3.1.4. (i) If X 0 is a Y -modification of X, then there are natural homeomorphisms Spa(Y, X 0 )f →Spa(Y, X) and ValY (X 0 )f →ValY (X). 0 (ii) If X is an open subscheme of X, then its preimage in ValY (X) (resp. Spa(Y, X)) is canonically homeomorphic to ValY 0 (X 0 ) (resp. Spa(Y 0 , X 0 )), where Y 0 = X 0 ×X Y . Caution: if Y 0 is an open qcqs subscheme of Y and the morphism Y 0 → X factors through an X-separated scheme X 0 of finite type, then Spa(Y 0 , X 0 ) e However, it may (and maps homeomorphically onto an open subspace of X. e is not contained in usually does) happen that the image of ValY 0 (X 0 ) in X X. This problem originates from the fact that a valuation, which has no proper primary specializations on Y 0 , may have such a specialization on Y . This observation enforces us to introduce a notion of open subdomain of X in section 3.3. Proposition 3.1.5. Assume that f : Y → X is a separated morphism of qcqs schemes. Then the spaces ValY (X) and RZY (X) are quasi-compact. e i ⊂ X, e such Proof. Let {Xi }i∈I be an open covering of X. Find open sets X e i ∩ X. Since any point of X e has a specialization in X, {X e i }i∈I that Xi = X e By quasi-compactness of X, e we can find a subcovering is a covering of X. e {Xi }i∈J with finite J, and then {Xi }i∈J is a finite covering of X. Thus, X is quasi-compact. Let us prove that for any Y -modification f : X 0 → X, the map φ : X → X 0 is continuous. It suffices to show that φ−1 (U ) is open, where U ⊂ X 0 is e is covered by open an open affine subscheme. The preimage of U in X 0 e sets Xi = Spa(Yi , X ), where {Yi } is an open affine covering of f −1 (X 0 ). e i ∩X, as required. Therefore, the preimage of U in X is covered by open sets X It follows, that the map ψ : X → RZY (X) is continuous. Since X is quasicompact and ψ is surjective, the space RZY (X) is quasi-compact. ¤ In the sequel, we will need to strengthen the classical valuative criterions of separatedness, see [EGA II], 7.2.3, and properness, see loc.cit., 7.3.8. Our aim is to show, that it suffices to consider valuative diagrams of specific types. We say that a morphism is compatible with a commutative diagram, if the diagram remains commutative after adjoining this morphism. Until the end of the section, f is not assumed to be separated. Lemma 3.1.6. Keep the notations of diagram (1) and set K 0 = k(i(η)) and R0 = R ∩ K. Then diagram (1) completes uniquely to a commutative 14 M. TEMKIN diagram Spec(K) / Spec(K 0 ) /Y ² ² / Spec(R0 ) ² /X Spec(R) and any morphism h : Spec(R) → Y compatible with the above diagram is induced by a morphism h0 : Spec(R0 ) → Y compatible with the diagram and determines h0 uniquely. Proof. The morphism Spec(K) → Y obviously splits through Spec(K 0 ). The morphism Spec(R) → X factors through Spec(OX,x ), where x is the image of the closed point of Spec(R). The image of Ox in R ⊂ K is contained in K 0 , hence the morphism Spec(R) → X factors through Spec(R0 ). By the same reasoning, a morphism h : Spec(R) → Y compatible with the diagram factors through h0 : Spec(R0 ) → Y , and they both are determined uniquely by the image of the closed point in Y . ¤ Lemma 3.1.7. Keep the notations of diagram (1) and assume that R ⊂ R0 ⊂ K is such that the morphism Spec(R0 ) → X admits a lifting g : e be the Spec(R0 ) → Y compatible with the morphism Spec(K) → Y . Let K 0 e e residue field of R and R the image of R in K. /Y w; w g www ww w ww Spec(K) e Spec(K) ² e Spec(R) ² / Spec(R0 ) ² / Spec(R) f ² /X e → Y compatible with the above diagram Then any morphism e h : Spec(R) is induced from a morphism h : Spec(R) → Y compatible with the diagram and determines h uniquely. e → Y compatible with the diagram. Proof. Consider a morphism e h : Spec(R) It suffices to show that it factors through Spec(R), the uniqueness is again e and x = f (y), then trivial. Let y be the image of the closed point of Spec(R) e h induces a homomorphism Oy → R. Since y specializes the image of the closed point of Spec(R0 ), we have also a homomorphism Oy → R0 . Finally, e = R0 /mR0 lies in R. e the compatibility implies that the image of Oy in K e Therefore, the image of Oy in R0 lies in R, and the homomorphism Oy → R factors through R. It gives the desired morphism h : Spec(R) → Y . ¤ Proposition 3.1.8. Let h : Z → Y and f : Y → X be morphisms of qcqs schemes, and ψ : Spa(Z, Y ) → Spa(Z, X) and ψ : ValZ (Y ) → ValZ (X) be the natural maps. Assume that h is dominant and g = f ◦ h is separated. RELATIVE RIEMANN-ZARISKI SPACES 15 (i) The morphism f is separated if and only if ψ is injective if and only if ψ is injective. (ii) Let us assume in addition that f is of finite type, then f is proper if and only if ψ is bijective if and only if ψ is bijective. Proof. The direct implications follows from the standard valuative criterions. Suppose that f is separated and not proper (resp. not separated), then by the standard valuative criterion and lemma 3.1.6, there exists an element y = (y, Ry , φy ) ∈ Spa(Y, X), such that the number of liftings of the morphism φy : Spec(Ry ) → X to Y is zero (resp. at least two). Let x denote the center of y on X. By [EGA I], 6.6.5, there exists a point z ∈ Z, such that y specializes h(z). Let R0 be any valuation ring of k(z), which dominates the image of Oy . It gives rise to an element (z, R0 , φ0 ) ∈ Spa(Z, Y ) centered on y. Choose e of the residue field K e of R0 , such that R e dominates the a valuation ring R e and define a valuation ring R of k(z) as the valuation ring Ry of k(y) ⊂ K, 0 e composition of R and R. The compatible homomorphisms Ox → Oy → R0 e induce a homomorphism Ox → R, and we obtain the and Ox → Ry → R following commutative diagrams. e Spec(K) ² e Spec(R) Spec(k(z)) /Z ² 0 / Spec(R0 ) φ ² /Y ² / Spec(R) φx ² /X e Spec(K) / Spec(k(y)) /Y ² ² / Spec(Ry ) ² /X e Spec(R) By lemma 3.1.6, there is a one-to-one correspondence between morphisms e → Y compatible with the right diagram. By Spec(Ry ) → Y and Spec(R) lemma 3.1.7, the latter morphisms are in one-to-one correspondence with the morphisms φ : Spec(R) → Y compatible with the left diagram. Notice that z = (z, R, φx ) is an element in Spa(Z, X), and any morphism φ as above gives a preimage of z in Spa(Z, Y ). We obtain that in the case (i), z has at least two preimages, and ψ is not injective. To prove (ii), we notice that the morphism φ0 is uniquely determined by morphisms Spec(k(z)) → Y and Spec(R0 ) → X, because Y → X is separated by our assumption. Therefore, any preimage of z in Spa(Z, Y ) comes from a morphism φ : Spec(R) → Y compatible with the left diagram. We obtain that z has no preimages under ψ, in particular ψ is not bijective. It remains to relate the fibers of ψ and ψ. Consider the minimal primary specialization z0 ∈ ValZ (X) of z. Then lemma 3.1.7 implies that the sets −1 ψ (z) and ψ −1 (z0 ) are naturally bijective. ¤ 16 M. TEMKIN 3.2. Y -blow ups of X. In this section we assume that f is affine. Under this assumption, there exists a large family of projective Y -modifications of X having good functorial properties. A Y -modification gi : Xi → X is called a Y -blow up of X if there exists a gi -ample OXi -module L provided with a homomorphism ε : OXi → L, such that fi∗ (ε) : OY f →fi∗ (L). It will be more convenient to say X-ample instead of gi -ample in the sequel. We call ε a Y -trivialization of L. Lemma 3.2.1. The following statements hold. (i) Suppose that Xj → Xi and Xi → X are Y -modifications such that Xj is a Y -blow up of X, then Xj is a Y -blow up of Xi . (ii) The family of Y -blow ups of X is filtered. (iii) The composition of Y -blow ups gij : Xj → Xi and gi : Xi → X is a Y -blow up. Proof. The first statement is obvious, because any X-ample OXj -module L is Xi -ample, and the notion of Y -trivialization of L depends only on the morphism Y → Xj . (ii) Let Xi , Xj be two Y -blow ups of X. Find X-ample sheaves Li , Lj with Y -trivializations εi , εj . Then the X-proper scheme Xij = Xi ×X Xj possesses an X-ample sheaf L = p∗i (L1 ) ⊗ p∗j (L2 ), where pi , pj are the projections. The natural isomorphism OXij f →OXij ⊗ OXij followed by fi∗ (εi ) ⊗ fj∗ (εj ) : OXij ⊗ OXij → L provides a Y -trivialization of L. Consider the schemetheoretic image X 0 of Y in Xij , and let L0 and ε0 be the pull backs of L and ε. Then (X 0 , L0 , ε0 ) is a Y -blow up of X which dominates Xi and Xj . (iii) Choose an X-ample OXi -sheaf Li and an Xi -ample OXj -sheaf Lj with ∗ (L⊗n ) is Y -trivializations εi nd εj . By [EGA II], 4.6.13 (ii), the sheaf Lj ⊗ gij i X-ample for sufficiently large n. It remains to notice that the composition ⊗n ∗ (ε⊗n ) is a Y -trivialization. of OXj f →OXj ⊗ OX with εj ⊗ gij ¤ i j We will need an explicit description of Y -blow ups. Let F = f∗ (OY ), and assume that E ⊂ F is a finitely generated OX -submodule containing the image of OX , the canonical homomorphism of OY -modules µY : f ∗ (E) → OY is surjective. By [EGA II], 4.2.3, we obtain a morphism h : Y → PX (E) = P , and the unit section homomorphism e : OX → E induces a homomorphism εP : OP → OP (1). Define XE to be the scheme-theoretic image of h and define L and ε to be the pull backs of OP (1) and εP to XE . Obviously, XE is a Y -modification of X and L is X-ample. It remains to notice that ε is a Y -trivialization, because µY ◦ f ∗ (e) : OY → OY is the identity. Lemma 3.2.2. Any Y -blow up of X is isomorphic to some XE . Proof. Let gi : Xi → X be a Y -blow up. Find an X-ample OXi -module L and its Y -trivialization ε. Replacing L by L⊗n and changing ε accordingly, we may assume that L is very ample, i.e. there exists a finitely generated quasi-coherent OX -module E and a closed immersion h : Xi → P = PX (E) RELATIVE RIEMANN-ZARISKI SPACES 17 ∗ such that φ : Lf →h (OP (1)). The morphism h with isomorphism φ correspond to an epimorphism H : gi∗ (E) → L by [EGA II], 4.2.3. Set E = E ⊕ OX , then the epimorphism H = H + ε : gi∗ (E) → L defines a morphism h : Xi → P = PX (E), which is a closed immersion because already E ⊂ E induces a closed immersion h. The second summand embedding e : OX → E defines a homomorphism εP : OP → OP (1), whose pull back to Xi is equal to ε; in particular, εP is a Y -trivialization of OP (1). The morphism λ = h ◦ fi : Y → P corresponds to a morphism E ⊗ F → F. Let E 0 be the image of E ⊗ OX in F, then λ factors through PX (E 0 ), and one can easily seethat Xi is isomorphic to the scheme-theoretic image of Y in PX (E 0 ), and ε is induced by the section e : OX → E 0 . ¤ Corollary 3.2.3. Assume that X 0 is an open subscheme of X and Y 0 = f −1 (X 0 ). Then any Y 0 -blow up Xi0 → X 0 extends to a Y -blow up Xi → X. Proof. Let f 0 : Y 0 → X 0 be the restriction of f , so F 0 = f∗0 (OY 0 ) is the restriction of F = OY on X 0 . By the lemma, a Y 0 -blow up of X 0 is determined by a finitely generated OX 0 -submodule E 0 ⊂ F 0 containing the image of OX 0 . By [EGA I], 6.9.7, one can extend E 0 to a finitely generated OX -submodule E ⊂ F. Replacing E by E + OX , if necessary, we can achieve that E contains the image of OX . Now, E defines a required extension of the blow up. ¤ The extension property of the corollary dictated our choice of the class of Y -blow ups of X. The following lemma is an analog of [BL], 4.4. Lemma 3.2.4. Given a quasi-compact open subset U ⊂ X = ValY (X), there exists a Y -modification X 0 → X and an open subscheme U ⊂ X 0 , such that U is the preimage of U in X. Proof. If X1 , . . . , Xn form a finite open affine covering of X and Yi = f −1 (Xi ), then Xi = ValYi (Xi ) form a covering of X. It suffices to solve our problem for each Xi separately, because any Yi -blow up of Xi extends to a Y -blow up of X, and Y -blow ups of X form a filtered family. Thus, we can assume that X = Spec(A), and then Y = Spec(B). We can furthermore assume that U = X ∩ Spa(Bb , A[a/b]), where a, b ∈ B, because as we saw in the proof of lemma 3.1.1, U is covered by finitely many subsets of this form. Now, the morphism Y → Proj(A[x0 , x1 , x2 ]), defined by (1, a, b), determines a required Y -blow up X 0 → X with U given by the condition x2 6= 0. ¤ Corollary 3.2.5. The map ψ : ValY (X) → RZY (X) is a homeomorphism. Proof. We already know that ψ is surjective and continuous. From other side, the lemma implies that ψ is injective and open. Indeed, for any pair of different points of X, there exists an open quasi-compact set U which distinguishes the points, and the lemma implies that they are distinguished already in some Y -modification X 0 of X. Hence ψ is injective, and its openness follows obviously. ¤ 18 M. TEMKIN We use the corollary to identify X with RZY (X), in particular, X is provided with a sheaf OX of regular functions. As another corollary, we obtain the following version of Chow lemma. Corollary 3.2.6. Any Y -modification X → X is dominated by a Y -blow up of X. Proof. Let U 1 , . . . , U n be an affine covering of X. By Yi and Ui we denote the preimages of U i in Y and X, respectively. By the lemma, we can find a Y -blow up X 0 → X and a covering {Ui0 } of X 0 , whose preimage in X coincides with {Ui }. Consider the scheme X 00 , which is the scheme-theoretic image of Y in X ×X X 0 . The preimages of U i and Ui0 in X 00 coincide, because their preimages in X coincide. Let Ui00 denote the corresponding subscheme of X 00 . For any 1 ≤ i ≤ n, the morphism hi : Ui00 → Ui0 is affine, because the morphism U i → X is affine (U i is affine and X-separated). From other side, hi is proper because it is a base change of the Y -modification h : X 00 → X 0 . Thus, hi is finite, and therefore h is finite. We claim that h is a Y -blow up (this claim is an analog of [BL], 4.5). Indeed, OX 00 is very ample relatively to h, because h is affine, and the identity homomorphism gives its Y -trivialization. By lemma 3.2.1 (iii), X 00 is a Y -blow up of X, and obviously X 00 dominates X. ¤ 3.3. Affinoid domains. Let f 0 : Y 0 → X 0 be a separated morphism of e 0 = Spa(Y 0 , X 0 ). Suppose that we are qcqs schemes, X0 = ValY 0 (X 0 ) and X given a compatible pair of morphisms gY : Y 0 → Y , gX : X 0 → X, then e 0 → X, e which takes (y 0 , R0 , φ0 ) to there is a natural continuous map ge : X 0 0 0 (y = gY (y ), R = R ∩ k(y ), φ). Proposition 3.3.1. Suppose that gY is an open immersion and gX is separated. Then ge(X0 ) ⊂ X if and only if the natural morphism h : Y 0 → Y ×X X 0 is a closed immersion. Proof. Suppose that h is a closed immersion. Let (y 0 , R0 , φ) ∈ X0 be a point e Consider the schemes η 0 = Spec(k(y 0 )) and and (y, R, φ) its image in X. 0 0 S = Spec(R ), the natural morphism η 0 → Y 0 ×X 0 S 0 is a closed immersion. Therefore the composition morphism η 0 → Y 0 ×X 0 S 0 → (Y ×X X 0 ) ×X 0 S0f →Y ×X S 0 induced by h is a closed immersion. It follows that the morphism Spec(k(y)) → Y ×X Spec(R) is a closed immersion too, as required. Suppose now that h is not a closed immersion. Notice, that h is a locally closed immersion, because gY is an open immersion. It follows that we can find a Y ×X X 0 -valuation y0 = (y 0 , R0 , φ) of Y 0 , such that the morphism φ : Spec(R0 ) → Y ×X X 0 cannot be lifted to a morphism Spec(R0 ) → Y 0 . Replacing y0 by its minimal primary specialization, we achieve that y0 is minimal and R0 ( k(y 0 ). Now, it is clear that y0 defines a minimal X 0 e is not minimal, because the morphism valuation on Y 0 , but its image in X 0 Spec(R ) → X lifts to a morphism Spec(R0 ) → Y . ¤ RELATIVE RIEMANN-ZARISKI SPACES 19 Let us assume that gY is an open immersion and gX is separated and of finite type. We saw that, if h is a closed immersion, then via ge X0 is naturally identified with a quasi-compact open subset of X. In this case we say that the map X0 → X is an open immersion and X0 is an open subdomain of X. If, in addition, X 0 and Y 0 may be chosen to be affine, then we say that X0 is an affinoid subdomain of X. Note also, that the situation described in the proposition appears in Deligne’s proof of Nagata compactification theorem under the name of quasi-domination. (Recall that by a quasi-domination of Y over X 0 one means an open subscheme Y 0 ⊂ Y and a morphism Y 0 → X 0 , such that the morphism Y 0 → Y ×X X 0 is a closed immersion, see [Con], section 2.) The notion of quasi-domination plays a central role in Deligne’s proof. Lemma 3.3.2. Given two open subdomains Xi = ValYi (Xi ) of X, their intersection X12 is an open subdomain of the form ValY1 ∩Y2 (X1 ×X X2 ). In particular, if X is separated and Xi are affinoid, then X12 is affinoid. Remark 3.3.3. Our definition of RZ spaces is a straightforward generalization of the classical one. It is also possible to define RZ spaces directly as follows: an affinoid space is a topological space X = ValB (A)f →RZB (A) provided with two sheaves of rings OX ⊂ MX , general spaces are pasted e 0 = Spa(B 0 , A0 ) of from affinoid ones. One can check that an affine subset X e = Spa(B, A) induces an open immersion ValB 0 (A0 ) → X iff X e 0 ∩ X is an X affinoid subset of X. We do not develop the outlined approach to Riemann-Zariski spaces in this paper, because it seems worth a separate study. The remark explains a motivation for defining affinoid subsets. The following example illustrates a difference between adic and Riemann-Zariski spaces. Let k be a field, e X e 0 are as above. Then X e0 A = B = A0 = k[T ], B 0 = k[T, T −1 ] and X, X0 , X, e in particular X e0 → X e is an open embedding, is a rational subdomain in X, 0 see [Hub2]. From other side, X is not an affinoid domain in X. One can show that the map X0 → X is bijective, but the sheaves MX and MX0 are not isomorphic. Theorem 3.3.4. The affinoid subdomains of X form a basis of its topology. Proof. If f is affine, then for any Y -modification X 0 → X, the morphism f 0 : Y → X 0 is affine. Therefore, for any open affine subscheme X 00 ⊂ X 0 and its preimage Y 0 = f −1 (X 00 ), ValY 0 (X 00 ) is an affinoid subset of X. Now, the case of affine f follows from corollary 3.2.5. We see also, that in the general case, it suffices to prove that X admits an affinoid covering, because being affinoid subset is a transitive property. If X is an open affine subscheme of X and Y = f −1 (X), then any affinoid subset in ValY (X) is affinoid in X. Thus, we can replace X and Y by X and Y and achieve that X is affine. By Thomason’s noetherian approximation theorem, there exists an affine morphism h : Y → T , whose target is of 20 M. TEMKIN finite Z-type. Consider the induced map λ : X → X0 = ValT (Z). We claim that, if ValT 0 (Z 0 ) is an affinoid subset of X0 , then its preimage in X is an affinoid subset ValY 0 (X 0 ), where Y 0 = T 0 ×T Y and X 0 = Z 0 ×Z X. Indeed, X 0 is obviously affine and of finite X-type, and the open subscheme Y 0 ⊂ Y is affine because the morphism Y → T is affine. Furthermore, since T 0 → T ×Z Z 0 is a closed immersion, the composition morphism Y 0 = Y ×T T 0 → Y ×T T ×Z Z 0 f →Z 0 ×Z Y f →Z 0 ×Z X ×X Y f →X 0 ×X Y is a closed immersion. It follows that it suffices to find an affinoid covering of X0 , and we can assume that X = Spec(Z) and Y is of finite Z-type. Since X is qcqs, it suffices to prove that a point y = (y, R, φ) ∈ X possesses an affinoid neighborhood X0 . Fix a non-zero element ω ∈ R with Rω = k(y), it exists because R is of finite height. Let g ∈ Oy be any lifting of ω. We will construct X0 in two steps. As a starting point, we consider the closed immersion α0 : Spec(k(y)) → Spec(R)×X Y . By diagram (2), there is a finer morphism α1 : Yy → T ×X Y , where T = Spec(Oy ) and Yy = Spec(Oy ). Step 1. The morphism α1 is a closed immersion. Notice that (Oy )g = Oy , hence the morphism Yy → T is an open immersion. The morphism α1 is a locally closed immersion, because it is a composition of a closed immersion Yy → Yy ×X Y with an open immersion Yy ×X Y → T ×X Y . Let Z be the scheme-theoretic image of α1 , then it suffices to prove that Yy f →Z. Observe that Z is a T -scheme and Yy is a schematically dense open subscheme. It follows that Z is glued from schemes Spec(Ci ) with Oy ⊆ Ci ⊆ Oy . Now, it is clear that either Yy f →Z, or Z contains a non-trivial specialization of y. The latter is impossible, because α1 (y) = α0 (y) is closed. Step 2. We can approximate Oy and Oy by finitely generated rings. Consider pairs (Yβ , Tβ ), where Yβ = Spec(Bβ ) is a neighborhood of y, such that g ∈ Bβ∗ , Tβ = Spec(Aβ ) and Aβ is a finitely generated subring of Bβ , such that the image of Aβ in k(y) lies in R, g ∈ Aβ and Bβ = (Aβ )g . It is easily seen, that ∪β Bβ = B and ∪β Aβ = A. We say that γ ≥ β if Yγ ⊂ Yβ and the image of Aβ in Bγ is in Aγ . For any such pair, we obtain the following commutative diagram, whose middle and left squares are Cartesian / Yγ /Y / Yβ Yy ² T ² / Tγ ² / Tβ ² /X For any β, consider the morphism hβ : Yβ → Tβ ×X Y of finitely presented Tβ -schemes. It is a locally closed immersion, because Yβ → Tβ is an open immersion. Since the middle and the left squares of the diagram are Cartesian, we have hβ ×Tβ Tγ = hγ and hβ ×Tβ T = α1 . Notice that proj lim Tβ = T , hence we can apply approximation results of [EGA IV], ch. 8. Since α1 is a closed immersion, hβ is a closed immersion for sufficiently large β. Then ValYβ (Tβ ) is a required affinoid neighborhood of y. ¤ RELATIVE RIEMANN-ZARISKI SPACES 21 Corollary 3.3.5. The space X is qcqs. Proof. Any open subdomain is quasi-compact by 3.1.5, hence it suffices to prove that the intersection of open subdomains is quasi-compact. The latter follows from 3.3.2. ¤ e → X, which to any X-valuation Let us define a retraction map π = πX : X e y ∈ X associates its minimal primary specialization. Then for any pair of schemes Y 0 , X 0 as in the beginning of the section, we can associate a map e0 → X e with π. g : X0 → X by combining the natural map X0 ,→ X e → X is continuous. In particular, g Corollary 3.3.6. The retraction π : X is continuous. Proof. It is easy to see, that for any open subdomain X0 = ValY 0 (X 0 ) of X, e It the set π −1 (X0 ) coincides with Spa(Y 0 , X 0 ), in particular it is open in X. remains to use the theorem. ¤ 3.4. Decomposable morphisms. Proof of theorem 1.1.3. We will prove a slightly more general result. Namely, we will prove that any quasi-compact open subset X of X = ValY (X) is an open subdomain of the form ValY (X), where the morphism Y → X is affine and schematically dominant. Applying this statement to X, one deduces the theorem, because then Y = Y and X is proper over X by the valuative criterion 3.1.8. By theorem 3.3.4, X admits a finite affinoid covering. In particular, X is covered by open subdomains Xi = ValYi (Xi ), 1 ≤ i ≤ n, where Yi are Xi -affine. Using induction on n we reduce the problem to the case of n = 2. Without loss of generality, we can assume that the morphisms Yi → Xi are schematically dominant. Set X12 = X1 ∩ X2 and Y12 = Y1 ∩ Y2 . In the sequel, we will act as in Step 3 of the proof of 1.1.2, the main difference is that we will use Y∗ blow ups instead of affine morphisms. For reader’s convenience, we give a commutative diagram containing the objects which were and will be defined. Y1 o ? _ Y12 ² ² ? _ X12 ² / X2 ² ? _ Z12 CC CC {{ CC {{ { CC { ! }{{ ² / Z2 X1 o ² Z1 o ² X1 o ? _X0 1 / Y2 X20 ² / X2 By lemma 3.2.4, we can replace Xi ’s by their Yi -blow ups, such that each Xi contains an open subscheme Xi0 , whose preimage in Xi coincides with 22 M. TEMKIN X12 . It may be impossible to glue Xi ’s along Xi0 ’s, but we know by lemma 3.1.4 (ii), that at least ValY12 (Xi0 )f →X12 for i = 1, 2. Let T be the schemetheoretic image of Y12 in X10 ×X X20 . Then ValY12 (T )f →X12 by lemma 3.3.2, 0 and, therefore, T is a Y12 -modification of Xi ’s by the valuative criterion. By 3.2.6, we can find a Y12 -blow up T 0 → X10 , which dominates T . It still may happen that T 0 is not a Y12 -blow up of X20 , but it is dominated by a Y12 -blow up Z12 → X20 . Then Z12 → T 0 is a Y12 -blow up by lemma 3.2.1 (i), hence Z12 → X10 is a Y12 -blow up by lemma 3.2.1 (iii). By lemma 3.2.3, we can extend the Y12 -blow ups Z12 → Xi0 to Yi -blow ups Zi → Xi . 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