Birational spaces

Accepted Manuscript Birational spaces

Uri Brezner

PII: DOI: Reference:

S0021-8693(16)30269-1 http://dx.doi.org/10.1016/j.jalgebra.2016.08.017 YJABR 15867

To appear in:

Journal of Algebra

Received date:

7 October 2015

Please cite this article in press as: U. Brezner, Birational spaces, J. Algebra (2016), http://dx.doi.org/10.1016/j.jalgebra.2016.08.017

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BIRATIONAL SPACES URI BREZNER

Abstract. In this paper we construct the category of birational spaces as the category in which the relative Riemann-Zariski spaces of [Tem11] are naturally included. Furthermore we develop an analogue of Raynaud’s theory. We prove that the category of quasi-compact and quasi-separated birational spaces is naturally equivalent to the localization of the category of pairs of quasi-compact and quasi-separated schemes with an affine schematically dominant morphism between them localized with respect to simple relative blow ups and relative normalizations.

1. Introduction In the 1930’s and 1940’s Zariski studied the problem of resolution of singularities for varieties of characteristic zero, and introduced the Riemann-Zariski space RZK (k) of a finitely generated field extension k ⊂ K. Namely, RZK (k) is the space of all valuations on K/k equipped with a (natural) topology, which can also be obtained as the projective limit of all projective models of K/k [Zar44]. Zariski’s original definition RZK (k) (for example [ZS60, Chapter VI §17]) did not include the trivial valuation of K, in contrast we follow Temkin [Tem11, §2] and include the trivial valuation in the Riemann-Zariski space (as a generic point). Given a separated morphism f : Y → X of quasi-compact and quasi-separated (qcqs) schemes, Temkin defined the relative Riemann-Zariski space, RZY (X), as the projective limit of the underlying topological spaces of all the Y -modifications of X [Tem10, Tem11]. He showed that RZY (X) is naturally homeomorphic to the space consisting of unbounded X-valuations on Y equipped with a natural topology. Furthermore the continuous map f : Y → X naturally factors as Y → RZY (X) → X. In the present paper we study the natural category that contains all qcqs schemes and the Riemann-Zariski spaces associated to them.We adopt the valuation point of view. Our approach is to first define for given rings A → B an affinoid birational space Val(B, A) of unbounded A-valuations on B. General birational spaces Val(Y, X) are glued from affinoid ones along affinoid sub-domains. Furthermore, we show that our construction is natural, by which we mean that it admits an analogue to Ryanaud’s theory: Let R be a valuation ring of Krull dimension 1, complete with respect to the J-adic topology generated by a principal ideal J = (π) ⊂ R, where π is some non-zero element of the maximal ideal of R, and K the fraction field of R. On the one hand we have the category of admissible formal R-schemes, while on the other hand we have the category of rigid K-spaces. It was Raynaud [Ray74] who suggested to view rigid spaces entirely within the framework Date: August 25, 2016. Key words and phrases. algebraic geometry. This work was supported by the Israel Science Foundation (grant No. 1018/11). 1

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of formal schemes. Elaborating the ideas of Raynaud, it is proved in [BL93] that the category of admissible formal R-schemes, localized with respect to the class of admissible formal blow ups, is naturally equivalent to the category of rigid Kspaces which are qcqs. We show that the category of pairs of qcqs-schemes with an affine, schematically dominant morphism between them, localized with respect to simple relative blow ups and relative normalizations, is naturally equivalent to the category of qcqs-birational spaces. We note that it should be possible to further define the categories of modules and coherent modules of a birational spaces and develop a theory of cohomology of (coherent) sheaves on birational spaces. A further attestment to the “correctness” of our construction will be an analogue to Grothendieck’s existence theorem [GD61b, 5.1.4], that is a complete classification of coherent modules of Val(Y, X) in terms of coherent modules of X and Y . We leave this work for a future paper. Henceforth, we restrict our attention to the case of affine, schematically dominant morphisms f : Y → X of qcqs-schemes, since it is essentially the same as assuming that f : Y → X is a separated morphism: Indeed, by Temkin’s decomposition theorem [Tem11, Theorem 1.1.3] any separated morphism f : Y → X of j

qcqs-schemes factors as Y → Z → X, where j : Y → Z is an affine, schematically dominant morphism and Z → X is proper. It will become clear from the construction that Val(Y, X) = Val(Y, Z) by the valuative criterion for properness, so our results hold for separated morphisms. The paper is organized as follows. Let A ⊂ B be commutative rings with unity. In Section 2 we construct a topological space Spa(B, A), consisting of all A-valuations of B, and its subspace Val(B, A), consisting of unbounded valuations, which is our main interest in Section 2. We call Val(B, A) the affinoid birational space associated to the pair of rings A ⊂ B. We study some of the topological properties of these spaces and endow Val(B, A) with two sheaves of rings OVal(B,A) ⊂ MVal(B,A) both making Val(B, A) a locally ringed space. The main highlight of Section 3 is the proof that the functor bir, that takes a pair of rings A ⊂ B to the affinoid birational space Val(B, A), gives rise to an antiequivalence from the localization of the category of pairs of rings with respect to relative normalizations to the category of affinoid birational spaces. Also in Section 3, we globalize the construction by introducing general birational spaces. These are topological spaces equipped with a pair of sheaves such that the space is locally ringed with respect to both sheaves and is locally isomorphic to an affinoid birational space. Furthermore we extend bir to a functor from the category consisting f of affine and schematically dominant morphism Y → X between qcqs schemes to the category of birational spaces. Section 5 is dedicated to the proof of Main Theorem. The category of pairs of qcqs schemes with an affine schematically dominant morphism between them, localized with respect to simple relative blow ups and relative normalizations, is naturally equivalent to the category of qcqs birational spaces. In order to achieve this goal, in Section 4 we further develop the theory of relative blow ups (called Y -blow ups of X in [Tem11, §3.4]). In particular we prove the universal property of relative blow ups.

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2. Construction of the Space Val(B, A) Throughout the paper all rings are assumed to be commutative with unity. 2.1. Valuations on Rings. In this Subsection we fix terminology and collect general known facts about valuations. Given a totally ordered abelian group Γ (written multiplicatively), we extend Γ to a totally ordered monoid Γ ∪ {0} by the rules 0 · γ = γ · 0 = 0 and 0 < γ

∀ γ ∈ Γ.

Definition 2.1.1. Let B be a ring and Γ a totally ordered group. A valuation v on B is a map v : B → Γ ∪ {0} satisfying the conditions • v(1) = 1 and v(0) = 0 • v(xy) = v(x)v(y) ∀x, y ∈ B • v(x + y) ≤ max{v(x), v(y)} ∀x, y ∈ B. Note that p = ker v = {b ∈ B | v(b) = 0} is a prime ideal in B. Remark 2.1.2. When B is a field the above definition coincides with the classical definition of a valuation with the value group written multiplicatively. Let v be a valuation on B with kernel p. Denote the residue field of p by k(p). Clearly v factors through the quotient B/p. Since v is multiplicative it further uniquely extends to k(p) and we obtain a valuation in the classical sense v¯ : k(p) → Γ ∪ {0}. On the other hand a prime ideal p ∈ Spec B and a valuation v¯ on the residue field k(p) uniquely determine a valuation v on B with kernel p by setting v(b) = v¯(¯b) where ¯b is the image of b in k(p). Hence giving a valuation v on B is equivalent to giving a prime ideal p and a valuation v¯ on the residue field k(p). Two valuations v1 , v2 on B are said to be equivalent if ker v1 = ker v2 = p and the induced valuations v¯1 , v¯2 on k(p) are equivalent in the classical sense, i.e. they have the same valuation ring or, equivalently, there is an order preserving group isomorphism between their images compatible with the valuations. Every valuation v1 on B is equivalent to a valuation v2 such that the value group Γ is generated, as an abelian group, by v2 (B − p). We will usually identify equivalent valuations without mention. With this convention a valuation v on B with kernel p uniquely defines a valuation ring contained in k(p) by Rv = {x ∈ k(p) | v¯(x) ≤ 1}. Hence a valuation v on B is equivalent to a diagram B

/ k(p) o

_ ?R v.

Definition 2.1.3. Let B be a ring, A a subring and v a valuation on B. We call v an A-valuation on B if v(a) ≤ 1 for every a ∈ A. Assume v is an A-valuation with kernel p. From the condition v(A) ≤ 1 the image of A → B → k(p) lies in Rv . We conclude that every A-valuation v on B

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uniquely defines a commutative diagram / k(p) O

BO (1)

? A

Φ

 / R? v .

Conversely any such diagram defines an A-valuation v on B and we are justified in identifying the A-valuation v on B with the 3-tuple (p, Rv , Φ).1 2.2. The Auxiliary Space Spa(B, A). For completeness and consistency of notation we collect here results regarding valuation spectra. The main reference of this subsection is [Hub93]. Definition 2.2.1. For any pair of rings A ⊂ B we set Spa(B, A) = {A-valuations on B}. Fix a pair of rings A ⊂ B. We provide Spa(B, A) with a topology. For any a, b ∈ B set Ua,b = {v ∈ Spa(B, A) | v(a) ≤ v(b) = 0}. The topology is the one generated by the sub-basis {Ua,b }a,b∈B . Given another pair of rings A ⊂ B  and a homomorphism of rings ϕ : B → B  that satisfies ϕ(A) ⊂ A , composition with ϕ gives rise to the pull back map ϕ∗ : Spa(B  , A ) → Spa(B, A). Specifically given an A -valuation v  = (p , Rv , Φ ) ∈ Spa(B  , A ), then v  ◦ ϕ is a valuation on B. Since ϕ(A) ⊂ A , v  ◦ ϕ is an A-valuation. So indeed ϕ∗ (v  ) = v  ◦ ϕ ∈ Spa(B, A). Clearly its kernel is p = ϕ−1 (p ). Now, we have a commutative diagram

BO

B w; O w w ϕ w ww ww w w



/ k(p) O

/ k(p ) ; O w w ww w w ww

(2) 

A

;A vv v v vv vv v v ϕ∗ (Φ )

Φ

/ Rv  ; w ww w w ww ww / Rv ◦ϕ .

It is clear that Rv ◦ϕ = Rv ∩ k(p) and that the ring map Φ = ϕ∗ (Φ ) is completely determined by Φ and ϕ. To conclude, ϕ∗ takes the point (p , Rv , Φ ) ∈ Spa(B  , A ) to the point (p, Rv ∩ k(p), Φ) ∈ Spa(B, A). Clearly Uϕ(a),ϕ(b) = ϕ∗ −1 (Ua,b ) for any a, b ∈ B. We obtain Lemma 2.2.2. Let A ⊂ B and A ⊂ B  be rings. For a homomorphism ϕ : B → B  satisfying ϕ(A) ⊂ A the pull back map ϕ∗ : Spa(B  , A ) → Spa(B, A) is continuous. 1Although Φ is completely determined by B → k(p) we prefer to keep Φ in the notation.

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Certain subsets of Spa(B, A) can be naturally identified with Spa of some pair of rings. Lemma 2.2.3. Let b, a1 , . . . , an ∈ B and assume that b, a1 , . . . , an generate the unit ideal. Let ϕb be the natural homomorphism B → Bb . Set B  = Bb and an  a1  ,..., A = ϕb (A) . Then b b (1) the pull back map Spa (B  , A ) → Spa(B, A) is injective (2) {v ∈ Spa(B, A) | v(ai ) ≤ v(b) ∀ 1 ≤ i ≤ n} = Spa (B  , A ) n  Uai ,b , in particular Spa (B  , A ) is open in Spa (B, A). (3) Spa (B  , A ) = i=1

Proof. (1) Obviously A ⊂ B  so Spa(B  , A ) is defined. We also obtain a commutative diagram BO

ϕb

? A

/ B O ? / A

which gives rise to the pull back map ϕ∗b : Spa(B  , A ) → Spa(B, A). If v = (p, Rv , Φ) ∈ Spa(B, A) is in the image and v  = (p , Rv , Φ ) ∈ Spa(B  , A ) maps to v, then necessarily b ∈ / p and p = pB  . Hence k(p ) = k(p), from which it follows that Rv = Rv and we obtain the commutative diagram  / / Spec B  i4 Spec A i i i i Φ iiii iiii i i i   iii Φ / Spec A. Spec Rv

Spec k(p)

Since Spec A → Spec A is separated, Φ is unique so v  = (p , Rv , Φ ) is unique. ai (2) Consider v  ∈ Spa(B  , A ) as an element of Spa(B, A). As ∈ A for all b i = 1, . . . , n, and b ∈ / ker v  , we see that v  (ai ) ≤ v  (b) = 0 for every i = 1, . . . , n.   Hence Spa(B , A ) ⊂ {v ∈ Spa(B, A) | v(ai ) ≤ v(b) ∀ 1 ≤ i ≤ n}. Now let v  ∈ {v ∈ Spa(B, A) | v(ai ) ≤ v(b) ∀ 1 ≤ i ≤ n}. Since (b, a1 , . . . , an ) = B there are c0 , c1 , . . . , cn in B such that 1 = c0 b + c1 a1 + . . . + cn an . Applying v  we obtain 1 = v  (1) = v  (c0 b + c1 a1 + . . . + cn an ) ≤ ≤ max{v  (c0 b), v  (c1 a1 ), . . . , v  (cn an )} ≤ v  (b) max {v  (ci )} 0≤i≤n



so necessarily v (b) = 0. Hence we can extend v to a valuation on B  by setting v  (b ) b v ( ) =  for any b ∈ B. Since v  (c) ≤ 1 ∀ c ∈ A we have v  (A ) ≤ 1 so b v (b) v  ∈ Spa(B  , A ). It follows that {v ∈ Spa(B, A) | v(ai ) ≤ v(b) ∀ 1 ≤ i ≤ n} ⊂ Spa(B  , A ), hence we have equality. (3) It is clear from the definition that {v ∈ Spa(B, A) | v(ai ) ≤ v(b)



∀ 1 ≤ i ≤ n} =

n  i=1

Uai ,b .



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Definition 2.2.4. For A ⊂ B  as in Lemma 2.2.3 we call Spa(B  , A ) a rational domain of Spa(B, A), denote it by R({a1 , . . . , an }/b) and regard it as a subset of Spa(B, A). The rational domains are similar in many ways to the basic open subsets D(b) of an affine scheme Spec B. Lemma 2.2.3 shows the analogue of the fact that D(b) ⊂ Spec B is open and canonically homeomorphic to Spec Bb . The basic opens also have the properties (1) D(b) ∩ D(c) = D(bc) (2) D(b) = ∅ ⇔ b is a nilpotent element (3) D(br ) = D(b) (4) if s ∈ B is invertible then D(b) = D(sb) Here is the precise formulation of the analogues for rational domains    Lemma 2.2.5. Let R ({ai }ni=1 /a0 ) and R {aj }m j=1 /a0 be rational domains of Spa(B, A). Then        = R {a /a a } /a a (1) R ({ai }ni=1 /a0 ) ∩ R {aj }m i j 0≤i≤n 0 0 j=1 0 0≤j≤m

(2) R ({ai }ni=0 /a0 ) = ∅ ⇔ a0 is a nilpotent element (3) R ({ari }ni=0 /ar0 ) = R ({ai }ni=0 /a0 ) for any r ∈ N (4) if s ∈ B is invertible then R ({ai }ni=1 /a0 ) = R ({sai }ni=1 /sa0 ) the unit  Proof. (1) By definition both a0 , . . . , an ∈ B and a0 , . . . , am ∈ B generate  ideal. It follows that {ai aj }0≤i≤n also generates the unit ideal and R {ai aj }i,j /a0 a0 0≤j≤m

is defined. By Lemma 2.2.3 R ({ai }ni=0 /a0 ) = ∩i Uai ,a0 and

,

   R {aj }m j=0 /a0 = ∩j Uaj ,a0

  R {ai aj }i,j /a0 a0 = ∩i,j Uai aj ,a0 a0 .

Since Ua,b ∩ Uc,d = {v ∈ Spa(B, A)|v(ad), v(bc) ≤ v(bd) = 0} the claim follows. (2) If a0 is nilpotent there is some r > 0 such that ar0 = 0. For any valuation v we have 0 = v(ar0 ) = v(a0 )r so v(a0 ) = 0. From this its follows that for any a1 , . . . , an ∈ B such that (a0 , a1 , . . . , an ) = B we have R({a1 , . . . , an }/a0 ) = ∅. If a0 is not nilpotent, there is a prime ideal p not containing a0 . Now for any a1 , . . . , an ∈ B such that (a0 , a1 , . . . , an ) = B the rational domain R({a1 , . . . , an }/ a0 ) contains the trivial valuation of k(p). (3) As a0 , . . . , an ∈ B generate the unit ideal so do ar0 , . . . , arn and R ({ari }ni=0 /ar0 ) is defined. By Lemma 2.2.3 R ({ai }ni=0 /a0 ) = {v ∈ Spa(B, A) | v(ai ) ≤ v(a0 )

∀ 1 ≤ i ≤ n},

R ({ari }ni=0 /ar0 )

∀ 1 ≤ i ≤ n}

and the equality is clear. (4) Obvious.

= {v ∈ Spa(B, A) |

v(ari ) 

≤

v(ar0 ) 



 Remark 2.2.6. Let R ({ai }ni=1 /a0 ) and R {aj }m j=1 /a0 be rational domains of   Spa(B, A). The intersection R ({ai }ni=1 /a0 ) ∩ R {aj }m j=1 /a0 consists of all valu  ations v satisfying the inequalities v(ai aj ) ≤ v(a0 a0 ) for i and j such that ij = 0.

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However, the elements {ai aj }ij=0 do not necessarily generate the unit ideal, for instance if a0 = a0 .

n By a rational covering of Spa(B, A) we mean an open cover R({ai }ni=1 /aj ) where a1 , . . . , an ∈ B generate the unit ideal. In [Hub93], Huber defines the valuation spectrum of a ring B

j=1

Spv(B) = {valuations on B}. He provides it with the topology generated by the sub-basis consisting of sets of the form {v|v(a) ≤ v(b) = 0} for all a, b ∈ B. Huber proves in [Hub93, 2.2] that Spv(B) is a spectral space. Clearly our Spa(B, A) is a subspace of Huber’s Spv(B). Lemma 2.2.7. The topological space Spa(B, A) is spectral. In particular it is quasi-compact and T0 . Proof. For a spectral space X we denote by Xcons the same set equipped with the constructible topology i.e. the topology which has the quasi-compact open sets and their complements as an open sub-basis (cf. [Hoc69, §2] and [Hub93, §2]). As the  / Spv B is continuous it is enough to show that the image injection Spa(B, A) of Spa(B, A) is closed in Spv(B)cons (see [Hoc69, §2]). Following Huber’s argument we just need to show that the set of binary relations | of φ(SpvB) that satisfy 1 | a ∀a ∈ A is closed in φ(SpvB). If | ∈ φ(SpvB) − φ(Spa(B, A)) then there is an element a ∈ A such that 1  a. The set Va of binary relations satisfying 1  a contains | and is open in {0, 1}B×B . Now, Va ∩ φ(Spv B) is open in φ(Spv B) (with the topology induced from {0, 1}B×B ), contains | and  (Va ∩ φ(Spv B)) φ(Spa(B, A)) = ∅. So φ(Spa(B, A)) is closed in φ(Spv B) (with the induced topology ). Hence Spa(B, A)cons is closed in Spv B cons .  2.3. The Space Val(B, A). We say that a valuation v : B → Γ ∪ {0} is bounded if there is an element γ ∈ Γ such that v(b) < γ for every b ∈ B. Definition 2.3.1. For any pair of rings A ⊂ B we set Val(B, A) = {v ∈ Spa(B, A) | v is unbounded} equipped with the induced subspace topology from Spa(B, A). As a subspace of a T0 space, Val(B, A) is also a T0 space. For a valuation v on B with abelian group Γ we denote by cΓv the convex subgroup of Γ generated by {v(b) | b ∈ B 1 ≤ v(b)}.  For any convex subgroup Λ v(b) if v(b) ∈ Λ of Γ we define a map v  : B → Λ ∪ {0} by v  (b) = . 0 if v(b) ∈ /Λ A direct verification shows that v  is a valuation on B if and only if cΓv ⊂ Λ. The valuation v  obtained in this way is called a primary specialization of v associated with Λ (see [HM94, §1.2]). Note that ker v ⊂ ker v  . Remark 2.3.2. By construction, for any valuation v on B, the primary specialization of v associated with cΓv is not bounded. It follows that a valuation v is not bounded if and only if it has no primary specialization other than itself.

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Lemma 2.3.3. Let A ⊂ B and A ⊂ B  be rings, and ϕ : B → B  a homomorphism satisfying ϕ(A) ⊂ A . Assume v, w ∈ Spa(B  , A ) such that w is a primary specialization of v. Then ϕ∗ (w) is a primary specialization of ϕ∗ (v). Proof. Assume that v : B  → Γ ∪ { 0} and that Λ isthe convex subgroup of Γ v(b ) if v(b ) ∈ Λ associated with w. Then for b ∈ B  we have w(b ) = . It now 0 if v(b ) ∈ /Λ  v(ϕ(b)) if v(ϕ(b)) ∈ Λ follows that for b ∈ B we have w(ϕ(b)) = .  0 if v(ϕ(b)) ∈ /Λ For v ∈ Spa(B, A), let Pv be the subset of all primary specializations of v. Specialization induces a partial order on Pv by the rule u ≤ w if u is a primary specialization of w for u, w ∈ Pv . Proposition 2.3.4. For any v ∈ Spa(B, A), the set Pv of primary specializations of v is totally ordered and has a minimal element. Proof. Let v : B → Γ ∪ {0} be a valuation on B. Let w : B → Λ ∪ {0} and u : B → Δ ∪ {0} be two distinct primary specializations of v. We may regard Λ and Δ as convex subgroups of Γ, so one is contained in the other. As both w and u are Δ ⊂ Λ. We want primary specialization of v, both Λ and Δ contain cΓv . Assume  w(b) if w(b) ∈ Δ to show that u is a primary specialization of w, i.e. u(b) = . 0 if w(b) ∈ /Δ For any b ∈ B if w(b) > 1 then v(b) = w(b), hence cΛw ⊂ cΓv . Conversely if v(b) > 1 then since cΓv ⊂ Λ we have w(b) = v(b), so cΛw = cΓv . It follows that Δ is a convex subgroup of Λ containing cΛw . For any b ∈ B, if w(b) ∈ Δ then w(b) = v(b) ∈ Δ. It follows that u(b) = v(b) = w(b). If w(b) ∈ / Δ then either w(b) = 0 or 0 = w(b) ∈ Λ. If w(b) = 0 then v(b) ∈ Λ, hence v(b) ∈ Δ and u(b) = 0. If w(b) = 0 then w(b) = v(b) ∈ / Δ, so u(b) = 0. The minimal element of Pv is the primary specialization associated with cΓv .  Next, we give an algebraic criterion for a valuation v ∈ Spa(B, A) to be in Val(B, A). Lemma 2.3.5. Let v = (p, Rv , Φ) ∈ Spa(B, A). Then v ∈ Val(B, A) if and only if the canonical map B ⊗A Rv → k(p) is surjective. Proof. Assume that B ⊗A Rv → k(p) is surjective. Since we assume that Γ is generated by the image of B − p, for any 1 < γ ∈ Γ there is 0 = f ∈ k(p) satisfying γ = v¯(f ). Let bi ∈ B , ri ∈ Rv for i = 1, . . . , n such that bi ⊗ ri ∈ B ⊗A Rv maps to f in k(p). Assume v(b1 ) = max{v(bi )}. Denote the image of bi in k(p) by b¯i . Now, as v¯(ri ) ≤ 1, we have γ = v¯(f ) = v¯(



Hence γ does not bound v.

v (b¯i ri )} ≤ max{¯ v (b¯i )} = v(b1 ). b¯i ri ) ≤ max{¯

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Conversely, for v = (p, Rv , Φ) ∈ Val(B, A) we have a diagram j5 k(p) jjjtj t: C j j j jjj t t   jjjj j t j j j /  BO j B ⊗AO Rv      / A Rv .

v ¯

/ Γ ∪ { 0}

For any f ∈ k(p), if v¯(f ) ≤ 1 then f ∈ Rv and 1 ⊗ f ∈ B ⊗A Rv maps to f ∈ k(p). Assume v¯(f ) > 1. As v ∈ Val(B, A) we see that Γ = cΓv , so there exists d ∈ B satisfying v¯(f ) ≤ v(d). It follows that f /d¯ ∈ Rv where d¯ is the image of d in k(p)  and d ⊗ f /d¯ ∈ B ⊗A Rv maps to f ∈ k(p). Remark 2.3.6. Since for any A ⊂ B and Rv we always have / / B ⊗A R v ,

B ⊗Z Rv

we can replace in the Lemma B ⊗A Rv with B ⊗Z Rv . Remark 2.3.7. Equivalently we can say that v is in Val(B, A) if and only if Spec k(p) → SpecB ×Spec A Spec Rv , or equivalently Spec k(p) → Spec B × Spec Rv , is a closed immersion. As we have seen, given another pair A ⊂ B  and a homomorphism BO

ϕ

/ B O ? / A

? A

composition with ϕ induces a map ϕ∗ : Spa(B  , A ) → Spa(B, A). However ϕ∗ does not necessarily restrict to a map Val(B  , A ) → Val(B, A). Lemma 2.3.8. In the situation described above, assume that the induced homomorphism B ⊗A A → B  is integral. Then composition with ϕ induces a map ϕ∗ : Val(B  , A ) → Val(B, A). Proof. Let v  = (p , Rv , Φ ) ∈ Val(B  , A ), set v = ϕ∗ (v  ). Then v = (p, Rv , Φ) where p = ϕ−1 (p ), Rv = Rv ∩ k(p) and Φ is the induced map AO 

Φ

/ Rv  O

Φ A _ _ _/ Rv .

By Lemma 2.3.5, in order to show that v ∈ Spa(B, A) lies in Val(B, A) we need to show that B ⊗A Rv → k(p) is surjective. The homomorphism ϕ gives rise to a

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diagram as in (2) from which we obtain the commutative diagram B  ⊗AO  Rv

/ / k(p ) O

B ⊗A Rv

/ k(p),

with the upper horizontal arrow surjective by Lemma 2.3.5. Let us construct a preimage of α ∈ k(p). If v(α) ≤ 1 then α is already in Rv (recall that v is the induced valuation on k(p) ). If v(α) > 1, then there is b ∈ B  such that v(α) ≤ v  (b ) since ϕ(α) ¯ ∈ k(p ) and v  is in Val(B  , A ). As B ⊗A A → B  is integral we have x0 , . . . , xn−1 ∈ Im (B ⊗A A → B  ) such that bn +xn−1 bn−1 +. . .+x0 = 0. There is some 0 ≤ i ≤ n − 1 such that v  (bn ) ≤ v  (xi bi ). It follows that v  (b ) ≤ v  (b )n−i ≤ v  (xi ). Now, there are a1 , . . . , am ∈ A and b1 , . . . , bm ∈ B such that aj ⊗ bj maps j

to xi , so v  (b ) ≤ max{v  (aj ) · v  ◦ ϕ(bj )} ≤ max{v(bj )}. The last inequality is due to the fact that v  (a) ≤ 1 for every a ∈ A . Choose b ∈ {b1 , . . . , bm } such that v(b) = max{v(bj )}. Now we have v(α) ≤ v(b). Denoting the image of b in k(p) by ¯b, we have v(α) ≤ v(¯b) or in other words v(α¯b−1 ) ≤ 1. Hence α¯b−1 ∈ Rv and b ⊗ α¯b−1 ∈ B ⊗A Rv maps to α.  2.4. Rational Domains. Set X = Val(B, A). For b, a1 , . . . , an ∈ B generating the unit ideal we defined a rational domain in Spa(B, A) as R({a1 , . . . , an }/b) = {v ∈ Spa(B, A) | v(ai ) ≤ v(b)}. We call the set R({a1 , . . . , an }/b) ∩ X a rational domain by

in X and adenote ait  1 n X({a1 , . . . , an }/b). Obviously X ({a1 , . . . , an }/b) = Val Bb , ϕb (A) ,..., b b where ϕb is the natural map B → Bb . Proposition 2.4.1. The rational domains of X form a basis for the topology. Proof. Let w ∈ X and U an open neighbourhood of w in Spa(B, A) (i.e. U ∩ X is an open neighbourhood of w in X). By the definition of the topology there is a = 0 for each i = 1, . . . , N natural number  N and ai , bi ∈ B such that v(ai ) ≤ v(bi )  such that w ∈ Uai ,bi ⊂ U . By taking the products ci where ci ∈ {ai , bi } i i  and b = bi , replacing N with a suitable natural number, the ai -s with the above i

products and shrinking U , we may assume that we have a1 , . . . , aN , b ∈ B satisfying w ∈ ∩i Uai ,b = U . As w(b) = 0 and w ∈ X we see that w(b) ∈ cΓw . Since w(b)−1 is not a bound of w, there exists d ∈ B such that w(b)−1 ≤ w(d). It follows that 1 = w(1) ≤ w(db) and w ∈ U ∩ U1,db = {v ∈ Spa(B, A) | v(ai ) ≤ v(b) = 0, 1 ≤ v(db)} = = R({da1 , . . . , dan , 1}/db). Hence w ∈ R({da1 , . . . , dan , 1}/db) ∩ X = X({da1 , . . . , dan , 1}/db) ⊂ U ∩ X.

BIRATIONAL SPACES

11

It remains to show that the rational domains satisfy the intersection condition of a basis. It follows from Lemma 2.2.5 that         = X {a /a a } /a a X ({ai }ni=0 /a0 ) ∩ X {aj }m i j 0≤i≤n 0 0 . j=0 0 0≤j≤m

In [Tem11], Temkin defines the semi-valuation ring Sv (p, Rv , Φ) on B. It is the pre-image of the valuation ring Bp . The valuation on B induces a valuation on Sv . We call ring of Sv . We briefly recall several properties of a semi-valuation [Tem11, §2]).

for a valuation v = Rv in the local ring Bp the semi-fraction ring (for details see

Remark 2.4.2. If Sv is a semi-valuation ring with semi-fraction ring Bp then (i) (ii) (iii) (iv) (v) (vi) (vii)

the maximal ideal pBp of Bp is contained in Sv . considering v as a valuation on Bp or Sv we have ker v = pBp . (Sv )ker v = Bp . Sv / ker v = Rv , in particular Sv is a local ring. for any pair g, h ∈ Sv such that v(g) ≤ v(h) = 0 we have g ∈ hSv . for any co-prime g, h ∈ Bp (i.e. gBp + hBp = Bp ), either g ∈ hSv or h ∈ gSv . the converse of (vi) is also true: for a pair of rings C ⊂ D, if for any two coprime elements g, h ∈ D either g ∈ hC or h ∈ gC then there exists a valuation on D such that C is a semi-valuation ring of v and D is its semi-fraction ring.

Remark 2.4.3. Let v = (p, Rv , Φ) ∈ X. By the multiplicativety the valuation v extends uniquely to a valuation on Bp . There is a unique semi-valuation ring Sv associated to the point (namely the pull-back of Rv to Bp ). The diagram (1) corresponding to v factors through the pair Sv ⊂ Bp i.e.

(3)

BO

/ Bp O

/ k(p) O

? A

 / S? v

 / R? v .

Furthermore note that Sv = {x ∈ Bp |v(x) ≤ 1}, and its maximal ideal is {x ∈ Bp |v(x) < 1}. Using these notions we can reformulate Lemma 2.3.5 in terms of semi-valuation and semi-fraction rings. Lemma 2.4.4. Let v = (p, Rv , Φ) ∈ Spa(B, A). Then v ∈ Val(B, A) if and only if the canonical map B ⊗A Sv → Bp is surjective. Proof. By 2.4.2 (i) the ideal pBp lies in Sv , so pBp is in the image of B ⊗A Sv → Bp . It follows that B ⊗A Sv → Bp is surjective if and only if B ⊗A Sv /B ⊗A pBp =  B ⊗A Rv → Bp /pBp = k(p) is. The result follows from Lemma 2.3.5. Let us study pullback of valuations and primary specialization with respect to semi-valuation rings.

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URI BREZNER

Lemma 2.4.5. Let A ⊂ B and A ⊂ B  be rings and ϕ : B → B  a ring homomorphism such that ϕ / B BO O / A A commutes. For a valuation v  ∈ Spa(B  , A ) there is a local homomorphism between the semi-valuation ring of ϕ∗ (v  ) and the semi-valuation ring of v  induced by ϕ. Proof. Let v  = (p , Rv , Φ ) ∈ Spa(B  , A ), set v = ϕ∗ (v  ) = (p, Rv , Φ). By applying the factorization as in diagram (3) to diagram (2) we obtain  B @ O

/ Bp O

BO

A

A ~? ~ ~~ ~~ ~ ~

/ Sv

/ B  > Op } } } } }} }}

{

{

{

/ k(p) O

/ Sv  {= / Rv .

/ k(p ) y< O y yy yy y y

/ Rv  x; x xx xx x x

The dashed arrow exists by the universal property of the pull back. Let x be an element of the maximal ideal of Sv . Then its image in Rv lies in the maximal ideal.  Since Rv → Rv is local, the image of x in Sv also lies in the maximal ideal. Given two valuations v = (p, Rv , Φ), w = (q, Rw , Ψ) ∈ Spa(B, A) with p ⊂ q, we would like to know if w a primary specialization of v. Assume v : B → Γ ∪ {0} and w : B → Δ ∪ {0}. By replacing v and w with equivalent valuations, we may assume that Γ = v(Bp× ) and Δ = w(Bq× ). Since p ⊂ q we have a canonical homomorphism j : B q → Bp . Lemma 2.4.6. With the above notation, w is a primary specialization of v if and only if the canonical homomorphism j : Bq → Bp restricts to a local homomorphism Sw → Sv of semi-valuation rings. Proof. If w is a primary specialization of v, then Δ is a convex subgroup of Γ  v(b) if v(b) ∈ Δ . Now, w extends uniquely to Bq , v containing cΓv and w(b) = 0 if v(b) ∈ /Δ extends uniquely to Bp , Sv = {x ∈ Bp |v(x) ≤ 1} and Sw = {x ∈ Bq |w(x) ≤ 1} (Remark 2.4.3). As p = ker v ⊂ q = ker w, for every s ∈ / q we have w(s) = v(s). b / q = ker w such that x = ∈ Bq . Then For any x ∈ Sw there are b, s ∈ B , s ∈ s  v(j(x)) if v(b) ∈ Δ b . The only non trivial case to j(x) = ∈ Bp and w(x) = s 0 if v(b) ∈ /Δ consider is when 0 = v(b) ∈ / Δ. In this case, since cΓv ⊂ Δ it must be that v(b) < Δ. As v(s) = w(s) ∈ Δ we obtain v(b) < v(s) so j(x) ∈ Sv . The same argument also shows that if x is in the maximal ideal of Sw then j(x) is in the maximal ideal of Sv .

BIRATIONAL SPACES

13

Conversely assume that we have a diagram / Bp Bq O O ? Sw

 / S? v

with the bottom arrow a local homomorphism. It is easy to see that Γ = Bp× /Sv× × and Δ = Bq× /Sw . Since Sw → Sv is local we obtain a homomorphism of groups b α : Δ → Γ. If x = ∈ Bq − Sw then its image under Bq → Bp lies in Bp − Sv , s since if w(x) > 1 and v(x) ≤ 1 then in particular x−1 ∈ Sw and w(x−1 ) < 1. By the locality of the homomorphism we have v(x−1 ) < 1. But this is impossible since it would imply that 1 = v(1) = v(xx−1 ) < 1. Now if x ∈ Bq× with v(x) = 1 i.e. its image under Bq → Bp has value 1, then x is already in Sw and by locality of the homomorphism we have w(x) = 1. Hence α is an injection and we may regard  Δ as a subgroup of Γ. v(b) if v(b) ∈ Δ It is now clear that w(b) = .  0 if v(b) ∈ /Δ There is an obvious retraction Spa(B, A) → Val(B, A). Definition 2.4.7. Let r : Spa(B, A) → Val(B, A) be the map sending every valuation in Spa(B, A) to its minimal primary specialization. Proposition 2.4.8. r : Spa(B, A) → Val(B, A) is continuous. Proof. Let U be an open subset of X = Val(B, A). As the rational domains form a basis for the topology it is enough to consider the case when U is a rational domain of Val(B, A). Let a1 , . . . , an , b ∈ B be elements generating the unit ideal. Set W = R({a1 , . . . , an }/b) and U = X({a1 , . . . , an }/b) = R({a1 , . . . , an }/b) ∩ X. We claim that r−1 (U ) = W . Let w ∈ Spa(B, A) such that r(w) ∈ U . By the definition of primary specialization r(w)(x) ≤ w(x) for every x ∈ B and as r(w)(b) = 0 it follows that w(b) = 0. Hence r −1 (U ) ⊂ W . Let w be a valuation in W with value group Γ and let v = r(w). If cΓw = Γ then v = w, that is, w ∈ Val(B, A) so w ∈ W Val(B, A) = U . If cΓw  Γ then v(ai ) ≤ v(b) since w(ai ) ≤ w(b). It remains to show that v(b) = 0. If v(b) = 0 then w(b) < cΓw and so w(ai ) < cΓw for every i = 1, . . . , n. There are c0 , c1 , . . . , cn ∈ B such that 1 = bc0 + a1 c1 + · · · + an cn and w(1) = 1 ∈ cΓw . By convexity of cΓw there is some i such that w(ai ci ) ∈ cΓw . Thus w(ai ci ) ≤ w(bci ) ∈ cΓw , so v(bci ) = 0. But since v(b) = 0 we also get  v(bci ) = 0, which is a contradiction. We conclude that v(b) = 0. Corollary 2.4.9. Val(B, A) is a qcqs topological space. Proof. As Spa(B, A) is a quasi-compact space by Lemma 2.2.7 and the retraction r : Spa(B, A) → Val(B, A) is continuous, Val(B, A) is quasi-compact. Any rational domain can be viewed as Val(B  , A ) for suitable rings A ⊂ B  , hence any rational domain is quasi-compact. As we saw in Proposition 2.4.1, the intersection of two rational domains is again a rational domain so in particular it is quasi-compact.

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URI BREZNER

Since the rational domains form a basis of the topology, any quasi-compact open subset of Val(B, A) is a finite union of rational domains. Now the intersection of any two quasi-compact open subsets of Val(B, A) is also a finite union of rational domains, thus quasi-compact and Val(B, A) is quasi-separated.  Example 2.4.10 (An Affine Scheme). Consider Val(B, B). Let v = (p, Rv , Φ) ∈ Val(B, B) = X, then v is an unbounded valuation on B such that v(B) ≤ 1. The only way this could be achieved is if v is a trivial valuation (i.e. Γ = {1} ) . Hence there is a 1 − 1 correspondents between points of Val(B, B) and prime ideals of B, that is points of Spec B. As for the topology,

a an  1 ,..., = X({a1 , . . . , an }/b) = Val Bb , ϕb (B) b b = Val(Bb , Bb ) ↔ Spec Bb = D(b) So there is a homeomorphism Val(B, B)  Spec B. For later use we define two canonical maps of topological spaces. Definition 2.4.11. Set σ : Spec B → X and τ : X → Spec A, where σ sends p ∈ Spec B to be the trivial valuation on k(p) (which is indeed in X = Val(B, A) since its valuation ring is k(p) and B ⊗A k(p) → k(p) is of course surjective), and τ sends v = (p, Rv , Φ) ∈ X to the image of the unique closed point of Spec Rv in Spec A. Proposition 2.4.12. (1) The composition τ ◦σ : Spec B → Spec A is the morphism corresponding to the inclusion of rings A ⊂ B. (2) σ is continuous and injective. (3) τ is continuous and surjective. Proof. (1) Given a prime p in B, σ(p) = (p, k(p), Φ). The maximal ideal of the valuation ring k(p) is the zero ideal, so τ (σ(p)) = ker Φ = ker (A → B → k(p)) = p ∩ A. (2) Let U = X({a1 , . . . , an }/b) be a rational domain in X and D(b) a basic open set in Spec B. For any p ∈ D(b) we have σ(p)(b) = 1 and σ(p)(ai ) = 0 or 1, so σ(D(b)) ⊂ X({a1 , . . . , an }/b). Conversely if v ∈ X({a1 , . . . , an }/b) and there is p ∈ Spec B that maps to v then we must have ker(v) = p and v is trivial on k(p). Hence σ is injective and σ −1 (X({a1 , . . . , an }/b)) = D(b). (3) Let D(a) ⊂ Spec A be a basic open set, v = (p, Rv , Φ) ∈ X and mv the / mv or equivalently maximal ideal of Rv . Then τ (v) lies in D(a) if and only if Φ(a) ∈ v(a) ≥ 1. In this case v(a) = 1 (since v(A) ≤ 1) and v ∈ X({1}/a) = {w ∈ Val(B, A) | w(a) = 1} −1

so τ (D(a)) = X({1}/a). As for surjectivety, first consider a maximal ideal q ∈ Spec A. Since the morphism Spec B → Spec A is schematically dominant there is p ∈ Spec B such that A∩p ⊂ q. Take p to be a prime of B maximal with respect to this property. The image i(A) of A under i : B → k(p) is a subring of k(p). The extended ideal i(q)i(A) is a proper ideal, since if 1 ∈ i(q)i(A) then there are a1 , . . . , an ∈ q and b1 , . . . , bn ∈ A such

BIRATIONAL SPACES

15

ai bi ∈ ker(i) = p. As 1 − ai bi ∈ A we have 1 − ai bi ∈ A ∩ p ⊂ q, i i i but since ai bi ∈ q we get that 1 ∈ q which is a contradiction. By [ZS60, VI §4.4]

that 1 −



i

there is a valuation ring Rv of k(p) containing i(A) such that its maximal ideal mv contains i(q)i(A). This gives us a valuation v = (p, Rv , i|A ) ∈ Spa(B, A). By the retraction we obtain a valuation v  = r(v) ∈ Val(B, A) with kernel p ⊃ p. If p = p then q is strictly contained in A ∩ p , by the choice of p. Since q is a maximal ideal we have 1 ∈ A ∩ p which is a contradiction. Hence p = p and v  = v that is v = (p, Rv , Φ) ∈ Val(B, A). Now τ (v) = Φ−1 (mv ) ⊃ q but since q is maximal we have τ (v) = q. Now, let q ∈ Spec A be some prime. Then qAq is the maximal ideal of Aq , and since Aq is flat over A, we have Aq ⊂ Bq = B ⊗A Aq . By the previous case there is a valuation v ∈ Val(Bq , Aq ) that is mapped to qAq by τ : Val(Bq , Aq ) → Spec Aq maps to . The canonical homomorphisms BO

/ Bq O

A

/ Aq

induces by Lemma 2.3.8 a morphism Val(Bq , Aq ) → X. As qAq ∈ Spec Aq is pulled back to q ∈ Spec A, the image of v in Val(B, A) is mapped by τ to q.  Remark 2.4.13. From the proof above we see that for any a ∈ A we have τ −1 (D(a)) = Val(Ba , Aa ) = X({1}/a), and for every rational domain X({a1 , . . . , an }/b) ⊂ X we have σ −1 (X({a1 , . . . , an }/b)) = D(b) Remark 2.4.14. By diagram (3) we can rephrase the definition of τ as the map that sends v = (p, Rv , Φ) ∈ X to the image of the unique closed point of Spec Sv under the map Spec Sv → Spec A. 2.5. Rational Covering. Any open cover of X can be refined to a cover consisting of rational domains, since the rational domains form a basis for the topology. Furthermore there is a finite sub-cover of X consisting of rational domains, as X is quasi-compact. Next we show that we can always refine this cover to a rational covering, that is there are elements T = {a1 , . . . , aN } ⊂ B generating the unit ideal such that the rational domains {X(T /aj )}1≤j≤N form a refinement of the finite sub-cover. Proposition 2.5.1. Any finite open cover of Val(B, A) consisting of rational domains can be refined to a rational covering. Proof. Let {Ui }N i=1 be a finite open cover of X = Val(B, A) consisting of rational (i) domains, i.e. for every i = 1, . . . , N we have a1 , . . . , a(i) ni ∈ B generating the unit ideal and (i)

(i)

(i)

(i)

Ui = X({aj }1≤j≤ni /a1 ) = {v | v(aj ) ≤ v(a1 )

j = 1, . . . , ni }.

16

URI BREZNER (i)

(i)

For every 1 ≤ k ≤ ni denote Vi,k = X({aj }1≤j≤ni /ak ). Note that Vi,1 = Ui and i X = ∪nk=1 Vi,k for each i = 1, . . . , N . Set I = {(r1 , . . . , rN ) ∈ NN | 1 ≤ ri ≤ ni i = 1, . . . , N }. For (r1 , . . . , rN ) ∈ I we denote (2) (N ) V(r1 ,...,rN ) = ∩1≤i≤N Vi,ri and a(r1 ,...,rN ) = a(1) r1 · a r 2 · . . . · a rN .

Note that V(r1 ,...,rN ) = {v ∈ X|v(aα ) ≤ v(a(r1 ,...,rN ) ) ∀α ∈ I} and that the elements {aα }α∈I generate the unit ideal. Hence {Vα }α∈I is rational covering of X.  2.6. Sheaves on Val(B, A). 2.6.1. MX and OX . We now define two sheaves on X = Val(B, A), both making X a locally ringed space. Notation. • For a pair of rings C ⊂ D we denote the integral closure of C in D by NorD C. • For qcqs schemes Y, X and an affine morphism f : Y → X, we denote the integral closure of OX in f∗ OY by Norf∗ OY (OX ) or NorY (OX ). • In the situation above we denote NorY X = SpecX (NorY OX ) and ν : NorY X → X the canonical morphism. Lemma 2.6.1. Let A ⊂ B be rings. Denote X = Spec A and Y = Spec B. Then X = Val(B, A) = Val(B, NorB A) and the canonical map τ : X → X factors through NorY X. Proof. The canonical morphism ν : NorY X → X is integral, hence universally closed and separated. Thus for any valuation v = (p, Rv , Φ) ∈ X we obtain a diagram by the valuative criterion (abusing notation and denoting by Φ both A → Rv and the induced morphism Spec Rv → Spec A) / NorY X k5 k ∃!Φ k k ν kk k   k Φ / X. Spec Rv

Spec k(p)

/Y 

That is, we obtain a unique (p, Rv , Φ ) ∈ Spa(B, NorB A). As v ∈ X the morphism Spec k(p) → Y × Spec Rv is a closed immersion by Remark 2.3.7. It follows, again form Remark 2.3.7, that (p, Rv , Φ ) ∈ Val(B, NorB A). Thus X = Val(B, NorB A). It is clear that the diagram of topological spaces Val(B, NorB A)

τ

/ NorY X

Val(B, A)

τ

 /X

ν

commutes.



The rational domains of X with inclusions as morphisms form a category. Declaring all coverings admissible we obtain a site. As the rational domains form a basis for the topology of X any sheaf on the site of rational domains uniquely extends, in a

BIRATIONAL SPACES

17

natural way, to a sheaf on the topological space X. Hence, it is enough to define the   sheaves only over the rational domains. Let  a U = X({a 1 , . . . , an }/b) = Val(B , A ) a 1 n ,..., where, as before, B  = Bb and A = ϕb (A) . We define two presheaves b b MX and OX on the rational domains of X by the rules U → MX (U ) = B 

U → OX (U ) = NorB  A .

Clearly OX (U ) ⊂ MX (U ). Theorem 2.6.2. With the above notation, the presheaves MX and OX are sheaves on the rational domains of X. Proof. Denote Y = Spec B. We know that the sets {D(b)}b∈B form a base for the topology of Y . Recall that we defined a map σ : Y → X such that for every rational domain X ({a1 , . . . , an }/b) ⊂ X we have σ −1 (X ({a1 , . . . , an }/b)) = D(b) (Remark 2.4.13). Now, by the definition of MX , for every rational domain U ⊂ X we have an isomorphism of rings Bb = MX (U )  σ∗ OSpec B (U ) = OSpec B (D(b)) = Bb . Let V ⊂ U ⊂ X be two rational domains of X. Suppose U = X ({a1 , . . . , an }/b) and V = X ({f1 , . . . , fm }/g). Then D(g) = σ −1 (V ) and D(b) = σ −1 (U ). For any p ∈ D(g) we have σ(p) ∈ V ⊂ U , hence p ∈ D(b). In other words we have a diagram D(g)

/ D(b)

/Y

 V

 /U

 / X.

σ

From the diagram we see that the restriction maps of MX commute with the restriction maps of OSpec B . We conclude that we have an isomorphism of presheaves between MX and σ∗ OSpec B as presheaves on the rational domains. Since OSpec B is a sheaf on Y , its restriction to the basic opens of Y is also a sheaf. It follows that MX is a sheaf on the rational domains of X. Let U be a rational domain in X and {Vi } an open covering of U consisting of rational domains. Let si ∈ OX (Vi ) be sections satisfying si |Vi ∩Vj = sj |Vi ∩Vj for every pair i, j. We already know that MX is a sheaf so there is a unique element s ∈ MX (U ) such that s|Vi = si for each i. We want to show that s is in OX (U ). We may assume that U = X and by Proposition 2.5.1 that {Vi } is a rational covering corresponding to elements b1 , . . . , br ∈ B, that is elements generating the unit ideal of B and Vi = X ({bj }j /bi ). We may further assume that none of b1 , . . . , br are nilpotent.   bj We denote Y = Spec B , X = Spec A , Bi = Bbi , Ai = ϕi (A) { }j (where ϕi bi is the canonical homomorphism B → Bi ), Yi = Spec Bi and Xi = Spec Ai . Then Vi = Val(Bi , A i ). Now, E = Abi is a finite A-module contained in B. Using the multiplication i

in B, we define E d as the image of E ⊗d under the map B ⊗d → B. Then E d is also a finite A-module contained in B for any d ≥ 1. Denoting E 0 = A we obtain a graded A-algebra E  = ⊕d≥0 E d and a morphism X  = Proj (E  ) → X. The affine charts of X  are given by Spec Ai , where Ai is the zero grading part of the

18

URI BREZNER

localization Eb i . Clearly Ai ⊂ Bi . Denoting Id = ker(E d → B → Bi ), we have  d  Ai = lim b−d E /Id which is exactly Ai . This means we have open immersions i d→∞

Yi 



/Y AA AA AA AA / X. / X

   Xi

As Y = ∪Yi , the schematically dominant morphisms Yi → Xi glue to a schematically dominant morphism Y → X  over X. Furthermore, for each i we have a commutative diagram Yi  Y

σ

/ Vi

τ

/ X i

σ

 /X

τ

 / X.

Denoting the normalization X  = NorY X  and taking the canonical morphism ν : X  → X  , then by Lemma 2.6.1 we have a diagram over X τ / Y PP / X B X  PPP BB PPP BBτ PPP BB ν PPPB  '  X.

We denote Xi = ν −1 (Xi ). Now, by construction si ∈ OX  (Xi ) = τ∗ OX (Xi ) = OX (Vi ) and si |Xi ∩Xj = sj |Xi ∩Xj for every pair i, j. Since OX  is a sheaf, they  glue to a section s ∈ OX  (X  ) = τ∗ OX (X). Remark 2.6.3. The construction above yields the same topological space and the same sheaves for both A ⊂ B and for NorB A ⊂ B. Remark 2.6.4. In fact the proof shows that OX = limτα ∗ OXα , where the injective −→

limit is over Xα = Proj(⊕Eαd ) → X and Eα is a finite A-module contained in B. 2.6.2. The stalks. Proposition 2.6.5. For any point v = (p, Rv , Φ) ∈ X, the stalk MX,v of the sheaf MX is isomorphic to Bp and the stalk OX,v of the sheaf OX is isomorphic to the semi-valuation ring Sv . Proof. Fix a point v = (p, Rv , Φ) ∈ X. The inclusion of sheaves OX ⊂ MX gives an inclusion of stalks OX,v ⊂ MX,v . By the definition of a semi-valuation ring we have a diagram (3). For any rational domain v ∈ W = X({ai }i /b) = Val(B1 , A1 )

BIRATIONAL SPACES

19

(we may assume that A1 integrally closed in B1 ) we have a unique factorization BO

A

/ MX (W ) = B1 O LL L

/ OX (W ) = A1 LL L

LL L&

LL L&

Bp O

Sv

/ k(p) = O { {{ { {{ {{

/ Rv . {= { {{ {{ { {

Taking direct limits we get a unique diagram for the stalks MX,v O

/ Bp O

/ k(p) O

? OX,v

 / S? v

 / R? v

and v induces a valuation in Val(MX,v , OX,v ). We claim that the horizontal arrows in the left square are isomorphisms. If v is a trivial valuation then Rv = k(p), so Sv = Bp . It follows that for every element b ∈ B we have v ∈ X({b}/1) and also for every element b ∈ / p we have v ∈ X({1}/b). Hence MX,v = OX,v = Bp . Assume v is not trivial. Let γ, η be co-prime elements in MX,v , i.e. there are elements ρ, τ ∈ MX,v such that γρ + ητ = 1. By Remark 2.4.2.(vii) if either γ ∈ ηOX,v or η ∈ γOX,v then OX,v is a semi-valuation ring and MX,v is its semifraction ring. Assume this is the case. In particular both MX,v and OX,v are local rings. Denote the maximal ideal of MX,v by m. From Remark 2.4.2 (ii), (iii) and (iv) it follows that m = ker v, m lies in OX,v , MX,v = (OX,v )m and OX,v /m = Rv . Hence OX,v = Sv and MX,v = Bp and we are done. It remains to show that either γ ∈ ηOX,v or η ∈ γOX,v . Clearly either v(γ) ≤ v(η) or v(η) ≤ v(γ). Assume v(γ) ≤ v(η). We claim that γ ∈ ηOX,v . By Proposition 2.4.1 the intersection of a finite number of rational domains is again a rational domain. Hence there is a rational domain U = Val(B  , A ) with g, h, r, t ∈ MX (U ) = B  such that g, h, r, t are representatives of γ, η, ρ, τ respectively. Then gr + ht is a representative of 1 ∈ MX,v . So there is a rational domain V = Val(B  , A ) ⊂ U such that gr + ht|V = 1 ∈ MX (V ) = B  . It follows that g|V , h|V ∈ MX (V ) = B  are representatives of γ, η and are co-prime. Furthermore v induces (canonically) a valuation on B  which has the same valuation ring as v. Replace g|V , h|V with g, h. Since v(γ) ≤ v(η) it follows that v(g) ≤ v(h). As V = Val(B  , A ) is a rational domain there are a1 , . . . , an , b ∈ B generating the an a1 unit ideal such that B  = Bb and A = ϕb (A)[ , . . . , ]. Now g, h ∈ B  so there b b is some natural number r such that br g, br h ∈ B and v(br g) ≤ v(br h). If v(b) > 1 we may shrink V to X ({1, a1 , . . . , an }/b). If v(b) ≤ 1 then there is an element d ∈ B such that v(b)−1 < v(d) since v is unbounded and not trivial. So 1 < v(bd) and v(d) = 0. It follows that v ∈ X ({1, a1 d, . . . , an d}/bd) ⊂ V . Shrinking V and replacing notation, we may assume that V = Val(B  , A ) = X ({1, a1 , . . . , an }/b).

20

URI BREZNER

Now 1 < v(b) so for some large enough j ≥ r we have bj g, bj h ∈ B , v(bj g) ≤ v(bj h) and 1 < v(bj+1 h). The elements 1, bj a1 , . . . , bj an , bj+1 g, bj+1 h generate  the unit j j j+1 j+1 ideal of B. We have v ∈ W = X {1, b a h, . . . , b a h, b g}/b h ⊂ V and 1 n    a1 an g 1 . Hence g|W ∈ h|W OX (W ) W = Val Bbj+1 h , ϕbj+1 h (A) j+1 , , . . . , , b h b b h and it follows that γ ∈ ηOX,v . By the same reasoning if v(η) ≤ v(γ) then  η ∈ γOX,v . Corollary 2.6.6. For any point p ∈ Spec B the stalks of the point σ(p) ∈ X are MX,σ(p) = OX,σ(p) = Bp . 3. Birational Spaces 3.1. Categories of Pairs. Definition 3.1.1. (i) By a pair of rings (B, A) we mean a ring B and a sub-ring A. (ii) A homomorphism of pairs of rings ϕ : (B, A) → (B  , A ) is a ring homomorphism ϕ : B → B  such that ϕ(A) ⊂ A . (iii) A homomorphism of pairs of rings ϕ : (B, A) → (B  , A ) is called adic if the canonical homomorphism B ⊗A A → B  is integral. (iv) The relative normalization of a pair of rings (B, A) is the pair of rings (B, NorB A) together with the canonical homomorphism of pairs ν = idB : (B, A) → (B, NorB A). f

(v) By a pair of schemes (Y → X) or (Y, X) we mean a pair of qcqs schemes Y, X together with an affine and schematically dominant morphism f : Y → X. (vi) A morphism of pairs of schemes g : (Y  , X  ) → (Y, X) is a pair of morphisms g = (gY , gX ) forming a commutative diagram Y

gY

/Y

gX

 / X.

f

 X

f

(vii) A morphism of pairs of schemes g : (Y  , X  ) → (Y, X) is called adic if the canonical morphism of schemes Y  → Y ×X X  is integral. f

(viii) The relative normalization of a pair of schemes (Y → X) is the pair of schemes (Y, NorY X) together with the canonical morphism of pairs ν = (idY , νX ) : (Y, NorY X) → (Y, X), where NorY X = SpecX (Norf∗ OY (OX )) and νX is the canonical morphism NorY X → X. We denote the category of pairs of rings with their morphisms by pa-Ring and the category of pairs of schemes with their morphisms by pa-Sch. Remark 3.1.2. As in Remark 2.3.6, a homomorphism of pairs of rings ϕ : (B, A) → (B  , A ) is adic if and only if B ⊗Z A → B  is integral.

BIRATIONAL SPACES

21 f

Remark 3.1.3. Since the structure morphism f of a pair of schemes (Y → X) is affine and schematically dominant we obtain embeddings of quasi-coherent sheaves of OX -algebras OX → Norf∗ OY OX ⊂ f∗ OY . It follows that (Y, NorY X) is indeed an object of pa-Sch. f

Definition 3.1.4. Let (Y → X) be a pair of schemes. Given an open (affine) subscheme X  ⊂ X its preimage Y  = f −1 (X  ) is an open (affine) subscheme of Y  . The restriction of f to Y  makes (Y  , X  ) a pair of schemes. We call (Y  , X  ) an open (affine) sub-pair of schemes. An affine covering of the pair (Y, X) is a collection of open sub-pairs {(Yi , Xi )} such that {Xi } are affine and cover X (then, necessarily their preimages {Yi } are affine and cover Y ) . Lemma 3.1.5. Assume the elements b, a1 , . . . , an ∈ B generate the unit ideal. Set an a1 B  = Bb . Let ϕb : B → Bb be the canonical map and denote A = ϕb (A)[ , . . . , ]. b b Then the homomorphism of pairs of rings ϕb : (B, A) → (B  , A ) is adic. Proof. Since b, a1 , . . . , an generate the unit ideal of B, a an  1 ,..., = B. B ⊗A A = ϕb (B) b b



Proposition 3.1.6. The category of pairs of schemes contains all fiber products. f

f

j

Proof. Let (Y → X) , (Y  → X  ) , (T → Z) be pairs of schemes and g : (Y  , X  ) → (Y, X) , h : (T, Z) → (Y, X) morphisms of pairs. Then f  × j : Y  ×Y T → X  ×X Z is an affine morphism. Denote the schematic image of f  × j by Z  and set T  = j

Y  ×Y T . Denote the induced morphism T  → Z  by j  . Clearly (T  → Z  ) is a pair of schemes equipped with canonical morphisms of pairs 

f (Y  → X  ) o

j

h

(T  → Z  )

g

j / (T → Z)

such that g ◦ h = h ◦ g  . Let (W → V ) be a pair of schemes with morphisms of pairs l



f (Y  → X  ) o

p

l

(W → V )

q

j / (T → Z)

such that g ◦ p = h ◦ q. By the universal property of fiber products of schemes there are unique morphisms W → T  and V → X  ×X Z. By the universal property of the scheme theoretic image V → X  ×X Z factors uniquely through Z  . Hence there is a unique morphism of pairs (W, V ) → (T  , Z  ).  3.2. Birational Spaces and the bir Functor. Definition 3.2.1. (i) A pair-ringed space (X, MX , OX ) is a topological space X together with a sheaf of pairs of rings (MX , OX ) such that both (X, MX ) and (X, OX ) are ringed spaces. (ii) A morphism of pair-ringed spaces (h, h ) : (X, MX , OX ) → (Y, MY , OY ) is a continuous map h : X → Y together with a morphism of sheaves of pairs of rings h : (MY , OY ) → (h∗ MX , h∗ OX )

22

URI BREZNER

such that both (h, h ) : (X, MX ) → (Y, MY ) and (h, h ) : (X, OX ) → (Y, OY ) are morphisms of ringed spaces. In Section 2 we constructed a pair-ringed space (X, MX , OX ) from a pair of rings (B, A), namely Val(B, A). From now on for any pair of rings (B, A) by Val(B, A) we mean the pair-ringed space (Val(B, A), MVal(B,A) , OVal(B,A) ). We will freely use the formulas for the stalks from Proposition 2.6.5. Definition 3.2.2. (i) An affinoid birational space is a pair-ringed space (X, MX , OX ) isomorphic to Val(B, A) for some pair of rings (B, A). (ii) A pair-ringed space (X, MX , OX ) is a birational space if every point x ∈ X has an open neighbourhood U such that the induced subspace (U, MX |U , OX |U ) is an affinoid birational space. (iii) A morphism of birational spaces (h, h ) : (X, MX , OX ) → (Y, MY , OY ) is a morphism of pair-ringed spaces such that the induced morphism of ringed spaces (h, h |OX ) : (X, OX ) → (Y, OY ) is a morphism of locally ringed spaces (but not necessarily (h, h ) : (X, MX ) → (Y, MY ); see Example 3.2.11 below). We denote the category of affinoid birational spaces with their morphisms by af-Birat and the category of birational spaces with their morphisms by Birat. Example 3.2.3. Given a ring B, we have a homeomorphism of topological spaces Val(B, B)  Spec B (Example 2.4.10). From Remark 2.6.6 it is clear that considered as an affinoid birational space Val(B, B) is exactly (Spec B, OSpec B , OSpec B ). Given another ring B  , a homomorphism of rings B  → B induces a morphism of schemes (f, f  ) : Spec B → Spec B  which can also be viewed as a morphism of affinoid birational space (f, f  ) : Val(B, B) → Val(B  , B  ). A scheme is locally isomorphic to an affine scheme so we obtain Lemma 3.2.4.

(1) Any scheme (X, OX ) can be viewed as a birational space V al(X, X) = (X, OX , OX ).

(2) The functor (X, OX ) → (X, OX , OX ) from qcqs schemes to birational space is fully faithful. (3) Any pair of schemes (Y, X) induces a birational space Val(Y, X) and continuous maps σ τ Y → Val(Y, X) → X as in Definition 2.4.11. Remark 3.2.5. (1) For a pair of schemes (Y, X), the points of Val(Y, X) are 3-tuples (y, Ry , Φ) such that y is a point in Y , Ry is a valuation ring of the residue field k(y) and Φ is a morphism of schemes Spec Ry → X making the diagram /Y Spec k(y) f

 Spec Ry

Φ

 /X

BIRATIONAL SPACES

23

commute, and Spec k(y) → Y ×X Spec Ry is a closed immersion (cf. [Tem11, §3]) . (2) Any affine covering {(Yi , Xi )} of (Y, X) gives rise to a covering of the birational space Val(Y, X) consisting of affinoid birational spaces {Val(Yi , Xi )}. Lemma 3.2.6. Let (Y, X) be a pair of schemes. Then Val(Y, X) is qcqs. 

Proof. Combine Remark 3.2.5(2) and Corollary 2.4.9.

Lemma 2.6.1 and Remark 2.6.3 easily extend from the affine case to the general one. Corollary 3.2.7. Let (Y, X) be a pair of schemes. Then Val(Y, NorY X) = Val(Y, X) as birational spaces and the canonical map τ : Val(Y, X) → X factors through NorY X. Proof. For every affine U ⊂ X we have (NorY OX ) (U ) = Norf∗ OY (U ) OX (U ). The result follows from Lemma 2.6.1 and Remark 2.6.3.  Example 3.2.8. For a finitely generated field extension K/k there is an obvious natural homeomorphism from RZ(K/k) to Val(K, k). At a valuation v the stalk is the pair of rings (K, Rv ). Let us now construct a functor bir : pa-Sch → Birat. Construction 3.2.9. We already saw the construction of a birational space Val(Y, X) from a pair of schemes (Y, X). We set bir(Y, X) = (Y, X)bir = Val(Y, X). By Remark 3.2.5(2) constructing a morphism of birational spaces V al(Y2 , X2 ) → V al(Y1 , X1 ) from a morphism of pairs (Y2 , X2 ) → (Y1 , X1 ) can be done locally on X and X  . For a homomorphism of pairs of rings ϕ : (B1 , A1 ) → (B2 , A2 ) we define the map of topological spaces ϕbir by the composition Spa(B2 , A2 ) O

ϕ∗

/ Spa(B1 , A1 ) r

 ? ϕbir Val(B2 , A2 ) _ _ _/ Val(B1 , A1 ) where ϕ∗ is the pull back map defined in section 2.2 and r is the retraction of Definition 2.4.7. We saw that both ϕ∗ (Lemma 2.2.2) and the retraction (Lemma 2.4.8) are continuous so ϕbir is continuous. For v = (p, Rv , Φ) ∈ Val(B2 , A2 ) we have ϕ∗ (v) = v ◦ ϕ ∈ Spa(B1 , A1 ) ϕbir (v) = r(v ◦ ϕ) =w = (q, Rw , Ψ) ∈ Val(B1 , A1 ). Since w is a primary specialization of the pullback valuation ϕ∗ (v) = v ◦ ϕ there is a natural homomorphism of the stalks MVal(B1 ,A1 ),w = (B1 )q O

/ (B1 )ϕ−1 (p) O

/ (B2 )p = MVal(B ,A ),v 2 2 O

OVal(B1 ,A1 ),w = Sw

/ Sv◦ϕ

/ Sv = OVal(B ,A ),v . 2 2

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URI BREZNER

By Lemma 2.4.5 and Lemma 2.4.6 both bottom arrows are local homomorphism. We obtain a morphism of affinoid birational spaces ϕbir : (B2 , A2 )bir → (B1 , A1 )bir . It follows from Remark 2.3.2 that bir respects identity homomorphisms. As for composition again it is enough to check only the affine case, which is the subject of next lemma. ϕ

ψ

Lemma 3.2.10. Let (B1 , A1 ) → (B2 , A2 ) → (B3 , A3 ) be homomorphisms of pairs of rings. Then ϕbir ◦ ψbir = (ψ ◦ ϕ)bir . Proof. For v = (p, Rv , Φ) ∈ Val(B3 , A3 ) we have (ψ ◦ ϕ)∗ (v) = ϕ∗ (ψ ∗ (v)) as elements of Spa(B1 , A1 ). By construction (ψ ◦ ϕ)bir (v) is a primary specialization of (ψ ◦ ϕ)∗ (v). Also, ψbir (v) is a primary specialization of ψ ∗ (v) as elements of Spa(B2 , A2 ), and ϕbir (ψbir (v)) is a primary specialization of ϕ∗ (ψbir (v)) as elements of Spa(B1 , A1 ). It follows from Lemma 2.3.3 that ϕ∗ (ψbir (v)) is a primary specialization of ϕ∗ (ψ ∗ (v)). Hence both ϕbir (ψbir (v)) and (ψ ◦ ϕ)bir (v) are primary specializations of (ψ ◦ ϕ)∗ (v). They are also both minimal primary specializations, since they are elements of Val(B1 , A1 ). By Proposition 2.3.4 we have  ϕbir (ψbir (v)) = (ψ ◦ ϕ)bir (v). The following example shows that the homomorphism on the stalks of M can indeed be not local. Example 3.2.11. Let K be an algebraically closed field. Consider A = A = B = K[T ] and B  = K[T, T −1 ]. Let ϕ : (B, A) → (B  , A ) be the obvious map. Clearly ϕ is not adic. Passing to birational spaces, we have ϕbir : X = Val(K[T, T −1 ], K[T ]) → Val(K[T ], K[T ]) = X. As we saw in Example 3.2.3, X = (A1 , OA1 , OA1 ). Let η be the generic point of A1 and of Spec K[T, T −1 ] ⊂ A1 . Let v be the valuation in X corresponding to the valuation ring Rv = K[T ](T ) ⊂ K(T ) = k(η). It is indeed in X as K[T, T −1 ] ⊗ K[T ](T ) → K(T ) is surjective. Since K[T ] ⊗ K[T ](T ) → k(η) = K(T ) is not surjective, the pullback v ◦ ϕ is not in X. Its primary specialization w is the trivial valuation on k(p) = K for the ideal p = (T ). The stalks are MX ,v = K(T ) and MX,w = K[T ](T ) . The induced homomorphism of stalks is the obvious injection K[T ](T ) → K(T ) which is not a local homomorphism. Lemma 3.2.12. Let (Y, X) , (Y  , X  ) be two pairs of schemes and let g : (Y  , X  ) → (Y, X) be a morphism of pairs. Then the diagram of topological spaces Y 

gY

σ

Val(Y  , X  )

σ

gbir

 / Val(Y, X)

gX

 /X

τ

 X commutes.

/Y

τ

BIRATIONAL SPACES

25

Proof. The question is local on X and X  hence it is enough to consider only the affine case. Let (B, A) , (B  , A ) be two pairs of rings and let g = (gB , gA ) : (B, A) → (B  , A ) be a homomorphism of pairs. We want to show that Spec B 

Spec gB

σ

 Val(B  , A )

σ

gbir

 / Val(B, A)

Spec gA

 / Spec A

τ

 Spec A

/ Spec B

τ

commutes. If p ∈ Spec B  , then σ  (p) is the trivial valuation on the residue field k(p). Similarly, σ ◦ Spec gB (p) ∈ Val(B, A) is the trivial valuation on the residue field −1 −1 (p)). The homomorphism gB : B → B  induces an injection k(gB (p)) → k(p) k(gB  , so the composition gbir ◦ σ (p) is just the trivial valuation on the residue field −1 k(gB (p)). If v = (p, Rv , Φ) ∈ Val(B  , A ), let gbir (v) = w = (q, Rw , Ψ). Denote by mv the maximal ideal of Sv and by mw the maximal ideal of Sw . The homomorphisms Φ : A → Rv and Ψ : A → Rw factor through Sv and Sw respectively. Denote the corresponding homomorphisms by Φ : A → Sv and Ψ : A → Sw . By Remark 2.4.14 we have τ  (v) = Φ−1 (mv ) and τ (gbir (v)) = Ψ−1 (mw ). Since the induced −1 (Φ−1 (mv )) = Ψ−1 (mw ). Hence homomorphism Sv → Sw is local we have gA −1 τ (gbir (v)) = Ψ−1 (mw ) = gA (Φ−1 (mv )) = Spec gA (τ  (v)).



The functor bir : pa-Rings → af-Birat is essentially surjective by the definition of the category of affinoid birational spaces. Given two pairs of rings (B, A), (B  , A ) and a morphism of affinoid birational spaces h : Val(B, A) → Val(B  , A ), by taking global sections we obtain a homomorphism of pairs of rings ϕ BO  ? NorB  A

ϕ

/B O  / Nor? B A.

Composition with the inclusion A ⊂ NorB  A gives only a homomorphism of the pairs ϕ : (B  , A ) → (B, NorB A), not of the original pairs (B, A) and (B  , A ). As Val(B, NorB A) = Val(B, A) and any homomorphism (B  , A ) → (B, A) uniquely extends to a homomorphism (B  , NorB  A ) → (B, NorB A), this discrepancy is superficial in some sense. Theorem 3.2.13. The functor bir is an anti-equivalence of categories between the category of pairs of rings pa-Rings localized at the class of relative normalizations and the category of affinoid birational spaces af-Birat. The functor of global sections is a quasi-inverse. Proof. Denote the class of relative normalization homomorphisms by M . By Lemma 2.6.1, bir factors through pa-RingsM and by definition of af-Birat it is essentially

26

URI BREZNER

surjective. It remains to show that bir : pa-RingsM → af-Birat is full and faithful. We start by proving fullness. Let (B  , A ), (B, A) be two pairs of rings and h a morphism of affinoid birational spaces h : (B, A)bir → (B  , A )bir . We may assume that A and A are integrally closed in B  and B respectively. Taking global sections we obtain a diagram ϕ /B BO  O ? ? /A A i.e. ϕ is a homomorphism of pairs of rings ϕ : (B  , A ) → (B, A). We claim that ϕbir = h. Given a valuation v = (p, Rv , Φ) ∈ Val(B, A) we denote h(v) = w = (q, Rw , Ψ) ∈ Val(B  , A ). Passing to stalks we obtain a diagram of pairs of rings (B  , A )  (Bq , Sw )

/ (B, A)

ϕ

 / (Bp , Sv ).

hv

Let n = pBp denote the maximal ideal of Bp , n = qBq the maximal ideal of Bq , mv the maximal ideal of Sv and mw the maximal ideal of Sw . Since Bq is a local −1

ring hv (n) ⊂ n . Pulling back to B  we see that ϕ−1 (p) ⊂ q. Hence the bottom line of the above diagram can be factored as Bq O

/ B  −1 = (Bq )  −1 h v (n) ϕ (p) O

/ Bp O

Sw

/ Sv◦ϕ

/ Sv .

By Lemma 2.4.5 and the definition of morphisms of birational spaces, we see that the bottom left arrow is a local homomorphism. By Lemma 2.4.6 w is a primary specialization of the pullback valuation v ◦ ϕ. As w is already in Val(B  , A ) it has no primary specialization other than itself. By Proposition 2.3.4 the primary specializations of v ◦ ϕ are linearly ordered and we conclude that r(v ◦ ϕ) = w where r is the retraction, or in other words ϕbir (v) = h(v). As for faithfulness, given two homomorphisms of pairs of rings ϕ, ψ : (B  , A ) → (B, A) such that ϕbir = ψbir we obtain a diagram Spec B

Spec ϕ Spec ψ

σ

σ

 Val(B, A)

// Spec B 

ϕbir ψbir

 / Val(B  , A ).

It follows from Proposition 2.4.12(2), Corollary 2.6.6, the definition of a morphism of birational spaces and Lemma 3.2.12 that Spec ϕ = Spec ψ as morphisms of

BIRATIONAL SPACES

27

schemes Spec B → Spec B  . Hence ϕ = ψ as homomorphisms of rings B  → B. As A and A are integrally closed in B  and B respectively, we also have that ϕ = ψ  as homomorphisms of pairs of rings (B  , A ) → (B, A). Since a birational space is locally affinoid we obtain the following result Corollary 3.2.14. Let X be a birational space and (B, A) a pair of rings. There is a natural bijection HomBirat (X, (B, A)bir )  Hompa-Rings ((B, NorB A), Γ(MX , OX )) . Proposition 3.2.15. The category of birational spaces contains all fiber products and the functor bir respects fiber products. Proof. Consider the full subcategory of pa-Rings consisting of all normalized pairs of rings, i.e. all pairs of rings A ⊂ B such that A = NorB A. Denote this category C. Note that this category contains all pushouts and that the pushout of the arrows / (B3 , A3 ) is the pair (B, A) where B = B2 ⊗B1 B3 and (B1 , A1 ) (B2 , A1 ) o A is the integral closure of the image of A2 ⊗A1 A3 in B. The embedding of C in pa-Rings has a left adjoint given by sending a pair of rings (B, A) to its relative normalization (B, NorB A). Hence C is a reflective subcategory of pa-Rings and is equivalent to the localization of pa-Rings with respect to relative normalizations. By Proposition 3.1.6 bir takes pushouts in pa-Rings to fiber products in af-Birat. It follows from the anti-equivalence of Theorem 3.2.13 that the category of affinoid birational spaces contains all fiber products. From Corollary 3.2.14 we see that a fiber product in af-Birat is also a fiber product in Birat. As a birational space is locally affinoid Birat contains all fiber products. Furthermore, since an affine covering (Yi , Xi ) of a pair of schemes (Y, X) induces an affinoid covering Val(Yi , Xi ) = (Yi , Xi )bir of Val(Y, X) = (Y, X)bir , the functor bir : pa-Sch → Birat respects fiber products.  Definition 3.2.16. Let X be a birational space and (Y, X) a pair of schemes . If (Y, X)bir = X we say that (Y, X) is a scheme model of X. Given another scheme model (Y  , X  ) of X, if there is a morphism of pairs of schemes g : (Y  , X  ) → (Y, X) such that gbir is the identity we say that (Y  , X  ) dominates (Y, X). 3.3. Characterization of Adic Morphisms. In this section we collect results regarding adic morphisms. These results are not used later on. Lemma 3.3.1. (i) Composition of adic morphisms of pairs of schemes is adic. (ii) Let g : (Y  , X  ) → (Y, X) and h : (Y  , X  ) → (Y  , X  ) be morphisms of pairs of schemes. If h ◦ g is adic and the induced morphism X  ×X  Y  → X  ×X Y is separated (e.g. if gY : Y  → Y is separated) then h is adic. (iii) Adic morphisms are stable under base change. (iv) Let g : (Y  , X  ) → (Y, X) be a morphism of pairs of schemes and {(Vi , Ui )} an −1 affine covering of (Y, X). If all the restrictions (gY−1 (Vi ), gX (Ui )) → (Vi , Ui ) are adic, then g is adic. We omit the proof since it is a direct application of the same results for integral morphisms. Let X be a scheme and x ∈ X a point. We know that there are canonical morphisms Spec k(x) → Spec OX,x → X where the image of (the singleton) Spec k(x) in X is x and the image of Spec OX,x is the set of generalizations of x. With this

28

URI BREZNER

in mind, let X be a birational space, v ∈ X a valuation and m the maximal ideal of MX,v . There are canonical homomorphisms of pairs of rings Γ (MX , OX ) → (MX,v , OX,v ) → (k(v), Rv ) , where k(v) = MX,v /m, which give rise to canonical morphisms of birational spaces Val (k(v), Rv ) → Val (MX,v , OX,v ) → X. Note that the homomorphism (MX,v , OX,v ) → (k(v), Rv ) is adic. Lemma 3.3.2. Let X be a birational space and v ∈ X a valuation. The canonical map Val (k(v), Rv ) → Val (MX,v , OX,v ) is injective and its image is the set of OX,v valuations on MX,v whose kernel is the maximal ideal of MX,v . Proof. Note that a point of Val (k(v), Rv ) corresponds to a valuation ring Rv ⊂ T ⊂ k(v), and the generalizing relation of points corresponds to the inclusion relation of valuation rings. Since (MX,v , OX,v ) → (k(v), Rv ) is adic, the image of T in Val(MX,v , OX,v ) is the outer square of the diagram MX,v O

/ k(v) O

k(v) O

OX,v

/ Rv

/T

obtained by composition. It follows that only a valuation w ∈ Val(MX,v , OX,v ) whose kernel is the unique maximal ideal of MX,v can be in the image. Furthermore for any such w, OX,v → T factors uniquely through Rv . Hence w is in the image and its preimage consists of one point.  Lemma 3.3.3. Let X be a birational space. Let v, v  ∈ X be points of X. Then v  is a generalization of v if and only if v  is in the image of the canonical morphism Val (MX,v , OX,v ) → X. Proof. Since every point of Val (MX,v , OX,v ) is a generalization of the unique closed point, which maps to v, we see that the image of Val (MX,v , OX,v ) in X consists of generalizations of v. Conversely, suppose that v  is a generalization of v. Choose an affinoid open neighborhood U = Val(B, A) of v. Then v  ∈ U . As there are no primary specializations in Val(B, A), it follows from [HM94, Remark 1.2.2 and Proposition 1.2.4] that ker v  = ker v and that the valuation ring of v  is a localization of the valuation ring of v at a prime ideal. Denote by R and R the valuation rings of v and v  respectively. The valuation in Val (MX,v , OX,v ) that corresponds to the diagram / k(v) O

MX,v O OX,v maps to v  .

/R

/ R 

Corollary 3.3.4. The image of the canonical map σ : Y → Val(Y, X) is a dense subset.

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29

Proof. For every v ∈ Val(Y, X) the generic point of Val (MX,v , OX,v ) is a trivial valuation which is in the image of σ.  Let g : (Y2 , X2 ) → (Y1 , X1 ) be a morphism of pairs of schemes. Considering Y1 as the birational space Y1 = Val(Y1 , Y1 ) we can form the fiber product of the maps gbir / Val(Y1 , X1 ) o σ1 Y1 . It is Val(Y, X) where Y = Y2 ×Y1 Y1 = Y2 Val(Y2 , X2 ) and X is the relative normalization of the scheme theoretic image of Y in X2 ×X1 Y1 . Denoting the scheme theoretic image of Y2 in X2 ×X1 Y1 by X2 we have a pull back square Val(Y2 , X2 )

/ Y1





Val(Y2 , X2 )

gbir

σ1

/ Val(Y1 , X1 )

(since Val(Y2 , X2 ) = Val(Y2 , NorY2 X2 ) ). It is now immediate that Proposition 3.3.5. A morphism of pairs of schemes g : (Y2 , X2 ) → (Y1 , X1 ) is adic if and only if the canonical diagram / Y1

Y2 

σ2

Val(Y2 , X2 )

gbir



σ1

/ Val(Y1 , X1 )

is a pullback square. Lemma 3.3.6. Let g : (Y2 , X2 ) → (Y1 , X1 ) be an adic morphism of pairs of schemes. (1) The morphism of ringed spaces (g, g  ) : (Val(Y2 , X2 ), MVal(Y2 ,X2 ) ) → (Val(Y1 , X1 ), MVal(Y1 ,X1 ) ) is a morphism of locally ringed spaces. (2) For every v ∈ Val(Y2 , X2 ) the induced homomorphism of stalks (MVal(Y1 ,X1 ),gbir (v) , OVal(Y1 ,X1 ),gbir (v) ) → (MVal(Y2 ,X2 ),v , OVal(Y2 ,X2 ),v ) is adic. Proof. It is enough to prove for the affine case. Assume Yi = Spec Bi , Xi = Spec Ai , Ai ⊂ Bi integrally closed in Bi , i = 1, 2 and g : (B1 , A1 ) → (B2 , A2 ) is adic (abusing notation and using g for the homomorphism of pairs of rings and the corresponding morphism of pairs of schemes). (1) By Lemma 2.3.8 the pullback of every valuation in Val(B2 , A2 ) is already in Val(B1 , A1 ). Hence gbir sends v ∈ Val(B2 , A2 ) to v ◦ g ∈ Val(B1 , A1 ). The induced homomorphism of stalks is the canonical map (B1 )g−1 (p) → (B2 )p which is local. (2) Let v ∈ X2 = Val(B2 , A2 ) with image w ∈ X1 = Val(B1 , A1 ). Let p = ker v and q = ker w. We claim that the induced homomorphism ((B1 )q , Sw ) → ((B2 )p , Sv ) is adic, where Sw and Sv are the semi-valuation rings of w and v. As

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in (1) above, q = g −1 (p). We have a commutative diagram B1  (B1 )q

/ B1 ⊗A1 A2  / (B1 )q ⊗Sw

/ B2

/ B 2 ⊗ A2 S v j v j j γ jjjj vvv j v j v jjjj vv  ujjj vv δ v / v (B2 )p Sv vv TTTT O vv TTTT v TTTT vv ζ TTTT vv v zv * (B1 )q ⊗Sw Sv ⊗B1 ⊗A1 A2 B2 . α

The claim is that δ is integral. Since g is adic α is integral. Hence  is integral as a base change of an integral homomorphism. Since v ∈ X2 , by Lemma 2.4.4, γ is surjective. It follows that ζ is surjective and δ = ζ ◦  is integral.  Theorem 3.3.7. Let g : (Y2 , X2 ) → (Y1 , X1 ) be a morphism of pairs of schemes. Then g is adic if and only if for any morphism of pairs of schemes h : (T1 , Z1 ) → (Y1 , X1 ) the induced morphism of ringed spaces  ) : (Val(T2 , Z2 ), MVal(T2 ,Z2 ) ) → (Val(T1 , Z1 ), MVal(T1 ,Z1 ) ) (˜ gbir , g˜bir

is a morphism of locally ringed spaces, where g˜ : (T2 , Z2 ) = (Y2 , X2 ) ×(Y1 ,X1 ) (T1 , Z1 ) → (T1 , Z1 ) is the pullback of pairs of schemes. Proof. Assume g is adic. As adic morphisms are stable under base change (Lemma 3.3.1(iii)) the claim follows from Lemma 3.3.6(1). For the opposite direction, by Lemma 3.3.1(iv) it is enough to consider only the affine case. Let g : (B1 , A1 ) → (B2 , A2 ) be a homomorphism satisfying the assumption. We want to show that g is adic i.e. that B1 ⊗A1 A2 → B2 is integral. This is the same as showing that A2 → B2 is integral where A2 is the image of B1 ⊗A1 A2 in B2 . Let R be a valuation ring with fraction ring K and a homomorphism Φ : A2 → R such that the diagram / Spec B2 Spec K  / Spec A2

 Spec R

commutes. As the assumption is stable under base change we may assume that this diagram is in fact a point of Val(B2 , A2 ) (if not take the fiber product with Val(R, R)). Let w = (p, R, Φ) ∈ Val(B2 , A2 ) be this point. By Proposition 3.2.15 / Spec B1

Val(B2 , A2 )  Val(B2 , A2 )

σ1

gbir

 / Val(B1 , A1 )

is a pullback diagram in the category of birational spaces. Hence w is a point of Val(B2 , A2 ) and gbir (w) is a trivial valuation of B1 . Denote gbir (w) = (q, k(q), Ψ). By assumption the induced homomorphism (B1 )q = MVal(B1 ,A1 ),gbir (w) → MVal(B2 ,A2 ),w = (B2 )p

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31

is local. It follows that q = g −1 (p). Since Sgbir (w) = OVal(B1 ,A1 ),gbir (w) → OVal(B2 ,A2 ),w = Sw is (always) local and Sgbir (w) = (B1 )q as gbir (w) is trivial, we obtain a local homomorphism (B1 )q → Sw → R = Sw /p(B2 )p . Hence the maximal ideal of Sw is in fact p(B2 )p . In other words for any b ∈ (B2 )p , if w(b) < 1 then w(b) = 0. So Sw = (B2 )p and w is a trivial valuation of B2 . This means that Val(B2 , A2 ) → Spec B1 factors through Spec B2 . Hence Val(B2 , A2 ) = Spec B2 as birational spaces and g is adic by Proposition 3.3.5.  For a morphism of pairs of schemes g : (Y2 , X2 ) → (Y1 , X1 ) it is not enough that  ) : (Val(Y2 , X2 ), MVal(Y2 ,X2 ) ) → (Val(Y1 , X1 ), MVal(Y1 ,X1 ) ) (gbir , gbir

is a morphism of locally ringed spaces for g to be adic as the next example shows. Example 3.3.8. Let k be a field. Set X = Val(k, k), X = Val(k(T ), k[T ](T ) ) and h : X → X the map induced by the obvious injection. X consists of one point x with stalk MX,x = k. X consists of two points x0 corresponding to the trivial valuation on k(T ) and x1 corresponding to the valuation ring k[T ](T ) , both mapping to x. The stalks are MX ,x0 = MX ,x1 = k(T ). The map h is clearly not adic but both induced maps of stalks MX,x → MX ,xi are an injection of fields which is local. 4. Relative Blow Ups In order to prove an analogue of Raynaud’s theory we need an analogue of admissible formal blow ups. For this purpose we introduce and study relative blow ups. f

4.1. Modifications. Let (Y → X) be a pair of schemes. A Y -modification of X is a factorization of f into a schematically dominant morphism h : Y → Z and a proper morphism g : Z → X (see [Tem10, §3.3]). Since f is affine (by the definition h

of a pair of schemes) and g is proper, h is also affine. Hence (Y → Z) is also a pair of schemes. In terms of pairs of schemes, a Y -modification of X is a morphism of pairs (gY , gX ) : (Y  , X  ) → (Y, X) such that gY is an isomorphism and gX is proper. Proposition 4.1.1. Let (Y, X) be a pair of schemes. Any Y -modification of X dominates (Y, X) as scheme models of Val(Y, X) . Proof. Let (gY , gX ) : (Y  , X  ) → (Y, X) be a Y -modification of X. By the valuative criterion of properness gbir : Val(Y  , X  ) → Val(Y, X) is topologically the identity.  : MVal(Y,X) → gbir ∗ MVal(Y  ,X  ) is the identity. For every It is also clear that gbir point v = (y, Ry , Φ) ∈ Val(Y, X) the stalk of OVal(Y,X) is the pullback of the maps MVal(Y,X),v

/ k(y) o

Ry ,

and the same is true for v  ∈ Val(Y  , X  ), so OVal(Y,X) → gbir ∗ OVal(Y  ,X  ) is an isomorphism and must be the identity as a restriction of MVal(Y,X) → MVal(Y  ,X  ) . 

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URI BREZNER f

4.2. Construction of a Relative Blow Up. Let (Y → X) be a pair of schemes and E a finite quasi-coherent OX -module (i.e. locally generated by finitely many sections) contained in f∗ OY . We construct a morphism of pairs πE = (πY , πX ) : (YE , XE ) → (Y, X) with πX projective. If we further assume that E contains the image of OX (in f∗ OY ) then we obtain a modification πE = (idY , πE ) : (Y, XE ) → (Y, X). Construction 4.2.1. Using the multiplication of the OX -algebra f∗ OY , for every d ≥ 0 we define the d-th power OX -module E d to be the image of E ⊗d → f∗ OY⊗d → f∗ OY , where the tensor product is over OX . The graded OX -module ⊕d≥0 E d is quasi-coherent and has a structure of a graded OX -algebra. Denote by I the image of E ⊗OX f∗ OY → f∗ OY (the ideal in f∗ OY generated by E). It is a finitely generated sheaf of ideals in f∗ OY . Note that I d , the d-th power of the ideal I, is the image of the multiplication map E d ⊗OX f∗ OY → f∗ OY . By the construction are natural homomorphisms of graded OX -algebras ⊕d≥0 E d → ⊕d≥0 (E d ⊗OX f∗ OY ) → ⊕d≥0 I d . Note that the second homomorphism is surjective. Thus we obtain a commutative diagram       / ProjX ⊕d≥0 E d / ProjX ⊕d≥0 E d ⊗O f∗ OY ProjX ⊕d≥0 I d  X  Y

f

 / X.

where the top left arrow is a closed immersion  [GD61a,  3.6.2] and the square is a pullback [GD61a, 3.5.3]. Set XE = ProjX ⊕d≥0 E d , YE = ProjX ⊕d≥0 I d . Denote the natural projective morphisms XE → X, YE → Y by πX , πY respectively and by fE the affine morphism which is the composition of the two top arrows in the diagram. Let U ⊂ X be an open affine subscheme such that E|U is generated by finitely many global sections. Assume U = Spec A and set E = Γ(U, E), B = Γ(U,  f∗ OY) and I = Γ(U, I). It follows that V = f −1 (U ) = Spec B and UE = Proj ⊕d≥0 E d ,   VE = Proj ⊕d≥0 I d are open subschemes of XE and YE respectively over U . Let s1 , . . . , sr be generators of the A-module E. The corresponding sections of degree 1 in ⊕d≥0 E d and their images in ⊕d≥0 I d are also generators of the A-algebras ⊕d≥0 E d and ⊕d≥0 I d respectively. For each i there are affine and schematically dominant open immersions (UE )si → UE and (VE )si → VE [GD61a, 3.1.4], fitting into commutative diagrams (VE )si

/ VE





(UE )si

fE

/ UE ,

and these immersions are compatible on the intersections. Furthermore, (UE )si =     Spec ⊕d≥0 E d (si ) , where ⊕d≥0 E d (si ) is the degree 0 part of the localization   of ⊕d≥0 E d with respect to si . Similarly (VE )si = Spec ⊕d≥0 I d (si ) , and the   restriction of fE to (VE )si corresponds to the homomorphism ⊕d≥0 E d (s ) → i   ⊕d≥0 I d (si ) . Since E d → I d is injective for all d ≥ 0, the graded homomorphism

BIRATIONAL SPACES

33

    ⊕d≥0 E d → ⊕d≥0 I d is injective hence ⊕d≥0 E d (si ) → ⊕d≥0 I d (si ) is injective as well. Thus (VE )si → (UE )si is schematically dominant. As UE = ∪i (UE )si , the morphism fE |VE : VE → UE is schematically dominant as well. Hence we conclude fE

that the map fE : YE → XE is affine and dominant, i.e. (YE → XE ) is a pair of schemes and πE = (πY , πX ) is a morphism of pairs. If we also assume that E contains the image of OX in f∗ OY then I = f∗ OY thus YE = Y . In other words πE : (Y, XE ) → (Y, X) is a Y -modification of X. Definition 4.2.2. We call the morphism of pairs πE : (YE , XE ) → (Y, X) constructed above the relative blow up of (Y, X) with respect to the module E. If E contains the image of OX we call the modification πE : (Y, XE ) → (Y, X) a simple relative blow up. Remark 4.2.3. Note that YE is just a usual blow up of Y . Explicitly, the quasicoherent ideal I ⊂ f∗ OY corresponds to a quasi-coherent ideal I ⊂ OY ( [GD61a, 1.4.10]) which in turn defines a closed subscheme Z ⊂ Y and YE = BlZ (Y ). As a special case of Proposition 4.1.1 we have Corollary 4.2.4. Let (Y, X) be a pair of schemes. Any simple relative blow up of (Y, X) dominates (Y, X) as scheme models of Val(Y, X). 4.3. Properties of Relative Blow Ups. Let h : (Y  , X  ) → (Y, X) be a morphism of pairs of schemes, i.e. a commutative diagram Y

hY

/Y

hX

 /X

f

 X

f

and let E be a finite quasi-coherent OX -module contained in f∗ OY . Then h∗X (E) is a finite quasi-coherent h∗X (OX ) = OX  -module. The inclusion E ⊂ f∗ OY induces a homomorphism h∗X (E) → h∗X (f∗ OY ) of OX  -modules. The morphism of schemes hY : Y  → Y is equipped with a homomorphism of sheaves OY → hY ∗ OY  on Y . Pushing forward to X and then pulling back to X  the homomorphism of OY modules OY → hY ∗ OY  we obtain a homomorphism of OX  -modules h∗X f∗ OY → h∗X (f ◦ hY )∗ OY  . Also, from the adjunction h∗X ◦ hX ∗ → Id, we have a natural homomorphism h∗X (f ◦ hY )∗ OY  = h∗X (hX ◦ f  )∗ OY  → f∗ OY  . Composing, we obtain a homomorphism of OX  -modules h∗X (E) → f∗ OY  . Definition 4.3.1. We call the sheaf theoretic image of the above morphism the inverse image module 2 of E (with respect to the morphism of pairs h) and denote it h−1 (E). The inverse image module h−1 (E) is a finite quasi-coherent OX  -module con tained in f∗ OY  . If furthermore E contains f  (OX ) then the homomorphism f  : OX  →  −1  −1  f∗ OY  factors through h (E). Since f is injective h (E) contains f (OX  ). 2by analogy with the inverse image ideal of the usual blow up

34

URI BREZNER

Proposition 4.3.2. Let πE : (YE , XE ) → (Y, X) be a relative blow up of (Y, X) with respect to the module E. Then the inverse image module πE−1 (E) is an invertible sheaf on XE . Proof. We have πE : (YE , XE ) → (Y, X) YE

πY

/Y

πX

 / X.

fE

 XE

f

∗ (E) in fE∗ OY . The question The inverse image module πE−1 (E) is the image of πX is local on X so we assume that X = Spec A, Y = Spec B, A ⊂ B and E = n bj ⊂ Bbi (ϕi the Abi with b0 , . . . , bn ∈ B. Denote Ai = ϕi (A) E(X) = bi j i=0   canonical homomorphism B → Bbi ). Then XE = Proj ⊕d≥0 E d . As we saw in Theorem 2.6.2 the affine charts are (XE )bi = Spec Ai . Denote by Ei the image of bj ∈ Ai we see E ⊗A Ai in Bbi under the map induced by multiplication. Since bi bj · bi ∈ Ei . So Ei is generated by the single element bi over Ai . In that bj = bi ∼ other words, multiplication by bi gives an isomorphism of modules OXE |(XE )bi →  πE−1 (E) |(XE )bi .

Proposition 4.3.3 (Universal property). Let (Y, X) be a pair of schemes with E f

as above. Let Y  → X  be another pair and h = (hY , hX ) : (Y  , X  ) → (Y, X) a morphism of pairs. If h−1 (E) is invertible on X  then h factors uniquely through πE . Proof. By construction h∗X (E) → h−1 (E) is a surjection to an invertible OX     d module. This extends to a map h∗X ⊕d≥0 E d → ⊕d≥0 h−1 (E) of graded OX   d  ⊗d algebras. Note that h−1 (E) = h−1 (E) since h−1 (E) is invertible. Let I  be  −1 the ideal of f∗ OY  generated by h (E), that is the image of h−1 (E)⊗OX  f∗ OY  → f∗ OY  . Then I  is the ideal h−1 (I) = h∗X (I) f∗ OY  , where I is the ideal of f∗ OY generated by E. As h−1 (E) is invertible, I  is locally generated by single element. We obtain a commutative diagram of quasi-coherent graded OX  -algebras   / ⊕d≥0 I  d h∗X ⊕d≥0 I d O O   h∗X ⊕d≥0 E d

  / ⊕d≥0 h−1 (E) d

where the horizontal arrows are surjections and the top row consists of f∗ OY  algebras. The follows from [GD61a, 3.2.4 and 3.7.1]  Lemma 4.3.4. Let (Y, X) be a pair of schemes. The families of relative blow ups and simple relative blow ups of (Y, X) are filtered.

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35

Proof. Let E  and E  be two finite quasi-coherent OX -modules contained in f∗ OY . Then E = E  · E  is also a finite quasi-coherent OX -module contained in f∗ OY . Note that E is the image of E  ⊗OX E  → f∗ OY ⊗OX f∗ OY → f∗ OY . If, furthermore, both E  and E  contain f  (OX ) then we have OX  OX ⊗OX OX → E  ⊗OX E  → E  · E  = E ⊂ f∗ OY . Since f  is injective E contains f  (OX ). Consider the relative blow up (YE , XE ). By Proposition 4.3.2, the inverse image module πE−1 (E) = πE−1 (E  ) · πE−1 (E  ) is an invertible sheaf on XE . Hence πE−1 (E  ) and πE−1 (E  ) are also invertible on XE . By the universal property of the relative blow up (Proposition 4.3.3) (YE , XE ) → (Y, X)  factors through both (YE  , XE  ) → (Y, X) and (YE  , XE  ) → (Y, X). Lemma 4.3.5. Let (Y, X) be a pair of schemes. Assume that X  is an open subscheme of X. Denote Y  = f −1 (X  ). Then a (simple) relative blow up of (Y  , X  ) extends to a (simple) relative blow up of (Y, X). Proof. In [Tem11, Corollary 3.4.4] Temkin shows that a simple relative blow up (Y -blow up of X in his terminology) of (Y  , X  ) extends to a simple relative blow up of (Y, X). The argument also applies to a general relative blow up .  Combining the Lemmas 4.3.4 and 4.3.5 we obtain Corollary 4.3.6. Let (Y, X) be a pair of schemes with open sub-pairs of schemes (Y1 , X1 ), . . . , (Yn , Xn ). For each i = 1, . . . , n let (Yi , Xi Ei ) → (Yi , Xi ) be simple relative blow up. Then (1) each simple relative blow up (Yi , Xi Ei ) → (Yi , Xi ) extends to a simple relative blow up (Y, XEi ) → (Y, X). (2) there is a simple relative blow up (Y, XE ) → (Y, X) which factors through each (Y, XEi ) → (Y, X). Lemma 4.3.7. Given a quasi-compact open subset U ⊂ X = Val(Y, X), there exists a simple relative blow up (Y, XE ) → (Y, X) and an open subscheme U ⊂ XE such that U = τ −1 (U ) = Val(fE−1 (U ), U ). Proof. Since U is quasi-compact and open, by Corollary 4.3.6 it suffices to show for any v ∈ U there exists a simple relative blow up (Y, XE ) → (Y, X) and an open subscheme U ⊂ XE such that v ∈ τ −1 (U ) = Val(fE−1 (U ), U ) ⊂ U. Fix v ∈ U. Let X be an open affinoid subspace of X containing v. Then v ∈ U ∩ X , which by Lemma 3.2.6 is also quasi-compact and open. Hence we can assume that X = Spec A and Y = Spec B. Furthermore, since the rational domains form a basis of the topology of Val(B, A) (Proposition 2.4.1) and each rational domain is quasi-compact(Corollary 2.4.9), we can assume that U is a rational domain. Let U = X ({a1 , . . . , an }/b). Then v satisfies the inequalities v(ai ) ≤ v(b) for all i and v(b) = 0. Since v is unbounded v(b)−1 is not a bound of v, so there exists d ∈ B such that v(b)−1 ≤ v(d). It follows that v(ai d) ≤ v(bd) for all i and 1 ≤ v(bd). Let E be the OX -module, contained in f∗ OY , generated by the global sections 1, a1 d, . . . , an d, bd. The open chart U = (XE )bd ⊂ XE satisfies v ∈ τ −1 (U ) ⊂ U.  Taken together with Corollary 4.3.6, we immediately obtain

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Corollary 4.3.8. Let (Y, X) be a pair of schemes and let Ω be a finite family of quasi-compact open subspaces of the associated birational space (Y, X)bir . Then  of open there is a simple relative blow up (Y, XE ) → (Y, X) together with a family Ω  subschemes of XE such that the associated family Ωbir coincides with Ω. Further covers XE . more if Ω covers (Y, X)bir , the family Ω 5. Birational Spaces in Terms of Pairs of Schemes 5.1. Statement of Theorem. By Corollaries 4.2.4 and 3.2.7 the functor bir takes simple relative blow ups and relative normalizations to isomorphisms of birational spaces. Hence we may regard bir as a functor from the localization of the category of pairs of schemes, with respect to the class of simple relative blow ups and relative normalizations, to the category of birational spaces. By Lemma 3.2.6 any birational space in the essential image of bir is qcqs. This section is dedicated to proving that Theorem 5.1.1. The bir functor provides an equivalence of categories between the localization of the category of pairs of schemes, with respect to the class of simple relative blow ups and relative normalizations, and the category of qcqs birational spaces. The proof splits to checking that bir is faithful, full and essentially surjective. These checks are Proposition 5.2.1 , Theorem 5.3.1 and Proposition 5.4.1 below. In light of Proposition 4.1.1 it would appear that we need to invert all modifications and not just simple relative blow ups. However by [Tem11, Corollary 3.4.8] any modification of (Y, X) is dominated by a simple relative blow up of (Y, X). It follows that the category of pairs of schemes localized with respect to the class of simple relative blow ups (and relative normalizations) is equivalent to the category of pairs of schemes localized with respect to the class of modifications (and relative normalizations). 5.2. Faithfulness. We start with proving that bir is faithful. f

f

Proposition 5.2.1. Let (Y → X) and (Y  → X  ) be pairs of schemes such that X = NorY X and X  = NorY  X  . Denote X = (Y, X)bir and X = (Y  , X  )bir . Let g1 , g2 : (Y, X) → (Y  , X  ) be two morphisms of pairs of schemes. If g1 bir = g2 bir as morphisms of the birational spaces X → X then g1 = g2 . Proof. Denote g1,bir = g2,bir = h. We have a commutative diagram g1,Y

Y

g2,Y

σ

σ

 X

h

 / X

g1,X

 // X  .

τ

τ

 X

// Y 

g2,X

It follows from Proposition 2.4.12 and Lemma 3.2.12 that g1 and g2 agree on the underlying topological spaces (both Y and X). Denote the topological part of g1 and g2 by g = (gY , gX ). Furthermore, as in the faithfulness part of Theorem 3.2.13, g1,Y and g2,Y agree as morphisms of schemes.

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Let {(Yi , Xi )}i∈I be an open affine covering of (Y  , X  ). Denote the topological pull back of each (Yi , Xi ) through g by (Yi , Xi ). For every i ∈ I there is an open affine covering {(Yij , Xij )}j∈Ji of (Yi , Xi ). It is enough to show that the restrictions g1,X |Xij and g2,X |Xij agree as morphisms of schemes Xij → Xi for each j ∈ Ji and i ∈ I. Hence we may assume that (Y  , X  ) and (Y, X) are affine pairs. This was already proved in the faithfulness part of Theorem 3.2.13.  5.3. Fullness. Next we prove that bir is full. Theorem 5.3.1. Let (Y, X) and (Y  , X  ) be pairs of schemes and let h : (Y  , X  )bir → (Y, X)bir be a morphism of birational spaces. Then there exist a simple relative blow up (Y  , XE ) → (Y  , X  ) and a morphism of pairs k : (Y  , NorY  XE ) → (Y, X) such that kbir = h◦gbir , where g is the morphism (Y  , NorY  XE ) → (Y  , XE ) → (Y  , X  ). In particular h is isomorphic to kbir . Proof. Consider the affine case. Let (B, A) and (B  , A ) be two pairs of rings. By Theorem 3.2.13 the morphism between the associated biratonal spaces h : (B  , A )bir → (B, A)bir is given by a morphism of the pairs of rings (B, A) → (B  , NorB  A ) = (B  , A ). We see that the required simple relative blow up is just the identity. For the general case, denote X = (Y, X)bir and X = (Y  , X  )bir . An affine covering {(Yi , Xi )} of (Y, X) gives an affinoid covering {Xi } of X. Each preimage h−1 (Xi ) can also be covered by finitely many open affinoid biratonal subspaces. Hence by Corollary 4.3.8, refining the coverings in a suitable way and replacing (Y  , X  ) with a suitable simple relative blow up, we may assume that we have coverings {Xi } of X and {Xi } of X consisting of finitely many open affinoid biratonal subspaces such that h(Xi ) ⊂ Xi for all i and they are represented by affine open coverings {(Yi , Xi )} of (Y, X) and {(Yi , Xi )} of (Y  , X  ). By the affine case we obtain for every i a simple relative blow up gi : (Yi , Xi Ei ) → (Yi , Xi ) and morphism ki : (Yi , Xi Ei ) → (Yi , Xi ) satisfying kibir = h|Xi ◦ gi bir . By Corollary 4.3.6, there is a simple relative blow up g : (Y  , XE ) → (Y  , X  ) such that the pairs (Yi , Xi Ei ) are open sub-pairs of (Y  , XE ) and form a covering. It follows from Proposition 5.2.1 that we can glue the ki and get a morphism k : (Y  , XE ) →  (Y, X) such that kbir = h ◦ gbir . Corollary 5.3.2. Let the pairs of schemes (Y, X) and (Y  , X  ) both be scheme models for the same birational space X. Then there is another scheme model (Y  , X  ) of X that dominates both via simple relative blow ups and perhaps a relative normalization. Proof. We replace (Y, X) and (Y  , X  ) with (Y, NorY X) and (Y  , NorY  X  ) respec∼ tively. We have an isomorphism h : (Y  , X  )bir → (Y, X)bir . As Y is embedded in the subset of X = (Y, X)bir of points v such that MX,v = OX,v , its scheme structure is determined by (X, MX ). The same is true for Y  , so h induces an isomorphism Y  Y  . We assume that Y  = Y . Applying Theorem 5.3.1 we obtain a simple relative blow up g : (Y, XE ) →  (Y , X  ) and a morphism of pairs k : (Y  , XE ) → (Y, X) such that kbir = h ◦ gbir , in particular kbir : (Y  , XE )bir → (Y, X)bir is also an isomorphism. Using Theorem −1 we obtain a simple relative blow up j : (Y, XF ) → (Y, X) and a 5.3.1 again for kbir

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−1 morphism of pairs q : (Y, XF ) → (Y  , XE ) such that qbir = kbir ◦ jbir .

(Y, XF ) (Y, XE ) o KKK KKKk g j KKK K%   (Y  , X  ) (Y, X). q

As j = k ◦ q : (Y, XF ) → (Y, X) is a simple relative blow up, q : (Y, XF ) → (Y  , XE ) is a simple relative blow up too. Thus the composition of simple relative blow ups g ◦ q : (Y, XF ) → (Y  , X  ) is also a simple relative blow up. So (Y, XF ) is the required scheme model.  5.4. Essential Surjectivety. Finally, let us prove that bir is essentially surjective. Proposition 5.4.1. Every qcqs birational space X has a scheme model. Proof. Consider a qcqs birational space X. We want to show that there is a pair of schemes (Y, X) satisfying (Y, X)bir  X. We proceed by induction on the number of open birational spaces which cover X and have scheme models. As X is quasicompact, it is enough to consider only the case of a covering consisting of two affinoid subspces. Assume that X is covered by two quasi-compact open subspaces U1 and U2 , which admit scheme models (V1 , U1 ) and (V2 , U2 ). Set W = U1 ∩ U2 . Since X is quasi-separated, an application of Corollary 4.3.8 shows that, after blowing-up, we may assume that the open immersions W ⊂ U1 and W ⊂ U2 are represented by open immersions of sub-pairs (T  , W  ) ⊂ (V1 , U1 ) and (T  , W  ) ⊂ (V2 , U2 ). Now, using Corollary 5.3.2, we can dominate the scheme models (T  , W  ) and (T  , W  ) by a third scheme model (T, W ) of W. Using Corollary 4.3.6 we extend the corresponding blow ups to (V1 , U1 ) and (V2 , U2 ), so we may view (T  , W  ) as an open sub-pair of (V1 , U1 ) and (V2 , U2 ). Gluing both along W yields the required scheme model (Y, X) of X. 

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, Relative riemann-zariski spaces, Israel Journal of Mathematics 185 (2011), 1– 42. [Zar44] O. Zariski, The compactness of the riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc. 50 (1944), no. 2, 683–691. [ZS60] O. Zariski and P. Samuel, Commutative algebra, vol. 2, van Nostrand, Princeton, 1960. E-mail address: [email protected] [Tem11]