MINIMAL MODELS OF RIGID ANALYTIC SURFACESKENTARO MITSUI

Contents
1. Introduction
2. Bimeromorphic Morphisms
2.1. Blowing-ups and Blowing-downs
2.2. Analytic Approach
2.3. Algebraic Approach
3. Minimal Models
4. Minimal Models of Fibered Surfaces
5. Appendix
5.1. Coherent Algebras and Finite Rigid Analytic Spaces
5.2. Cohomology Groups
5.3. Weil Divisors and Cartier Divisors
5.4. Proper Rigid Analytic Spaces of Dimension One
References

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1. Introduction
The main purpose of the present paper is to give the existence theorem of
relatively minimal regular models of proper regular rigid analytic surfaces.
We study bimeromorphic morphisms between quasi-compact regular rigid
analytic surfaces. Then we calculate certain cohomology groups to show
the existence of relatively minimal regular models. We also study minimalmodels of fibered surfaces.
We show that any bimeromorphic morphism between quasi-compact regular rigid analytic surfaces is a finite succession of blowing-downs. Hopf
proved the corresponding theorem in the case of compact smooth complex
analytic surfaces in [10]. Ueno stated the theorem in the case of propersmooth rigid analytic surfaces without proof in [23].
We give two proofs of the theorem. The first proof is similar to Hopf’s.
The second proof is based on the fact that we may algebraize bimeromorphic morphisms locally.
This work was supported by Grant-in-Aid for JSPS Fellows (21-1111).
1

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KENTARO MITSUIWe explain the details of the second proof. First, since any proper rigid
analytic space of dimension one is projective (see Appendix), extending a
relatively ample divisor, we may algebraize any bimeromorphic morphism
over a sufficiently small neighborhood of any fundamental point. Then, toconclude the proof of the theorem, we use Köpf’s relative GAGA theoremsin [15] and Shafarevich’s study of birational transformations between twodimensional Noetherian schemes in [21].
Notations and Conventions. We fix a complete non-Archimedean valuation field K with a non-trivial valuation and assume that rigid analytic
spaces are defined over K. We mainly use the terminologies and notations of[4]. However, we make a modification of the definition of smoothness. Let
X be a rigid analytic space. When the local ring OX,x is regular for all x ∈ X,
we say that the rigid analytic space X is regular. When the base change
X ×K K ′ for any extension K ′ /K of complete valuation fields is regular, we
say that the rigid analytic space X is smooth (or geometrically regular). If
the base field K is perfect, then regularity is equivalent to smoothness (see[12]).
The dimension of a rigid analytic space X is the supremum of the Krull
dimension of the local ring OX,x for all x ∈ X. When the Krull dimension
is constant for all x ∈ X, we say that the rigid analytic space X is of pure
dimension.
Let us denote the residue field at a point x on a rigid analytic space by
K(x), which is a finite extension of K. We set a complete valuation on K(x)which is the unique extension of that on K.
We call the absolute value of a non-zero element of K raised to a rational
number power√ a special real number. We denote the set of all special real
numbers by |K × |.
Acknowledgment. The author thanks Kenji Ueno for support and suggestions, and Fumiharu Kato for support.
2. Bimeromorphic Morphisms
2.1. Blowing-ups and Blowing-downs. A proper surjective morphism φ : X →Y of rigid analytic spaces is called a bimeromorphic morphism if there exist
analytic subsets S and T of X and Y of codimension at least one respectively such that the restriction φ| X − S → Y − T is an isomorphism. Notethat Corollary 7 in [4, 9.1.4] implies that the subsets X − S and Y − T are
admissible open subsets.
A surface is a reduced separated rigid analytic space of pure dimension
two. Let φ : X → Y be a bimeromorphic morphism between quasi-compact
surfaces with the analytic subsets S and T in the above definition. Then theanalytic subset S is the union of a finite number of points and proper curves.

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

3

By Proposition 4.6 in [3], the fiber φ−1 (y) is a point or a proper connectedcurve for each y ∈ Y. By the same proposition, we may assume that theanalytic subset T is a finite number of points and that the analytic subset S
is equal to the preimage φ−1 (T ). We call an element y of T a fundamental
point for φ and the curve φ−1 (y) an exceptional curve for φ.
Conrad defined and studied blowing-ups of rigid analytic spaces in [7,4.1] (see also [20] for algebraically closed base field cases). The blowingups provide examples of bimeromorphic morphisms since the blowing-ups
are proper (Corollary 2.3.9 in [7]). Any blowing-up on an affinoid space isisomorphic to the analytification of the algebraic one. We only treat regulare → X be the blowing-up
surfaces and their blowing-ups at a point. Let π : Xof a regular surface X at a point p. Then the resulting surface is also regular.
e along the
In this case, we say that the morphism π is the blowing-down of X
−1
divisor π (p). The following extension theorem shows the uniqueness ofsuch contractions of divisors.
Theorem 2.1 (extension theorem). Let T be a nowhere dense analytic subset of a normal rigid analytic space Y. Assume that a morphism f : Y −T →Z of rigid analytic spaces satisfies the following condition: There exists anadmissible covering {Ui }i∈I of Y such that each image f (Ui −T ) is containedin an admissible affinoid open subset of Z. Then there exists a unique morphism g : Y → Z such that the restriction g| Y − T → Z is equal to f .
Proof. First, we assume that the rigid analytic spaces Y and Z are affinoid
spaces Sp A and Sp B respectively. Applying the Riemann extension theorem to the pull-backs of the analytic functions on Sp B under f ([2, §3]),
we obtain the unique K-algebra homomorphism B → A. The uniquenessfollows from the uniqueness in the Riemann extension theorem. Therefore,
we obtain the desired morphism g : Sp A → Sp B. The construction of gis compatible with the restriction of Sp A to any admissible affinoid open
subset. Thus, by assumption, the general case follows from Proposition 1
in [4, 9.3.3].

In the following subsections, we prove the following theorem.
Theorem 2.2 (Hopf’s theorem). Any bimeromorphic morphism between
quasi-compact regular surfaces is a finite succession of blowing-downs.
2.2. Analytic Approach. In this subsection, we prove Hopf’s theorem via
an analytic approach.
We prove Hopf’s theorem by induction on the number of the irreducible
components of the union of all the exceptional curves. Let y be one of
the fundamental points for a bimeromorphic morphism φ : X → Y. We
take an admissible affinoid open subset V and two analytic functions f ′ and4

KENTARO MITSUIg′ on V that generates the maximal ideal of the local ring OY,y . We may
assume that the subset V does not contain the other fundamental points. Put
U := φ−1 (V). Since any proper morphism is quasi-compact, the admissible
open subset U is quasi-compact. Put f := φ∗ f ′ and g := φ∗ g′ .
Claim. The principal divisors ( f ) and (g) are given by C + E and D + F
respectively where the divisors C, D, E, and F satisfy the following conditions.
(1) The divisors C and D are prime divisors that are not contained inφ−1 (y).
(2) The supports of the divisors E and F equal φ−1 (y).
Proof. Since the restriction φ| U − φ−1 (y) → V − {y} is an isomorphism, theclaim follows from Corollary 2.2.9 in [5].
Claim. Choosing suitable analytic functions f ′ and g′ , we may assume thatthe divisor E equals F in the above claim.
Proof. Since the admissible open subset U is quasi-compact, by Theorem
5.9, we have a finite locally principal covering of U for all prime divisorsthat appear in the finite sum ( f ) or (g). Then the above claim follows fromthe following lemma.
Lemma 2.3. Let I be an ideal of an affinoid algebra A. Let f and g be twoelements of A. Assume that at least one of the two elements is not containedin I. Then there exists a positive real number ǫ satisfying the following
condition. For all two non-zero elements α and β of K whose absolutevalues are less than ǫ, neither f + αg nor β f + g is contained in I.
Proof. Let ψ : B(1)2 → A2 be the continuous map defined by ψ(s, t) = ( f +sg, t f + g). If the ideal I contains f or g, then the condition is fulfilled
on an open neighborhood of (0, 0) minus {s = 0} or {t = 0} respectively.
Therefore, there exists a desired positive real number ǫ in this case. Assume
that the ideal I contains neither f nor g. By Proposition 2 in [4, 3.7.2],
the subset I × A ∪ A × I of A2 is closed. Therefore, the complement ofψ−1 (I × A ∪ A × I) is an open neighborhood of (0, 0). This proves the secondcase.
Claim. The ratio ( f : g) defines a morphism from U to P1V over V.
Proof. We have only to show that the two divisors C and D do not intersect.
Suppose that these two divisors intersected. Let p be one of the intersections. Then the point p is contained in φ−1 (y).
Let π : Dn → D be the normalization of D. The restriction φ ◦ π| Dn −(φ ◦ π)−1 (y) → {g′ = 0} − {y} is an isomorphism. Since the normalization Dn
is separated, there exists a finite family {Wq }q∈π−1 (p) of disjoint admissible

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

5

affinoid
open subsets of Dn such that each element Wq contains q. Put W :=S∗nq∈π−1 (p) Wq and h := π f . Taking an admissible affinoid covering of D
and applying Lemma 2.3 in [14], we obtain a special real number ǫ suchthat the admissible affinoid open subset {z ∈ Dn | |h| ≤ ǫ} is contained in
W. Put G := {x ∈ X | | f ′ | ≤ ǫ, g′ = 0}. Then the preimage of G under
φ is an admissible affinoid open subset of Dn . Thus, Theorem 2.1 showsthat the rigid space G is isomorphic to an admissible affinoid open subset of
Dn . Therefore, the pull-back h is a parameter at the point π−1 (p) while theanalytic function π∗ g vanishes.
We may make the same argument with respect to C. The two results
show that the two analytic functions f and g generate the maximal idealof the local ring OX,p This contradicts the assumption that the point y is afundamental point for φ.
e → Y be the blowing-up of Y at the point y. The above claim
Let π : Ye of rigid analytic spaces suchenables us to define the morphism e
φ: X → Ythat φ = π ◦ e
φ. Then the number of the irreducible components of the unionof all the exceptional curves for e
φ is less than that for φ. Therefore, Hopf’s
theorem follows by induction on this number.
2.3. Algebraic Approach. In this subsection, we prove Hopf’s theorem byan algebraic approach. To algebraize bimeromorphic morphisms locally, we
prepare the following lemma.
Lemma 2.4. Let π : X → Y be a proper morphism of rigid analytic spaces.
Assume that the fiber at a point p on Y is of dimension at most one. Then
there exists an admissible open subset U of Y that contains p such that thebase change π ×Y U : X ×Y U → U is projective.
Proof. Put X p := π−1 (p). By Theorem 5.12, the fiber X p is a projective
curve over K(p).
P By Proposition 7.1.32 in [17], we have an effective ample
Cartier divisor i∈I ai Di on X p .
We fix i ∈ I. Since the morphism π is proper, there exists two admissible
affinoid open subsets P and Q of the preimage of an admissible affinoid
open subset V of Y under π such that the relations Di ∈ P and P ⋐V Q hold.
Since the affinoid open subset Q does not contain the whole fiber X p , there
exists an effective Cartier divisor E on X p whose support is not contained in
Q. Then, by Corollary 7.3.23 in [17], there exist a meromorphic function f
on X p and a positive integer b such that the inequality ( f ) ≥ Di − bE holds.
We may assume that the admissible affinoid open subset P is given by{x ∈ Q | |g1 (x)| ≤ ǫ, . . . , |gn (x)| ≤ ǫ} where the family {g1 , . . . , gn } is anaffinoid generating familySof Q over V and ǫ is a special real number that isless than one. Put R := j=1,...,n Qhg1 /ǫ, . . . , gn /ǫ, ǫ/g j i. Replacing ǫ with

6

KENTARO MITSUIa larger special real number, we may assume that the restriction of R to X pdose not contain the support of ( f ).
We choose an analytic function h on Q whose restriction to the analytic
subset Q ∩ X p is equal to the restriction of f to it. Then the relations X p ∩R ⊂ R − {h = 0} ⊂ R hold. Applying Lemma 1.1 in [14], we obtain aconnected admissible affinoid open subset Wi of V that contains p such thatthe restriction {h = 0} ∩ π−1 (Wi ) is contained in (Q − R) ∩ π−1 (W). Then we
obtain the Cartier divisor π−1 (Wi ) ∩ P ∩ {h = 0} on Q. Thus, by Lemma 1.1
in [24], we obtain the Cartier divisor Fi on π−1 (Wi ), whose restriction to X p
is greater than Di .
Repeating the same procedure for each i ∈ I, we obtain the intersection
W ′ of the admissible open subsets Wi and the sum F ′ of the restriction of the
Cartier divisors Fi to π−1 (W ′ ). By Proposition 7.5.5 in [17], the restrictionof the Cartier divisor F ′ to X p is ample. Applying Theorem 3.2.9 in [7] tothe line bundle defined by the Cartier divisor F ′ , we conclude the proof ofthe lemma.
Proof of Hopf’s theorem. Let φ : X → Y be a bimeromorphic morphism between quasi-compact regular rigid analytic spaces of pure dimension two.
Since the statement is local with respect to Y, we may assume that the morphism φ has a single fundamental point y.
By the above lemma and Chow’s theorem (Theorem 5.3), we may algebraize the bimeromorphic morphism φ over an admissible affinoid open
subset which contains y. Therefore, the theorem follows from the Hopf’s
theorem for two-dimensional Noetherian regular schemes ([21, 4, p.55]).

3. Minimal Models
A relatively minimal regular surface (resp. relatively minimal smoothsurface) is a proper regular surface X (resp. proper smooth surface) suchthat any bimeromorphic morphism from X to a proper regular surface (resp.
proper smooth surface) is an isomorphism. A relatively minimal regularmodel (resp. relatively minimal model) of a proper regular surface X (resp.
proper smooth surface) is a relatively minimal regular surface Y (resp. relatively minimal surface) such that there exists a bimeromorphic morphismfrom X to Y.
To show the existence of relatively minimal regular models and relatively
minimal models, we prove the following proposition.
e → X be the blowing-up of a proper regular
Proposition 3.1. Let π : X
surface X at a point p. Then the inequalitye Ω1 ) > dimK H 1 (X, Ω1X )dimK H 1 (X,eX

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

7

holds. For any line bundle L, the equalitye π∗ L) = dimK H q (X, L)
dimK H q (X,holds for all q.

Proof. We take an admissible affinoid open subset U of Y such that two
analytic functions on U generate the maximal ideal of the local ring OY,p .
Then the assertion follows from the Mayer-Vietoris sequence (Proposition
5.6) and the local calculation in [21, 5, pp.59–65].
Proposition 3.1 and Hopf’s theorem (Theorem 2.2) show the existence ofrelatively minimal regular models and relatively minimal models.
Theorem 3.2 (existence of relatively minimal regular models). Any proper
regular surface admits a relatively minimal regular model.
Theorem 3.3 (existence of relatively minimal models). Any proper smoothsurface admits a relatively minimal model.
4. Minimal Models of Fibered Surfaces
A (regular) fibered surface with fibers of arithmetic genus g (over a regular curve) is a triple (X, S , π) where X is a regular surface, S is a regularcurve, and π is a proper flat morphism from X to S satisfying the following
condition. There exists a nowhere dense analytic subset S 0 of S such thatfor all p ∈ S − S 0 , the fiber π−1 (p) is an irreducible curve of arithmetic
genus g over K(p).
Remark. The flatness of the morphism π is equivalent to the surjectivity ofthe morphism π.
We define algebraic fibered surfaces over algebraic regular curves in thesame way. An (algebraic) fibered surface (X, S , π) is said to be smoothly
fibered if the projection π is smooth. An (algebraic) fibered surface (X, S , π)is said to be proper if the total space X and the base space S are proper.
A relatively minimal fibered surface is a fibered surface (X, S , π) satisfying the following condition. For any fibered surface (Y, S , ρ), any bimeromorphic morphism from X to Y over S is an isomorphism. A relatively
minimal model of a fibered surface (X, S , π) is a relatively minimal fiberedsurface (Y, ρ, S ) with a bimeromorphic morphism from X to Y over S . If
all the relatively minimal models of X are isomorphic to Y over S , we calleach relatively minimal model the minimal model of X over S or the minimal fibered surface. A prime divisor E on a fibered surface (X, S , π) is said
to be exceptional if there exists a fibered surface (Y, S , ρ) and a bimeromorphic morphism β : X → Y over S satisfying the following conditions.
(1) The restriction β|X−E is an isomorphism.

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KENTARO MITSUI(2) The image β(E) is a point on Y.
We also define algebraic (relatively) minimal fibered surfaces in the sameway except for replacing bimeromorphic morphisms by proper birational
morphisms.
Theorem 4.1 (Castelnuovo’s criterion). A prime divisor E on a fibered surface (X, S , π) is exceptional if and only if the following conditions are satisfied.
(1) The divisor E is contained in a fiber at a point on S .
(2) The cohomology group H 1 (E, OE ) vanishes.
(3) The degree of the normal bundle OX (E)|E of E is equal to − dimK C(E).
In this case, the divisor E is isomorphic to the projective line over C(E).
Proof. For any bimeromorphic morphism φ : X → Y over S between fiberedsurfaces, the image of exceptional curves of φ under π is a nowhere dense
analytic subset of S . Therefore, Lemma 2.4, Chow’s theorem (Theorem5.3), and the GAGA theorems (see [15]) enable us to use algebraic results.
Thus, the theorem follows from the algebraic Castelnuovo’s criterion (Theorem 9.3.8 in [17], [21, 6, p.102], Theorem 3.10 in [16]).
Lemma 4.2. An algebraic fibered surface over an affinoid algebra is minimal if and only if its analytification is minimal.
Proof. We have only to show the only if part. We assume that the analytification is not minimal. Then Hopf’s theorem (Theorem 2.2) and Castelnuovo’s criterion (Theorem 4.1) show that the analytification contains an
exceptional divisor. Thus, the lemma follows from the GAGA theorems
and the algebraic Castelnuovo’s criterion.

The following lemma shows that the analytification commutes with basechange.
Lemma 4.3. Let Y be a locally of finite type scheme over an affinoid algebra A. Then, for a K-algebra homomorphism A → B of affinoid algebras,
there exists a canonical isomorphism∼ (Y ×Yan ×
Sp B →
Spec B)an
Sp A

Spec Aanwhere Y is the analytification of A-scheme Y and (Y ×Spec A Spec B)an isthe analytification of B-scheme Y ×Spec A Spec B.

Proof. We can prove this lemma by the same method as in the proof of Satz
1.9 in [15].
Lemma 4.4. The image of any flat morphism of affinoid spaces is a finite
union of affinoid subdomains.

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

9

Proof. This lemma is shown in [19, 3.4.8] (see also Proposition 3.1.7 1 in[8]).

A morphism ρ : Y → T is said to be locally projective if there exists anadmissible covering {Ui }i∈I of T such that each base change ρ ×T Ui : Y ×T
Ui → Ui is projective.
Lemma 4.5. For any proper flat surjective morphism ρ : Y → T to a regular curve with fibers of dimension at most one, there exist two morphisms
α1 : T 1 → T and α12 : T 2 → T 1 satisfying the following conditions.
(1) The morphism α1 is an admissible affinoid covering.
(2) The morphism α12 is étale quasi-compact, and surjective.
(3) The base change ρ ×T T 2 is locally projective.
Proof. This lemma is a special case of Theorem 2.1.4 in [6].

Lemma 4.6. For any fibered surface π : X → S and any étale morphism
α : T → S , the base change π ×S T : X ×S T → T is also a fibered surface.
Moreover, if the arithmetic genus of the fibers of π is at least one and themorphism α is surjective, the fibered surface X is minimal if and only if thebase change π ×S T : X ×S T → T is minimal.
Proof. Since the morphism α is étale, by (21.D) Theorem 51 in [18], thebase space T and the fiber product X ×S T are regular. This implies the firststatement.
We show the second statement. By Castelnuovo’s criterion (Theorem
4.1) and Lemma 4.4, it suffices to show the case when the base spaces S
and T are affinoid spaces. Moreover, by Lemma 2.4, we may assume thatthe projection π is projective. By Chow’s theorem (Theorem 5.3), Lemma
4.3, and Lemma 4.2, the second statement follows from Proposition 9.3.28
in [17].

Theorem 4.7 (existence of minimal models). Any fibered surface with fibersof arithmetic genus at least one over a quasi-compact regular curve admits
a minimal model.
Proof. The last two lemmas and Theorem 9.3.21 in [17] show that the imageof the exceptional divisors is a finite number of points. Thus, Castelnuovo’s
criterion (Theorem 4.1) implies that there exists a relatively minimal modelof the fiberd surface. Lemma 2.4 and Corollary 9.3.24 in [17] show that therelatively minimal model is a minimal model of the fiberd surface.

5. Appendix
5.1. Coherent Algebras and Finite Rigid Analytic Spaces. In this subsection, we study coherent OX -algebras on a rigid analytic space X and finite

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KENTARO MITSUI

rigid analytic spaces over X. By CohAlg(X) we denote the category of coherent OX -algebras on a rigid analytic space X. By Fin(X) we denote thecategory of finite rigid analytic spaces over X. We prove that the category
CohAlg(X) is equivalent to the category Fin(X). Then we prove that we canalgebraize finite rigid analytic spaces over a projective rigid analytic spaceover an affinoid space.
First, recall that for any K-homomorphism A → B between affinoid algebras and any finite A-module M, the completion tensor product M b⊗A B isisomorphic to a finite B-module M ⊗A B (Proposition 6 in [4, 3.7.3]). Therefore, for an arbitrary morphism f : X → Y of rigid analytic spaces and an
arbitrary coherent OY -module F , we may define the pull-back f ∗ F of F
under f , which is a coherent OX -module. In particular, we obtain a functorµXY : CohAlg(Y) → CohAlg(X). We also obtain a functor νXY : Fin(Y) →Fin(X) by the base change of finite rigid analytic spaces over X via f .
By local calculation, we obtain the projection formula.
Proposition 5.1 (projection formula). Let π : X → Y be a finite morphismof rigid analytic spaces. Then, for any coherent OX -module F and any coherent OY -module G, there exists a canonical isomorphism π∗ (F ⊗OX π∗ G) π∗ F ⊗OY G.
For a finite rigid analytic space X ′ over X, the push-forward of the structure sheaf of X ′ is a coherent OX -algebra (Proposition 5 in [4, 9.4.2]). This
gives a functor φX : Fin(X) → CohAlg(X). Since any finite algebra over anaffinoid algebra is an affinoid algebra (Proposition 5 in [4, 6.1.1]), from a
coherent OX -algebra we obtain a finite rigid analytic space over X by pasting spaces and morphisms (Proposition 1 in [4, 9.3.2] and Proposition 1 in
[4, 9.3.3]). This gives a functor ψX : CohAlg(X) → Fin(X). Then we havea following theorem.

Theorem 5.2. For an arbitrary rigid analytic space X, the functors φX andψX give an equivalence of the category of coherent OX -algebras and the category of finite rigid analytic spaces over X. The equivalence commutes withbase change in the following sense. For an arbitrary morphism f : X → Y
of rigid analytic spaces, the diagram
CohAlg(Y)∼µXY

Fin(Y)
νXY



CohAlg(X)∼Fin(X)

is commutative.
In the rest of this subsection, we prove that we can algebraize finite rigid
spaces over a projective rigid analytic space over an affinoid space. We fix

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

11

an affinoid algebra A. Köpf gave the analytification functor from the category of locally of finite type schemes over an affine scheme Spec A to thecategory of rigid analytic spaces over an affinoid space Sp A in [15, §1].
Let us denote each analytification of a locally of finite type scheme X overSpec A, a morphism f over Spec A between such schemes, and a coherent OX -module F by Xan , f an , and F an respectively. When the morphism
f : X → Y is proper, we have the isomorphism(q)(Rq f∗ F )an  Rq f∗an F an

for any coherent OX -module F and all q (Folgerung 3.13 in [15]). First, notethat the following analogue of Chow’s theorem follows from the GAGA
theorems (Satz 5.1 in [15], Satz 4.11 in [15]).
Theorem 5.3 (Chow’s theorem). The analytification functor gives an equivalence between the category of projective schemes over an affine scheme
Spec A, where A is an affinoid algebra, and the category of projective rigid
analytic spaces over an affinoid space Sp A.
By Fin(X) we denote the category of finite schemes over a scheme X.
Theorem 5.4. For any projective scheme X over an affinoid algebra, theanalytification functor yields an equivalence between the category Fin(X)and the category Fin(Xan ). In particular, any finite rigid analytic space overa projective rigid analytic space over an affinoid space is projective.
Proof. Let F be the composite of functors∼ Fin(Xan ),∼ CohAlg(X) →∼ CohAlg(Xan ) →Fin(X) →where the second analytification functor gives an equivalence by the GAGA
theorems. We have only to show that there exists a natural transformationfrom the functor F to the analytification functor of finite schemes over X.
This follows from Theorem 5.2 and the above isomorphism (0). The laststatement follows from Corollaire 6.1.11 in [9].


5.2. Cohomology Groups. We review the theory of cohomology groups
and give Mayer-Vietoris sequences of cohomology groups of abelian sheaves.
First, we review some facts of cohomology groups of abelian sheaves on
rigid analytic spaces. The category of abelian sheaves on a rigid analytic
space is enough injective (see [25]). For a quasi-separated paracompact
rigid analytic space, Čech cohomology agrees with cohomology (Lemma
2.5.7 in [8] and Remark 2.5.5 in [8]). If a quasi-separated paracompact rigid
analytic space is of pure dimension d, then the q-th cohomology group of
any abelian sheaf vanishes for q > d (Corollary 2.5.10 in [8]). The q-th cohomology group of any coherent module on an affinoid space vanishes for
q > 0 (Theorem 8,7 in [22], Satz 2.4 in [13]). Therefore, using Leray-Cartan

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KENTARO MITSUI

spectral sequence (Corollaire 3.3 [1]) with respect to an admissible affinoid
covering, we can calculate cohomology groups of any coherent module on
a separated paracompact rigid analytic space. The q-th cohomology groupsof any OX -coherent module F on a proper rigid analytic space X is a finite
dimensional K-vector space (Theorem 3.3 in [11]). Let us denote the dimension of this K-vector space by hq (F ). In this case, we define the Euler
characteristic χ(F ) of F in the following way:∞X(−1)q dimK H q (X, F ).
χ(F ) :=
q=0

Since the pull-back of an admissible affinoid covering by a finite morphism
is again an admissible affinoid covering (Proposition 1 in [4, 9.4.4]), we
have the following proposition.
Proposition 5.5. Let π : X → Y be a finite morphism between separated
paracompact rigid analytic spaces X and Y. Then, for a coherent OX module F , there exists an isomorphism
H q (X, F )  H q (Y, π∗ F )for all q ≥ 0. In particular, if the rigid analytic spaces X and Y are proper,
then the equality χ(F ) = χ(π∗ F ) holds.
Proposition 5.6 (Mayer-Vietoris sequence). Let {U1 , U2 } be an admissible
covering of an admissible open subset of a rigid analytic space X. Then,for any abelian sheaf F , there exists a canonical exact sequence:
0 −→ H 0 (U1 ∪ U2 , F |U1 ∪U2 ) −→ H 0 (U1 , F |U1 ) ⊕ H 0 (U2 , F |U2 ) −→−→ H 0 (U1 ∩ U2 , F |U1 ∩U2 ) −→ H 1 (U1 ∪ U2 , F |U1 ∪U2 ) −→ · · · .

Proof. We use geometric points on rigid analytic spaces (see [25]). Let
ı0 : U1 ∪ U2 → X, ı1 : U1 → X, ı2 : U2 → X, and ı3 : U1 ∩ U2 → X be theinclusion morphisms. Theorem 1 in [25, 4] implies that the sheaf sequence
0 −→ ı0! (F |U1 ∪U2 ) −→ ı1! (F |U1 ) ⊕ ı2! (F |U2 ) −→ ı3! (F |U1 ∩U2 ) −→ 0.

is exact. Therefore, we obtain the long exact sequence:

0 −→ H 0 (X, ı0! (F |U1 ∪U2 )) −→ H 0 (X, ı1! (F |U1 )) ⊕ H 0 (X, ı2! (F |U2 )) −→−→ H 0 (X, ı3! (F |U1 ∩U2 )) −→ H 1 (X, ı0! (F |U1 ∪U2 )) −→ · · · .

Let G be an abelian sheaf on an admissible open subset U of X. Let ı : U →X be the inclusion morphism. For any geometric point x on X, we have theisomorphism:G x , x ∈ U,
ı! G x  0,otherwise.

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

13

Therefore, if a sheaf sequence G → I• is the canonical injective resolution,
so is the extension ı! G → ı! I• of the sheaf sequence. Thus, we obtain theisomorphism
H q (U, G)  H q (X, ı! G)
for all q. This proves the proposition.
For a rigid analytic space X, by O×X we denote the abelian sheaf of units
in the sheaf of the ring OX . Then there exists a canonical isomorphism
Pic X  H 1 (X, O×X ).
Lemma 5.7. Let I be a coherent OX -ideal on a paracompact quasi-separated
rigid analytic space X. Assume that I2 = 0. Put X0 := (X, OX /I). Let
π : X0 → X be the natural closed immersion. Then the sheaf sequence
0 −→ I −→ O×X −→ π−1 O×X0 −→ 0

is exact where, on an admissible affinoid open subset U of X, the morphism
I → O×X is given by a 7→ 1 + a and the the morphism O×X → π−1 O×X0 is
induced by the natural projection OX (U) → OX (U)/I(U). Therefore, we
obtain the long exact sequence:· · · −→ H 1 (X, I) −→ Pic X −→ Pic X0 −→ H 2 (X, I) −→ · · · .
In particular, if the rigid analytic space X is of dimension one, the morphismπ∗ : Pic X → Pic X0 is surjective.
Proof. Since the sheaf sequence is exact on any affinoid open subset, it isexact at any geometric point on X. Therefore, the sheaf sequence is exact.

5.3. Weil Divisors and Cartier Divisors. We define Weil divisors andCartier divisors on normal rigid analytic spaces. Then we prove that these
are the same notion on regular rigid analytic spaces.
We refer to [3] and [5] for the definition and fundamental results of Weildivisors on normal rigid analytic spaces. We refer to [3] for these of meromorphic functions on rigid analytic spaces. Let us denote the sheaf of meromorphic functions on a rigid space X by MX . A Cartier divisor on X isa global section of the abelian sheaf M×X /O×X . The Cartier divisor classgroup of X is the quotient group Γ(X, M×X /O×X )/Γ(X, M×X ). Let us denote
this abelian group by CaCl(X). The long exact sequence induced by thesheaf exact sequence
0 −→ O×X −→ M×X −→ M×X /O×X −→ 0

gives the injective group homomorphism CaCl(X) → Pic(X), D 7→ OX (D).
We may represent a Cartier divisor by the family {(Ui , fi )}i∈I where thefamily {Ui }i∈I is an admissible covering of X and fi is an element of M×X (Ui ).

14

KENTARO MITSUIThe meromorphic function fi is called a defining function of D on Ui . The
Cartier divisor D is said to be effective if the Cartier divisor D is represented
by the family {(Ui , fi )}i∈I where each meromorphic function fi is an analytic
function on Ui . An effective Cartier divisor defines the closed subspace. We
sometimes identify the closed subspace with the Carier divisor.
When the rigid space X is normal, we may describe Cartier divisors in
the following way. Note that we may restrict Weil divisors to an arbitrary
admissible open subset. A locally principal covering for a Weil divisor D
on a normal rigid analytic space X is an admissible covering {Ui }i∈I of X
such that the restriction of D|Ui is a principal divisor ( fi ) on Ui . Then a
Cartier divisor on X is a Weil divisor on X that admits a locally principal
covering. Note that we may assume that a locally principal covering is anadmissible affinoid covering if exists.
Lemma 5.8. A prime Weil divisor D on a regular affinoid space Sp A admits
a finite locally principal Zariski covering.
Proof. Let I be the ideal of A corresponding to D. We take a point x on
Sp A. Let m be the corresponding maximal ideal. Then the localization Am
is a unique factorization domain since it is a regular local ring. Therefore,
we may write Im = f x Am where f x ∈ Am . Let U x be the subset of Sp A thatis the complement of the support of the divisor D − ( f x ). By Corollary 7 in
[4, 9.1.4], the subset U x is an admissible open subset of Sp A. We take suchan admissible open subset for each point x on Sp A. Since the admissible
open subset U x contains x, the family {U x } x∈Sp A is a Zariski covering of
Sp A. We take a finite subcovering U of this covering. By Corollary 7 in [4,9.1.4], the covering U is an admissible covering of Sp A. Since the equality

D|U x = ( f x ) holds, the covering U is locally principal for D.
The above lemma implies the global case.
Theorem 5.9. All Weil divisors on a regular rigid analytic space X are
Cartier divisors on X.
5.4. Proper Rigid Analytic Spaces of Dimension One. A curve is a reduced separated rigid analytic space of pure dimension one. In this section,
we assume that C is a proper curve. We prove the Riemann-Roch theorem for proper curves. Using this theorem, we show that any proper rigid
analytic space of dimension one is projective.
Since any affinoid algebra is excellent (see [5, 1.1]), the singular locus
of quasi-compact curve is the union of a finite number of points. A regulardivisor on a proper curve is a formal finite sum of regular points. A prime
regular divisor is a divisor which
is defined by aPsingle regular point. The
P
degree of a regular divisor P aP P is the sum P aP dimK K(P). Let us

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

15

denote this integer by degC D. Since we may regard a regular divisor as a
Cartier divisor, a regualr divisor D on C defines the line bundle on C. Let
us denote this line bundle by OC (D).
To calculate the dimensions of cohomology groups of line bundles, let us
show the following lemma.
Lemma 5.10. Let P be a prime regular divisor on a proper curve C. Then,for any line bundle L, the following inequalities and equality hold:(1) 0 ≤ h0 (L ⊗ OC (P)) − h0 (L) ≤ dimK K(P);(2) 0 ≤ h1 (L) − h1 (L ⊗ OC (P)) ≤ dimK K(P);(3) χ(L ⊗ OC (P)) = χ(L) + dimK K(P).
Proof. We define the coherent OC -module F by the following sheaf exact
sequence:
0 −→ OC (−P) −→ OC −→ F −→ 0.
Tensoring the line bundle L, since the support of cokernel F is the point P,
we obtain the sheaf exact sequence:
0 −→ L −→ L ⊗ OC (P) −→ F −→ 0.

Since, by Proposition 5.5, the equalitydimK K(P),hi (C, F ) = 0,

i = 0,
i≥1

holds, the lemma follows from the long exact sequence induced by theabove exact sequence.

Theorem 5.11 (Riemann-Roch theorem for proper curves (the first form)).
For any regular divisor D on a proper curve C, the equality
χ(OC (D)) = χ(OC ) + degC D
holds.
P
P
Proof. We write D = P aP P. By induction on | P aP |, the theorem follows
from the equality of Lemma 5.10.
Theorem 5.12. Any proper rigid analytic space of dimension one is projective.
Remark. If the base field K is separably closed, the theorem follows from
Theorem 2.1.4 in [6].
Proof. It suffices to show that any proper rigid analytic space of pure dimension one is projective. Moreover, by Lemma 5.7 and Corollary
S 3.1.6[7], we have only to show that any proper curve is projective. Let i∈I Ci be
the irreducible decomposition of a proper curve C. Choose a regular point

16

KENTARO MITSUI

Pi on each Ci . By Lemma 5.10 and Theorem 5.11, there exists a positive
integer ni such that the line bundle OC (ni Pi ) admits a section that is not asection of OC ((ni − 1)Pi ). The section gives the meromorphic function fi ,which is analytic except for the pole at the point Pi . Let f be the summationof all the meromorphic functions fi . Then the restriction of f to each Ci is
non-constant. The meromorphic function f gives the morphism φ : C → P1K
of rigid analytic spaces.
Proposition 4 in [4, 9.6.2] and the following sentences show that the morphism φ is proper. Therefore, the proper mapping theorem (Proposition 3
in [4, 9.6.3]) shows that the image φ(C) is an analytic subset of P1K . We put
the reduced structure on φ(C). Since the morphism φ is proper, we have theStein factorization µ ◦ λ : C → S → φ(C) (Proposition 5 in [4, 9.6.3]). By
Lemma 4 in [4, 9.6.3], the morphism λ is surjective and for any point s on
S , the preimage λ−1 (s) is a connected analytic subset of C. If the preimage
λ−1 (s) is not a point, then the meromorphic function f is constant on a irreducible component of C. This is absurd. Therefore, the preimage λ−1 (s)is a point on C. Thus, Lemma 4 in [4, 9.6.4] shows that the morphism λ is
an isomorphism. Since the morphism µ is finite, Theorem 5.4 implies thatthe rigid analytic space S is projective. Therefore, the proper curve C is
projective.

Theorem 5.13 (Riemann-Roch theorem for proper curves (the second form)).
Let π : C n → C be the normalization of a proper curve C. For any line bundle L on C, the equality
holds.

χ(L) = χ(OC ) + degC n π∗ L

Proof. We define a coherent OC -module F by the following sheaf exact
sequence:
0 −→ OC −→ π∗ OC n −→ F −→ 0.
Since the singular locus of each Ui is a finite number of points, the supportof F is a finite number of points. Then, tensoring L, we obtain the sheaf
exact sequence:
0 −→ L −→ π∗ OC n ⊗ L −→ F −→ 0.

These sequences give the equalitiesandχ(π∗ OC n ) − χ(OC ) = χ(F )χ(π∗ OC n ⊗ L) − χ(L) = χ(F ).
Eliminating χ(F ), we obtain the equality:
χ(L) − χ(OC ) = χ(π∗ OC n ⊗ L) − χ(π∗ OC n ).

MINIMAL MODELS OF RIGID ANALYTIC SURFACES

17

Since, by the projection formula (Proposition 5.1), we have the isomorphismπ∗ OC n ⊗ L  π∗ π∗ L,
the right-hand side of the above equality is equal to χ(π∗ π∗ L) − χ(π∗ OC n ).
Therefore, Proposition 5.5 gives the equalityχ(π∗ π∗ L) − χ(π∗ OC n ) = χ(π∗ L) − χ(OC n ),which is equal to degC n π∗ L by the Riemann-Roch theorem for proper regular curves (Theorem 5.11).

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Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502,
Japan
E-mail address: mitsui@math.kyoto-u.ac.jp