KENTARO MITSUI C ONTENTS
The main purpose of the present paper is to give a criterion for relative minimality of proper regular rigid analytic surfaces.  We have proved the existence of relatively minimal regular models of proper regular surfaces in [8].  First, we study coverings of analytic subsets.  Second, we study deformations of divisors.  Finally, we apply these studies to the study of proper regular rigid analytic surfaces. Ueno studied these objects in [10] in the case of smooth rigid analytic spaces over algebraically closed base fields of characteristic zero.  We give detailed proofs of the theorems concerned with them in the general case. First, to analyze analytic subsets on rigid analytic spaces, we study cov- erings of analytic subsets.  Further, we study small tubular neighborhoods of smooth divisors on smooth rigid analytic spaces.  In the affinoid case, these tubular neighborhoods were studied by Kiehl in [4]. Second, we study deformations of divisors on rigid analytic spaces. We prove that we can deform effective Cartier divisors on quasi-compact sepa- rated rigid analytic spaces whenever certain obstructions vanish. The study of coverings enables us to generalize the result to the case when the rigid analytic spaces are not quasi-compact.  The point of the proof is to show the convergence of certain series. We show this convergence by using the non-Archimedean open mapping theorem. Finally, we give a criterion for relative minimality of proper regular sur- faces which is similar to Castelnuovo's criterion in the case of algebraic surfaces and complex analytic surfaces. The study of deformations enables us to prove that we can blow down exceptional curves of the first kind on regular rigid analytic surfaces. Then, by the study of bimeromorphic mor- phisms in [8], we obtain the desired criterion. When we study deformations and blowing-downs, we refer to the arguments of complex analytic cases ([7], [6]). In the appendix, we give methods of calculating intersection numbers and prove the Riemann-Roch theorem for proper smooth rigid analytic surfaces. Notations and Conventions. We fix a complete non-Archimedean valu- ation 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 [1] and [8]. Acknowledgment. The author thanks Kenji Ueno for support and sugges- tions, and Fumiharu Kato for support and helpful comments. In this section, we show the existence of small tubular neighborhoods for proper smooth divisors on separated smooth rigid analytic spaces of pure dimension.  More precisely, we show that, when a proper smooth divisor on such a rigid analytic space satisfies a certain condition, there exists a small admissible open subset that contains the divisor and admits a finite admissible affinoid covering that consists of tubular neighborhoods of the divisor.  If the rigid analytic space is proper, then the condition is fulfilled for any smooth divisors. We start with preparing terminologies and notations.  For a special real number r (see [8]), let us denote a one-dimensional closed disk with the radius r by B(r). Let V be an affinoid subdomain of an affinoid space U . When V is relatively compact in U (over K ), we write V ∈ U. Let { U i } i ∈ I and { V i } j ∈ ] be two finite families of admissible affinoid open subsets of a rigid analytic space X .  When V i is relatively compact in U i for all i , we write {V i}iel ∈ {U i}iel . If the family {V i}iel covers a subset S of X, then we say that the family { U i } jel is a relatively big covering of S  and that the family { V i } i ει is a relatively small covering of S  associated to the relatively big covering { U i } i ∈ I . When S  equals X , we say that X admits a relatively big covering. Let {V i } j E I be a relatively small covering of a subset S  of a rigid analytic space X associated to the relatively big covering { U j } j ∈ I . When X is quasi- separated, for any subset J of I , the family { U j } of an admissible open subsetリj of X by the following lemma. When 2 X is separated, the family {U ic I U {X – U ic I V i} is an admissible covering of X by Lemma 1.1 in [11]. For two coverings U and V of a rigid analytic space X and a subset W of X , let us denote {U ∩ V | U ∈ U, V ∈ V} and {U ∩ W | U ∈ U} by U ∩ V and U ∩ W respectively. Lemma 2.1. Let U be a family of admissible affinoid open subsets of a quasi-separated rigid analytic space X.  Assume that X admits an admis- sible affinoid covering V all of whose covering elements intersect at most finite number of elements of U.  Then the family U ∩ V is an admissible covering of an admissible open subset ∪ U of X.  Therefore, the family U is an admissible covering of | | U. Proof. Put U := ∪ U . Let V be an element of V . Since U ∩ V is a finite union of an admissible affinoid open subsets of V , Corollary 4 in [1, 9.1.4] shows that U ∩ V is an admissible open subset of V . Thus, the union U is an admissible open subset of X . The same corollary shows that U ∩ V has a refinement which is an admissible covering of U ∩ V . Therefore, since V ∩ U is an admissible covering of U , the family U ∩ V is an admissible covering of U . This proves the first assertion. Since the admissible covering U ∩ V is finer than the covering U , the last assertion follows. □ The next lemma enables us to enlarge the elements of relatively small coverings. Lemma 2.2. Assume that two finite families U and V of admissible affinoid open subsets satisfy the relation V ∈ U. Then there exists a finite family W of admissible affinoid open subsets such that the relations V ∈ W ∈ U hold. Proof. It suffices to show that for an affinoid subdomain V which is rela- tively compact in an affinoid space U , there exists an affinoid subdomain W of U such that V ∈ W ∈ U. Proposition 4 [1, 6.2.1] implies that we may write U = Sp K { f 1 , . . . , f n } such that for a suitable special real number ε, the affinoid subdomain V is contained in the set U ( fi / ε, . . . , f n / ε ). Take a special real number ε 0 such that the inequalities ε < ε 0 < 1 holds. Then we have only to put W := U(fi/œ, . . . , fn/œ). □ The next lemma shows that the relatively compactness is stable under the pull-backs by closed immersions. Lemma 2.3.  Let U and V be two admissible affinoid open subsets of a rigid analytic space X such that V is relatively compact in U. Let Y be an analytic subset of X.  Then V ∩ Y is relatively compact in U ∩ Y.  Therefore, if an analytic subset admits a relatively big covering, then the analytic subset is proper. Proof. We may write U = Sp K 〈 f 1 , . . , f n 〉 such that V is contained in the subset {x ∈ U | | fi(x)| < 1, . . . , | fi(x)| < 1}. For each i, let g i be the image of f i by the homomorphism corresponding to the closed immersion U ∩ Y → U. Then U ∩ Y = Sp K 〈 g 1 , . . , g n 〉 , and V ∩ Y is contained in the subset {x ∈ U ∩ Y | | fi(x)| < 1, . . . , | fii(x)| < 1}, which is equal to the subset {x ∈ U ∩ Y | |g 1 (x)| < 1, . . . , |g n (x)| < 1}. This proves the lemma. The following lemma shows that the relatively compactness is stable un- der product and intersection. Lemma 2.4. Let V i be an affinoid subdomain of an affinoid space U i for i = 1,2. Assume that each V i is relatively compact in U i . Then the product V 1 × V 2 is relatively compact in the product U 1 × U 2 .  If V 1 and V 2 are admissible affinoid open subsets of a separated rigid analytic space X, then the intersection V 1 ∩ V 2 is relatively compact in the intersection U 1 ∩ U 2 . Proof. The first statement follows from Lemma 1 in [1, 9.6.2]. Since X is separated, the second statement follows from Lemma 2.3.口Lemma 2.5.  Let V  be an affinoid subdomain of an affinoid space Y .  Assume that affinoid subdomains U and W are relatively compact in the affinoid spaces X and V respectively. Then for any morphism φ : X → Y of rigid analytic spaces, the affinoid subdomain U ∩ φ− ¹ (W) is relatively compact in the affinoid subdomain φ-¹( V ).  In particular, when an affinoid subdomain T is relatively compact in an affinoid subdomain S , for a family { g i } 1<i<m+n of analytic functions on X and a family { ε i , ε j } 1≤ i≤m+n of special real numbers such that ε i < ε ′, for all i, the affinoid subdomain T(g 1 / €1, . . . , g m / €m, €'m+1, . . . , €'m+n/gm+n) is relatively compact in the affinoid subdomain: S(g 1 /E 1 , . . . , g m /E m , E m +1 /8 m +1 , . . . , E m +n /8 m +n ) . Proof.  The graph of φ is an analytic subset of the product X×Y . By Lemma 2.4, the product U×W is relatively compact in the product X×V . Therefore, the first statement follows from Lemma 2.3.  The last statement follows from the first statement if we put X := S(g1/E1,...,gm/Em,Em+1/gm+1,...,Em+n/gm+n), Y := B(1)m+n, and
4 Lemma 2.6. Assume that an analytic subset Y of a rigid analytic space X is defined by analytic functions f 1 , . . , f n on X. Let U be an admissible open subset of X that contains Y. Then there exists a special real number ε such that the affinoid subdomain X(fi/e,...,fn/e) is contained in U. Proof.  This lemma is shown in Lemma 2.3 in [5]. See also Lemma 1.1.4 in ● [3]. We can replace relatively big coverings with enough smaller coverings. Lemma 2.7. Let Y be an analytic subset of a rigid analytic space X. Assume that there exists a relatively big covering of Y. Let U be an admissible open subset of X that contains Y. Then there exists a relatively big covering of Y that is contained in U. Proof. Let {V i}iel be a relatively small covering associated to a relatively big covering { U i } i ∈ I of Y .  We fix an arbitrary element i of I .  We may write U i = Sp K (f 1 , . . , f m ). Put Y i := Y ∩ U i . We have a finite number of defin- ing functions g 1 , . . , g n of Y i on U i . By Lemma 2.6, there exists a special real number ε such that U i (g 1 /ε, . . ,g n /ε) ⊂ U . It suffices to show that for special real numbers ϵ 0 , ϵ 1 such that ϵ 0 < ϵ 1 < ϵ , the affinoid subdo- main V i (g 1 / ε 0 , . . ,g n / ε 0 ) is relatively compact in U i (g 1 / ε 1 , . . ,g n / ε 1 ). This follows from Lemma 2.5. □ We can refine relatively big coverings by Zariski coverings. Lemma 2.8.  Let U be a Zariski covering of an affinoid space X.  Let V be an affinoid subdomain that is relatively compact in X.  Then there exists a relatively big covering of V that refines U. Proof.  Refining U , we may assume that X is covered by a finite number of admissible open subsets of the type { x ∈ X | | f ( x ) | > 0} . Further, by Lemma 6 in [1, 9.1.4], we may assume that the covering elements are rational sub- domains of the type X( ε / f ). Therefore, it suffices to show that for special real numbers & and ε 1 such that ε 0 < ε 1 < ε , the affinoid subdomain V ( ε 1 / f ) is relatively compact in X ( ε 0 / f ). This follows from Lemma 2.5. Proposition 2.9. Assume that the family {U i}iel is a relatively big covering of an analytic subset D of a separated rigid analytic space X.  We take a family { f ij } j } j } of defining functions of D on each U j .  Put U j c := {x ∈ U | ∀ j ∈ J i, [f ij (x)| ≤ ε}.  Then there exist an admissible open subset U o of X and a special real number ε satisfying the following conditions. (1) The family {U i}ieI U {U o} is an admissible covering of X. (2)  For all i ∈ I, the restriction U iε is disjoint from U 0. Proof. Let V be the union of the relatively small covering of {Uj} j j j j j . Put U o := X − V . The first condition is fulfilled. Since Lemma 2.1 implies that V is an admissible open subset of X , Lemma 2.6 shows that there exists a special real number ε such that V contains U iε for all i ∈ I .口Proposition 2.10. Let π : X → Y be a morphism of rigid analytic spaces. Assume that the rigid analytic space X is quasi-separated. Suppose that the preimage of a point p on Y admits a relatively big covering.  Then there exists an admissible open subset V of Y that contains p such that the re- striction π to the preimage of V under π is proper. Proof. Let D be the preimage of the point p under π . We choose an ad- missible affinoid open subset W of Y that contains p and a family { g } J J of defining functions of p on W .  For a special real number ε , we put W , := {ν ∈ Y | Vi ∈ I, |g;(x)| ≤ ∈}. We take a relatively small covering { V i } ies of D associated to the rela- tively big covering { U i } j ∈ I . Since the family { π * g i } j ∈ I is a family of defining functions of D on the preimage π− 1 ( W ), by Lemma 2.6, we obtain a special real number ε such that the preimage of the admissible affinoid open subset W ε under π is contained in the union of the relatively small covering { V i } j j ε l ; Put Z := π-¹(W ε ). Since the relation V i ∩ Z ∈ w U i ∩ Z holds for all i ∈ I , the restriction π to Z is proper. Thus, by Lemma 2.1, the admissible open subset W ε is a desired one. □ From now on, we assume that a rigid analytic space X is separated and normal. Let D be a smooth prime divisor on X . Let U be an affinoid open subset Sp A of X . Put R := DNU . We say that U is a local tubular neighbor- hood of D if there exist a defining function f of D on U and an isomorphism U ≈ R × B (1) such that the corresponding K -algebra homomorphism φ : A / f A (X) → A satisfies the following conditions. (1) The homomorphism φ sends X to f . (2) The restriction of φ to A / f A is a section of the natural homomor- phism A → A / f A . In this case, we call R an extendable open subset of D.  An admissible affinoid open subset of an extendable open subset is also extendable.  For a special real number ε ≤ 1, let us denote the admissible open subset Sp A/fA(X/ε) of X by U ε or R(ε).  When a smooth prime divisor D ad- mits a relatively big covering that consists of extendable open subsets, we say that the divisor D admits an extendable covering . A local tubular neigh- borhood covering of a smooth prime divisor D is a relatively big covering of D such that each covering element is a local tubular neighborhood of 6 D. A tubular neighborhood of a smooth prime divisor D is an admissible open subset P of X such that there exist an analytic function F on P and an admissible covering {U j } ε ) of P satisfying the following conditions. (1) The family { U i } iel is a local tubular neighborhood covering of D . (2) For each i ∈ I , the restriction F | U , is the composite of the isomor- phism U i → (U i ∩ D) × B (1) and the projection (U i ∩ D) × B (1) → B(1). For a special real number ε ≤ 1, let us denote the preimage F − 1 ( B ( ε )) by P ε. Theorem 2.11. Let f 1 , . . , f n be a generating family of an ideal I of an affinoid algebra A.  Assume that the affinoid space Sp A is smooth on the closed subspace Sp A/I.  Further, suppose that the closed subspace Sp A/I is smooth. Then there exist three special real numbers ε, ε 1 , and ε 2 , and a finite family (g j )נבןof elements of A satisfying the following conditions. For each j ∈ J, put B j := A (f 1 / ε, . . . , f n / ε, ε 0 / g j ). (1) The inequality ∈ ξ ε 1 holds. (2) The family {Sp A{є ı /g j } j (3) For all j ∈ J, there exists an integer n j and an isomorphism φ j : (B j /IB j )(X j 1 , . . ,X jn j ) ⇒ B j where the left-hand side is a free affinoid algebra over B i /IB j. (4) For all j ∈ J, the family {φ j (X jk)}1<k≤n j generates the ideal IB j . (5) For all j ∈ J, the restriction φ j on B j /IB j is a section of the natural homomorphism B j → B j /IB j . If the affinoid space Sp A is of pure dimension m and the affinoid space Sp A/I is of pure dimension m – n, then we can require the following addi- tional conditions. (6) For all j ∈ J, the integer n j is equal to n. (7) For all j ∈ J and for all integer k such that 1 ≤ k ≤ n, there exists a non-zero element ε i of K such that the isomorphism φ i maps X ik to fk/Ej. Proof. This theorem follows from Theorem 1.18 in [4] and the following remark of the theorem.  We have only to note that, when we construct a family {Sp B i} of affinoid subdomains of Sp A , we restrict a refinement of a Zariski covering of Sp A to Sp A { f i / ε, . . . , f n / ε ⟩ . □ We give some equivalent conditions of the existence of local tubular neighborhood coverings. Theorem 2.12.  An extendable covering of a smooth prime divisor D on X can be extend to a local tubular neighborhood covering of D.  Conversely, if the rigid analytic space X is smooth on D and the divisor D admits an extendable covering, then there exists a local tubular neighborhood cover- ing of D. Therefore, if the rigid analytic space X is smooth on D, then the following conditions are equivalent. (1) There exists an extendable covering of D. (2) There exists a local tubular neighborhood covering of D. (3) There exists a relatively big covering of D. Remark. If the rigid analytic space X is proper, the third condition is ful- filled for any divisors. Proof. First, we show the first statement.  Let {S i}iel be a relatively small covering associated to an extendable covering { R i } j ∈ I of D . Take a special real number δ that is less than one. Then, by Lemma 2.4, S i (δ) ∈ R i (1) for all i .  Therefore, the family { R j (1)} j e l is a local tubular neighborhood covering of D . Next, we show the second statement.  Let { Yj}jeJ be a relatively small covering associated to a relatively big covering { X i } j ∈ L of D . By Lemma 5.8 in [8] and Lemma 2.8, we may assume that the family { Xj}jeJ is a locally principal covering. Further, by Theorem 2.11, we may assume that there exists a local tubular neighborhood of D on each X j . Put R j := X j ∩ D and S i := Y i ∩ D for each i . Lemma 2.3 shows that S i ∈ R i for all i . Therefore, the divisor D admits an extendable covering { R i } j ∈ I . □ Proposition 2.13. Let {U i}iel be a local tubular neighborhood covering of a smooth prime divisor D on the rigid analytic space X. Then the following statements hold. (1) There exists a local tubular neighborhood covering {V i}ie1 such that V i ∈ U i for each i. (2) For all family {δ i } iε1 of special real numbers such that δ i ≤ 1 for all i, the family {U is,}ie1 is a local tubular neighborhood covering of D. (3) Let U be an admissible open subset of X that contains D.  Then there exists a special real number ∈ such that U i ε is contained in U for each i. (4) Let U be an admissible affinoid open subset of X.  Let V be an admissible open subset of U that contains U ∩ D. Then there exists a special real number ε such that V contains U iε ∩ U for each i. Proof. Put R i := D ∩ U i .  By Lemma 2.3 and Lemma 2.2, we have two coverings { S i } iei, { T i } iei of D such that T i ∈ S i ∈ R i for all i. Take two special real numbers ϵ 0 , ϵ 1 such that ϵ 0 < ϵ 1 < 1. By Lemma 2.4, T i ( ε 0 ) ∈ S i ( ε 1 ) ∈ U i for all i . Therefore, the first statement holds. Take a special real number ε i such that ε i < δ i . Then the relation S i ( ε i ) ∈ U iδ , holds for all i . 8 Therefore, the second statement holds. By Lemma 2.6, the third and fourth statements hold if we replace the above special real numbers ε i and δ j . □ From the above proposition, we deduce the following proposition. Proposition 2.14. Let P be a tubular neighborhood of a smooth prime di- visor D on the rigid analytic space X. Then the following statements hold. (1) For any special real number ∈ such that ∈ 1, the subset P c is a tubular neighborhood of D. (2)  Let U be an admissible open subset of X that contains D. Then there exists a special real number ε such that P ε is contained in U. (3) Let U be an admissible affinoid open subset of X.  Let V be an admissible open subset of U that contains U ∩ D. Then there exists a special real number ε such that V contains P ε ∩ U. In this section, we prove that we can deform a quasi-compact effective Cartier divisor whenever a certain cohomology group vanishes. Let D be a quasi-compact reduced Cartier divisor on a quasi-compact separated rigid analytic space X . Let { U j } j ∈ I be an admissible affinoid cov- ering of X . Put U := {U i } j ∈ I . For each i ∈ I , put D i := D ∩ U i . We may write D i  =  Sp B i  and U i  =  Sp A i .  Assume that the kernel of the natural surjective homomorphism θ i : A i → B i is generated by the single non-zero- divisor h i of A i . For each i, j, k ∈ I, put A ij := A i ⊕ A j, U ij := Sp A i ∩ Sp A j, U ijk := Sp A i ∩ Sp A j ∩ Sp A k , B ij := B i ⑧ B j, and D ij := D ∩ U ij = Sp B ij . Let h ij be the invertible element h i /h i of A ij . Let δ be a special real number. For each i ∈ I , put V iδ := U i × Sp K ( T / δ ) . Put Y δ := X × Sp K { T / δ } , and V δ := { V iδ } iει . All complete K -algebra norms on any affinoid algebra is equivalent to each other (Proposition 2 in [1, 6.1.3] and Corollary 3 in [1, 2.1.8]).  For each i, j ∈ I , we set complete K -algebra norms on affinoid algebras Aj, Ajj, B j , and B j . Using a coherent O x -ideal O x (− D ), we put O D := O x /O x (− D ).  The normal bundle of D is the line bundle O X ( D ) | D on D . Let us denote this line bundle by N D / X .  For a presheaf F of abelian groups on X , let us denote the q -cochain group (resp. q -cocycle group) of U with coefficient in F by C9(U,F) (resp. Z9(U,F)). Theorem 3.1. We use the above notations. Let { t } iel be a section of the normal bundle N DIX of D. Assume that the cohomology group H 1 (D, N DIX ) vanishes. Then there exist a special real number δ, s = { s i } i ∈ C 0 (V δ, O γ δ), and f = { fij }i,jel E C 1 (V δ , O Y k ) satisfying the following conditions: (1) For all i ∈ I, the first two terms in the expansion of s i with respect to T is equal to h i + s i1 T where the image of s i1 under θ i is equal to (2) For all i, j,k ∈ I, the two equalities s i = f ijs j and f ik = f ijf jk hold. (3) For all i, j ∈ I, the analytic function fij is invertible. We prove the above theorem. For each i , j ∈ I , we write and First, note that it suffices to construct s ∈ C ( V 1 , O γ i ) and f ∈ C 1 (V 1 , O γ i ) that satisfy the conditions (1)–(2). Indeed, if we choose a special real num- ber δ such that the inequality
holds for all i, j ∈ I , then the condition (3) is fulfilled. We construct a formal solution.  For formal power series F, G ∈ A [[T]] over an affinoid algebra A , if F ≡ G (mod T µ + 1 ), then we write F ≡ µ G . Put s i0 := h i , s i1 := t i , and f ij0 := h ij. For μ ≥ 0, put
and
Claim. There exist sm = { sim}iel ∈ C 0 (U, O x ) and fm = { fijm}i.jel ∈ C 1 (U, O x ) for all m ≥ 0 such that the equation
holds for all μ ≥ 0. If the above equation ( μ ) is satisfied, then the congru- ence
holds. 10 Proof.  Clearly, the equation (0) holds. For a positive integer μ , assume that the families { s m } o≤m≤μ− 1 and { f m } o≤m≤μ− 1 satisfy the equation ( μ − 1). We define s u and f u such that the equation ( μ ) is fulfilled. The equation ( μ ) is equivalent to the equation fijuhj = Sip — hijSjp + gijp on Uij where g ijμ ∈ A ij is defined by the following congruence: gijµTµ =µ sµ-¹ — fµ-¹ sµ-¹ on Uij. The cochain {g ijμ | D ij } i.j ∈ I satisfies the one-cocycle condition: 8iku = 8iju + hijgjku on Dijk First, we show that there exists s μ such that the equality (*) Sip – hijSjp + gijp = 0 on Dij holds. If μ is equal to one, then the equality g ijμ = 0 holds. For each i ∈ I , put S iμ := ι i ( t i ) where ι is a set-theoretical section of the surjective homomor- phism θ i : A i → B i . Then, since t 1 is a section of the normal bundle N D I X , the equation (*) is fulfilled. If μ is greater than one, by the assumption H 1 ( D , N D ( x ) = 0, there exists tiu ∈ B i such that the equality tip – hijtjμ + gijμ = 0 on Dij holds.  For each i ∈ I , put s iμ  := ι i (t iμ ).  Then s μ satisfies the equation (*).  Since the analytic function siμ − hijs jμ + gijμ vanishes on Dij and the restriction h j | U j , is a defining function of D ij , the analytic function s iμ − hijSju + giju is divisible by hj. Set fijµ := (Siµ - hijµSjµ + Sijµ)/hj on Uij Then the pair s μ and f μ is a solution of the equation ( μ ). ● Lemma 3.2.  Let h be a defining function of an effective Cartier divisor D on a affinoid space Sp A.  Then there exists a positive real number α such that if an element f of A vanishes on D, then the equality \f/h\ a ≤ α\f\ a holds. Proof. Since, for all K -linear maps between K -normed spaces, continuity is equivalent to boundedness (Corollary 3 [1, 2.1.8]), Banach's theorem (Theorem 1 in [2, I, §3]) shows that the isomorphism A → fA is homeo- morphism. This proves the lemma. □ For each i, j ∈ I , applying the above lemma to the affinoid space Sp A jj and the defining function h j | ∪ i , of D | ∪ i , we obtain the positive real number α jj . Put α := max j j ∈ I { j : We introduce K-Banach space norms on cochain groups.  Put A q  := C 9 (U | D , O X ) and C 9 := C 9 (U | D , N D | X ). Since the family U | D is a finite admissible affinoid covering of D , for each q , we may regard the q -cochain group A 9 as an affinoid algebra that is the finite direct sum of the affinoid algebras.  Then the q -cochain group A q is a K -Banach algebra and the q - cochain group C q is an A q -Banach space.  For each q = 0,1, we define q-coboundary operator δ q : C q → C q +1 as follows. For u = { u i } i ∈ I ∈ C 0 , put (δ 0 u) ij := h iju j − u j on U ij For v = { vij}i,j€1 E C 1 , put (δ 1 ν)ijk := hijVjk = Vik + Vij on Uijk· By Z 1 and B 1 we denote the K-normed subspaces Ker δ 1 and Im δ 0 of C 1 respectively. Claim. There exists a positive real number β satisfying the following con- dition. For any ν ∈ C 1 such that δ 1 ν = 0, there exists u ∈ C 0 such that the equation ν = δ 0 u and the inequality | u | c 0 < β | ν | c 1 hold. Proof. Since addition, subtraction, multiplication, and restriction are con- tinuous, the coboundary operators are continuous K -linear maps.  There- fore, the kernel Z 1 of δ 1 is a K -Banach space. Put η := δ 0 : C 0 → Z 1 . Since the cohomology group H 1 (D, N D I X ) vanishes, the equality Z 1 = B 1 holds. Then the K -linear map η between K -Banach spaces is surjective and con- tinuous.  Thus, Banach's theorem (Theorem 1 in [2, I, §3]) shows that the K -linear map η is open. Therefore, we obtain a desired positive real number β. □ We take β in the above claim. Put t μ := { t iμ } iει ∈ C 0 and g μ := { g i jμ } i.jει ∈ C 1 . If μ is greater than one, then we may assume that | t μ | S ρ C 0 < β | G μ | S ρ C 1 . By Banach's theorem (Theorem 1 in [2, I, §3]), we may take a special real number γ such that the inequality | ι ( t )| ι,  < γ | t | B, holds for all i ∈ I and all t ∈ B i . By R we denote a real number field or an affinoid algebra with a norm. For two formal power series F = ∑ ∞=0 a m T m ∈ R [[T]] and G = ∑ ∞=0 b m T m ∈ R[[ f ]], we write F ≪ G if | a m | ≤ | b m | for all m ≥ 0. We set a formal power series over R
CRITERION FOR MINIMALITY OF RIGID ANALYTIC SURFACES where b and c is a positive real number. Then the relation
holds.  We choose four positive real numbers a , b , c , and d satisfying the following inequalities: (1) max | s i 1 l a ; < b/5; (2) max | h i j | A i j < d ; (3) α(1 + (ab/c) + d) < a; (4) βγ(1 + (ab/c) + d) < c. Claim.  For all i, j ∈ I and all μ ≥ 0, the relations (i, j, μ) fii – hij ≪ aP(T and (i,μ) hold.
Proof.  Clearly, the relations ( i, j, 0) and ( i, 1) hold. Suppose that the integer μ is positive.  Assume that for all i, j ∈ I , the relations ( i, j, μ − 1) and ( i, μ ) hold. We show that for all i, j ∈ I , the relations ( i, j, μ ) and ( i, μ + 1) hold. The equation ( μ ) implies that the congruence fijµTfhj ≡µ sfi – fij-l sfi holds. The right side of the above congruence is equal to s" – h i – (f"-1 – h ij)(s"j – h j) – h ij(s"j – h j) + h i – (f"j" – h ij)h j. Since the last two terms do not contribute to the term containing T r for each ν ≥ μ, the relation
holds. Thus, the relation fijµTµhj ≪ (1 + (ab/c) + d)P(T) holds. By the choice of α , the relation fijµTµ ≪ α(1 + (ab/c) + d)P(T) holds.  Therefore, the relation ( i, j, μ ) holds.  By the definition of g ijμ+1 , the congruence gijµ+1 Tµ+1 =µ+1 Sµ=µ holds. The right side of the above congruence is equal to s" = hi = (fij = hij)(s" = hj) = hij(s" j = hj) + hi = (fij = hij)hj. Since the last two terms do not contribute to the term containing T v for each ν ≥ μ, the relation
holds. Thus, the relation gijµ+1Tµ+1  ≪ (1 + (ab/c) + d)P(T) holds.  Since the inequality | siμ+1 | A , < Y | tiμ+1 | B , holds, by the choice of β , the relation
holds. Therefore, the relation ( i, μ + 1) holds.
Since P(T) converges on a neighborhood of 0, the formal solutions converge for some special real number δ, i.e., s ∈ C ( V δ, O γ δ ) and f ∈ C 1 (V δ, O γ δ). This proves Theorem 3.1. We say that a Cartier divisor D admits a global defining function if there exist an admissible open subset P of X that contains D and a defining func- tion of D on P. Theorem 3.3. Let D be an effective Cartier divisor on a quasi-compact separated rigid analytic space X. Assume that the following conditions are satisfied. (i) The cohomology group H 1 (D, N D ( x ) vanishes. (ii) The normal bundle NDIX admits a nowhere vanishing section. Then the divisor D admits a global defining function. Proof. There exist δ, s , and f satisfying the conditions in Theorem 3.1. By the condition (2)–(3) in the same theorem, we obtain the divisor E δ on Y δ . By lemma 2.6, for all sufficiently small special real number η , the section s ii is nowhere vanishing on Sp A ′ for all i ∈ I where A ′ ; = A i 〈 h i /η〉 . We take a special real number δ such that the inequality
holds for all i ∈ I . Take a special real number μ that is less than one. For each i ∈ I, put U ′ := Sp A ′ 〈 s i0 / s i1δμ 〉 . Since the condition (1) in Theorem 3.1 is satisfied, the restriction of the projection π : E δ → X to the preimage π− 1 (U ′ ) is an isomorphism.  Put P := U j ∈ I U ′ .  Then, by Lemma 2.1, the union P is an admissible open subset of X . Thus, the composite P → B( δ ) of the restriction of the inverse of the morphism π to π − 1 ( P ) and the second projection Y δ → B(δ) is a global defining function of D .口14 The supremum semi-norm on an affinoid algebra A is the K-algebra semi- norm | · |Sp A : A → R ≥ 0 defined by the equality:
Corollary 2 in [1, 3.8.2] shows the following lemma. Lemma 3.4. For an affinoid algebra A, the inequality | f | s ρ A  ≤  | f | A holds for all f ∈ A. Let D be an effective Cartier divisor on a separated rigid analytic space X that admits a relatively big covering. By Lemma 2.8 and Proposition 2.9, we obtain the admissible covering of U ∪ { U 0 } of X and the special real number ε such that the family U is an admissible covering in Theorem 3.1. Put J := I ∪ {0}, Vos := Uo × Sp K〈T/δ〉 and V := {V := } := : : : Theorem 3.5. We use the above notations.  Assume that the cohomology group H 1 (D, N DIX ) vanishes. Then we may take a special real number δ in Theorem 3.1 satisfying the following additional condition. (4) For all i ∈ I, the analytic function s i dose not vanish on V is ∩ V os. Proof. There exist δ , s , and f satisfying the conditions in Theorem 3.1. By the above lemma, we may take a special real number δ such that the inequality
holds for all i ∈ I . This proves the theorem. ● Theorem 3.6. Let D be an effective Cartier divisor on a separated rigid analytic space X that admits a relatively big covering.  Assume that the following conditions are satisfied. (i) The cohomology group H 1 (D, N D ( X ) vanishes. (ii) The normal bundle NDJx admits a nowhere vanishing section. Then the divisor D admits a global defining function. Further, if the rigid analytic space X and the divisor D are smooth, then there exists a tubular neighborhood P of D. Proof.  The first assertion follows from the above two theorems. We show the second assertion. By Lemma 2.12, we may assume that the family U is a local tubular neighborhood covering.  We may write D i = Sp B i and U i = Sp A i where A i = B i 〈 X i 〉 . Further, we may assume that each h i is the corresponding element to X i in A j . We define a real valued function | · | C(x) on a free affinoid algebra C(X) over an affinoid algebra C by the following equality:
Then the function | · | C ( X ) is a complete K -algebra norm on C ( X ) . For each i ∈ I , we set such a complete K -algebra norm on free affinoid algebras A i and A i 〈T〉 over B i and B i〈T〉 respectively. Let { t i } j ∈ I be a nowhere vanishing section of the normal bundle N DIX of D . Then there exist δ , s , and f satisfying the conditions in Theorem 3.1 and Theorem 3.5. We may assume that the inequality
holds for all i ∈ I . Then the K -algebra homomorphism φ i : A i 〈T 〉 〈 s i 〉 ⇒ B i 〈 T 〉 〉 is an isomorphism for all i ∈ I . Further, we may take special real numbers δ and η in the following way such that the K-algebra homomorphism ψ i : B i 〈X i 〉η〉 ⇒ B i 〈X i 〉η, T / δ 〉 〈 S i 〉 is an isomorphism for all i ∈ I . By Lemma 2.6, for all sufficiently small η , the section s i1 is nowhere vanishing on D i ( η ) for all i .  Since the condition (1) in Theorem 3.1 is satisfied, we have only to take special real numbers δ and η such that the inequalities
and
hold for all i ∈ I. By the conditions (2)–(4) in Theorem 3.1 and Theorem 3.5, the cochain s defines a divisor E on Y δ . Let Ψ be the projection E → X . For each i ∈ I , put E i  := Ψ− 1 (U in ).  Since the restriction Ψ | E i → U in corresponds to the isomorphism ψ i , the restriction is an isomorphism. The isomorphism φ i gives an open immersion Φ i : D i × B(δ) → E. By Lemma 2.6, there exists a special real number δ ′ such that D i × B(δ ′ ) is contained in Φ − 1 ( E i ) for all i ∈ I . For each i ∈ I , put W i := Ψ ◦ Φ i ( D i × B(δ ′ )). Then the family { W i } iel is a local tubular neighborhood covering of D . Let P be the union U j ∈ J W i . Gluing the composites of the isomorphisms (Ψ ◦ Φ i )-'| W i → D i × B(δ') and the second projections D i × B(δ') → B(δ') (Proposition 1 in [1, 9.3.3]), we obtain the analytic function F : P → B(δ ′ ). Therefore, the admissible open subset P is a tubular neighborhood of D .    □ 4.  BLOWING-DOWNS OF EXCEPTIONAL CURVES OF THE FIRST KIND In this section, we show that we can blow down exceptional curves of the first kind on regular surfaces. An exceptional curve of the first kind is a prime divisor D on a regular surface X satisfying the following conditions. By C ( D ), we denote the field of constant functions H 0 ( D , O D ) on D , which is a finite extension of K . (1) The divisor D is isomorphic to a projective line over C( D ). (2) The degree of the normal bundle of D is equal to − dim κ C ( D ). When X is proper, by Proposition 5.3, the last condition is equivalent to that the self-intersection number of the curve is − dim κ C(D) (see Appendix). A local calculation shows that the center of the blowing-up is an exceptional curve of the first kind. In this section, we show the converse. Theorem 4.1 (Castelnuove's criterion). Let D be an exceptional curve of the first kind on a regular surface X that admits a relatively big covering. Then we can blow down X along D to a regular surface. Hopf's theorem (Theorem 2.2 in [8]) and the above theorem give a nec- essary and sufficient condition for relative minimality. Corollary 4.2 (criterion for relative minimality).  A proper regular surface is relatively minimal if and only if the surface does not contain any excep- tional curves of the first kind. Proof of Castelnuovo's criterion.  To prove the above theorem, we have only to show that the divisor D contracts to a regular point. We choose two dis- tinct C(D)-rational points 0 and ∞ on D .  We take an admissible affinoid open subset U and two analytic functions f and g on U satisfying the fol- lowing conditions. (1) The subset U contains 0. (2) The analytic function f defines the divisor D on U . (3) The two analytic functions f and g generate the maximal ideal of the local ring Oxo Since the rigid space X is regular and the divisor D is quasi-compact, we have a finite covering of D on whose element the restriction of the divisor D is defined by a single analytic function. Since the rigid space X is separated, by Lemma 2.1, the union Y of the covering elements is an admissible open subset of X .  Applying Lemma 2.6 to the intersection W of the covering element U and another covering element V , and the restriction of defining functions of 0 on V to W , we may assume that the single element U of the covering contains 0. By the same method, we may assume that the divisor (g) dose not intersect the other covering elements.  Adding two divisors D and (g) on Y , we obtain the divisor E on Y . Since the normal bundle of the divisor E is trivial, by Theorem 3.3, we obtain a global defining function φ of E on a quasi-compact admissible open subset of X . Repeating the same procedure for the point ∞ , we obtain two analytic functions φ and ψ on a quasi-compact admissible open subset W of X satisfying the following conditions. (1) A divisor (φ) − D intersects D at the single point 0. (2)  A divisor (ψ) − D intersects D at the single point ∞. Then the two analytic functions φ and ψ define the divisor D.  We may define a morphism π : W → B(r) × B(r) for a sufficiently large special real number r .  By Proposition 2.10 and Lemma 2.4 in [8], for a sufficiently small special real number ε , the restriction of π to B(ε) × B(ε) is projective. Chow's theorem (Theorem 5.3) shows that the preimage of B(ε)×B(ε) under π is the analytification of a regular Noethering scheme of pure dimension two.  Therefore, the theorem follows from Castelnuovo's criterion for such schemes [9, 6, p102]. □ In this section, we assume that X is a proper normal surface. We give some methods for calculating intersection numbers of divisors and line bun- dles.  Then we prove the Riemann-Roch theorem for proper smooth sur- faces. For two line bundles L 1 and L 2 on X , we define the intersection number of L 1 and L 2 by the integer: x(Ox) - x(L) - x(L) + x(L)  $ L). Let us denote this integer by L 1 · L 2 . For divisors D 1 and D 2 on X , de- fine D 1 · D 2 as O x (D 1 ) · O x (D 2 ). The following proposition is immediate consequence of the definition. Proposition 5.1. For any two line bundles L 1 and L 2 , the following equal- ities hold: (1)  L 1 · O x = O x · L 1 = 0;
(2)  L 1 · L 2 = L 2 · L 1 · Lemma 5.2. For any effective divisor D on X and any line bundle L on X, the following equality holds: O x (D) · L = x (O D ) − x (O D ® L Y ). CRITERION FOR MINIMALITY OF RIGID ANALYTIC SURFACES Proof.  Since we have the sheaf exact sequence 0 → O x (−D) → O x → O D → 0, tensoring L v , we obtain the sheaf exact sequence: 0 → O x (−D) & L V → L V → O D & L V → 0. These sequences give the equalities X(O D ) = X(O X ) − X(O X(−D)) and X(O D & L Y ) = X(L Y ) − X(O X(−D) & L Y ). Subtracting the second equality from the first equality, we obtain the desired equality. The fundamental calculation methods are given by the following propo- sition. Proposition 5.3. Let π : D" → D be a normalization of a prime divisor D on X.  Then, for any two line bundles L 1 and L 2 on X, the following equalities hold: (1) O x (D) · L = deg Dn π* L; (2) O x (D) · L 1 S L 2 = O x (D) · L 1 + O x (D) · L 2 . Proof. The first equality follows from the previous lemma and Theorem 5.13 in [8]. Therefore, the second equality holds since the equality deg D n π * ( L 1 & L 2 ) = deg D n π * L 1 + deg D n π * L 2 holds. Lemma 5.4. For any effective divisor ∑ iel aiDi on X and any line bundle L on X, the following equality holds: Proof. By induction on ∑ j ∈ l α i , it suffices to show that for any effective di- visor D 1 , any prime divisor D 2 , and any line bundle L on X , the following equality holds: O x (D 1 + D 2 ) · L = O x (D 1 ) · L + O x (D 2 ) · L Since we have the isomorphism O D 2 & O x (−D 1 ) ≡ O x (−D 1 )/O x (−D 1 − D 2 ), we obtain the sheaf exact sequence: 0 → O D 2   O X (−D 1 ) → O D 1 +D 2  → O D 1  → O . Tensoring L V , we obtain the exact sequence: 0 → O D 2  $ O X (−D 1 ) $ L V  → O D 1 +D 2  $ L V  → O D 1  $ L V  → O . These sequences give the equalities X(OD1+D2) = X(OD1) + X(OD2 ® OX(-D1)) and X(OD1+D2 & LV) = X(OD1 & LV) + X(OD2 & OX(-D1) & LV). Subtracting the second equality from the first equality, we obtain the desired equality by Lemma 5.2 and Proposition 5.3. By a direct calculation of intersection numbers, we obtain the following lemma. Lemma 5.5.  For any three line bundles L 1 , L 2 , and L 3 on X, the following equations are equivalent: (1)  L 1 · L 2 ® L 3 = L 1 · L 2 + L 1 · L 3 ;
(2) χ(O x ) − χ(L 3 ) − χ(L 3 ) + χ(L i ® L 2 ) + χ(L 2 ® L 3 ) + χ(L 3 ®
L 3 ) − χ(L i ® L 3 ) = 0. Lemma 5.6.  Let L 1 , L 2 and L 3 be line bundles on X.  Assume that one of these line bundles is isomorphic to O x (D) where D is an effective divisor. Then the following equality holds: L 1 · L 2 8 L 3 = L 1 · L 2 + L 1 · L 3 · Proof.  By Lemma 5.5, it suffices to show that the following equality holds: O x (D) · L 1 & L 2 = O x (D) · L 1 + O x (D) · L 2 · This equality follows from Lemma 5.4 and Proposition 5.3. ● Proposition 5.7. Let L 1 , L 2 and L 3 be three line bundles on X. Assume that one of these line bundles is isomorphic to O x (D) where D is a divisor. Then the following equality holds: L1 · L2 8 L3 = L1 · L2 + L1 · L3 Proof.  By Lemma 5.5, it suffices to show that the equality Ox(D) · L 1 & L 2 = Ox(D) · L 1 + Ox(D) · L 2 - holds. We may write D = D 1 − D 2 where D 1 and D 2 are effective divisors. Lemma 5.6 shows that the equalities O x (D) · L = O x (D 1 ) · L + O x (−D 2 ) · L and 0 = O x (D 2 ) · L + O x (−D 2 ) · L 20 CRITERION FOR MINIMALITY OF RIGID ANALYTIC SURFACES hold. Subtracting the second equality from the first equation, we obtain the equality:
Thus, the desired equality follows from Lemma 5.6.
Corollary 5.8. Let D be a divisor ∑ jє ļ α jD i on X. Let π i : D i  → D i be a normalization of a prime divisor D i . Then, for any line bundle L on X, the following equality holds:
Corollary 5.9. For any two divisors ∑ jε J ajDj and ∑ jDj, the following equality holds: We show that local calculations yield the intersection number of two dis- tinct prime divisors. The local intersection number of two distinct prime di- visors D 1 and D 2 at a point p on D 1 ∩ D 2 is the integer dim κ ( O X / O X ( − D 1 ) + O x (−D 2 )) ρ . Let us denote this number by I ( p, D 1 , D 2 ). Since the intersec- tion D 1 ∩ D 2 is a finite number of points, the sum ∑ PED10D 2 I(p,D 1 ,D 2 ) is finite. Proposition 5.10.  For any two distinct prime divisors D 1 and D 2 on X, the following equality holds:
By definition, the equality
holds. Since we have the exact sequence we obtain the equality: Since the right side of the above equality is equal to D 1 · D 2 , we obtain the desired equality. ■ 21 The line bundle Ω 2 ⁄ of regular two-forms on a smooth surface Y is called the canonical line bundle on Y . Let us denote this line bundle by K Y . We prove the Riemann-Roch theorem for proper smooth surfaces. Theorem 5.11 (Riemann-Roch theorem for proper smooth surfaces). As- sume that X is smooth. For any line bundle L on X, the following equality holds:
Proof.  The Serre duality theorem (Theorem in [11, 5.1]) gives the equality: L ·K ¥ = x(L) - x(L ¥ ). Adding the above equality to the equality
we obtain the desired equality.
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