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Documenta Mathematica

Gegr¨ undet 1996 durch die Deutsche Mathematiker-Vereinigung

Extra Volume

A Collection of Manuscripts Written in Honour of

Kazuya Kato

on the Occasion of His Fiftieth Birthday

Editors:

S. Bloch, I. Fesenko, L. Illusie, M. Kurihara,

S. Saito, T. Saito, P. Schneider

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ver¨offentlicht Forschungsarbeiten aus allen mathe- matischen Gebieten und wird in traditioneller Weise referiert.

Artikel k¨onnen als TEX-Dateien per E-Mail bei einem der Herausgeber eingereicht werden. Hinweise f¨ur die Vorbereitung der Artikel k¨onnen unter der unten angegebe- nen WWW-Adresse gefunden werden.

Documenta Mathematicapublishes research manuscripts out of all mathema- tical fields and is refereed in the traditional manner.

Manuscripts should be submitted as TEX -files by e-mail to one of the editors. Hints for manuscript preparation can be found under the following WWW-address.

http://www.mathematik.uni-bielefeld.de/documenta Gesch¨aftsf¨uhrende Herausgeber / Managing Editors:

Alfred K. Louis, Saarbr¨ucken louis@num.uni-sb.de

Ulf Rehmann (techn.), Bielefeld rehmann@mathematik.uni-bielefeld.de Peter Schneider, M¨unster pschnei@math.uni-muenster.de Herausgeber / Editors:

Don Blasius, Los Angeles blasius@math.ucla.edu Joachim Cuntz, Heidelberg cuntz@math.uni-muenster.de Patrick Delorme, Marseille delorme@iml.univ-mrs.fr Bernold Fiedler, Berlin (FU) fiedler@math.fu-berlin.de Edward Frenkel, Berkely frenkel@math.berkeley.edu Kazuhiro Fujiwara, Nagoya fujiwara@math.nagoya-u.ac.jp Friedrich G¨otze, Bielefeld goetze@mathematik.uni-bielefeld.de Wolfgang Hackbusch, Leipzig (MPI) wh@mis.mpg.de

Ursula Hamenst¨adt, Bonn ursula@math.uni-bonn.de Lars Hesselholt, Cambridge, MA (MIT) larsh@math.mit.edu Max Karoubi, Paris karoubi@math.jussieu.fr Rainer Kreß, G¨ottingen kress@math.uni-goettingen.de Stephen Lichtenbaum, Providence Stephen Lichtenbaum@brown.edu Eckhard Meinrenken, Toronto mein@math.toronto.edu

Alexander S. Merkurjev, Los Angeles merkurev@math.ucla.edu Anil Nerode, Ithaca anil@math.cornell.edu

Thomas Peternell, Bayreuth Thomas.Peternell@uni-bayreuth.de Takeshi Saito, Tokyo t-saito@ms.u-tokyo.ac.jp

Stefan Schwede, Bonn schwede@math.uni-bonn.de Heinz Siedentop, M¨unchen (LMU) hkh@mathematik.uni-muenchen.de Wolfgang Soergel, Freiburg soergel@mathematik.uni-freiburg.de G¨unter M. Ziegler, Berlin (TU) ziegler@math.tu-berlin.de

ISSN 1431-0635 (Print), ISSN 1431-0643 (Internet) SPARC

Leading Edge

Documenta Mathematicais a Leading Edge Partner of SPARC, the Scholarly Publishing and Academic Resource Coalition of the As- sociation of Research Libraries (ARL), Washington DC, USA.

Address of Technical Managing Editor: Ulf Rehmann, Fakult¨at f¨ur Mathematik, Universit¨at Bielefeld, Postfach 100131, D-33501 Bielefeld, Copyright c° 2003 for Layout: Ulf Rehmann.

Typesetting in TEX.

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Documenta Mathematica

Extra Volume: Kazuya Kato’s Fiftieth Birthday, 2003

Preface 1

Prime Numbers 3

Ahmed Abbes and Takeshi Saito Ramification of Local Fields

with Imperfect Residue Fields II 5–72

Kenichi Bannai

Specialization of the p-adic Polylogarithm to

p-th Power Roots of Unity 73–97

Laurent Berger

Bloch and Kato’s Exponential Map:

Three Explicit Formulas 99–129

Spencer Bloch and H´el`ene Esnault

The Additive Dilogarithm 131–155

David Burns and Cornelius Greither Equivariant Weierstrass Preparation

and Values of L-functions at Negative Integers 157–185 John Coates, Peter Schneider, Ramdorai Sujatha

Links Between Cyclotomic

andGL2 Iwasawa Theory 187–215

Robert F. Coleman

Stable Maps of Curves 217–225

Christopher Deninger

Two-Variable Zeta Functions

and Regularized Products 227–259

Ivan Fesenko

Analysis on Arithmetic Schemes. I 261–284 Jean-Marc Fontaine

Presque Cp-Repr´esentations 285–385

Takako Fukaya

Coleman Power Series for K2

andp-Adic Zeta Functions

of Modular Forms 387–442

Y. Hachimori and O. Venjakob

Completely Faithful Selmer Groups

over Kummer Extensions 443–478

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Kato Homology of Arithmetic Schemes and Higher Class Field Theory

over Local Fields 479–538

Masato Kurihara

On the Structure of Ideal Class Groups

of CM-Fields 539–563

Nobushige Kurokawa, Hiroyuki Ochiai, and Masato Wakayama

Absolute Derivations and Zeta Functions 565–584 Barry Mazur and Karl Rubin

Studying the Growth of Mordell-Weil 585–607 Shinichi Mochizuki

The Absolute Anabelian Geometry

of Canonical Curves 609–640

Hyunsuk Moon and Yuichiro Taguchi

Refinement of Tate’s Discriminant Bound and Non-Existence Theorems for Mod p

Galois Representations 641–654

Arthur Ogus

On the Logarithmic

Riemann-Hilbert Correspondence 655–724

Bernadette Perrin-Riou

Groupes de Selmer et Accouplements;

Cas Particulier des

Courbes Elliptiques 725–760

Christophe Soul´e

A Bound for the Torsion in the K-Theory

of Algebraic Integers 761–788

Akio Tamagawa

Unramified Skolem Problems and

Unramified Arithmetic Bertini Theorems

in Positive Characteristic 789–831

Takeshi Tsuji

On the Maximal Unramified Quotients of p-Adic ´Etale Cohomology Groups

and Logarithmic Hodge–Witt Sheaves 833–890 Nobuo Tsuzuki

On Base Change Theorem

and Coherence in Rigid Cohomology 891–918

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Preface

The editors of this volume wish to express to Kazuya Kato their gratitude and admiration for his seminal contributions to arithmetic algebraic geometry and the deep influence he has been exerting in this area for over twenty years through his students and collaborators throughout the world.

S. Bloch, I. Fesenko, L. Illusie, M. Kurihara, S. Saito, T. Saito, P. Schneider

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Prime Numbers

Kazuya Kato

The song of prime numbers sounds Tonnkarari, We can hear if we keep our ears open,

We can hear their joyful song.

The song of prime numbers sounds Chinnkarari, Prime numbers sing together in harmony The song of love in the land of prime numbers.

The song of prime numbers sounds Ponnporori, Prime numbers are seeing dreams,

They sing the dreams for the tomorrow.

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Sosuu no uta wa tonnkarari Mimi o sumaseba kikoe masu Tanoshii uta ga kikoe masu Sosuu no uta wa chinnkarari Koe o awasete utai masu Sosuu no kuni no ai no uta Sosuu no uta wa ponnporori Sosuu wa yume o mite imasu Ashita no yume o utai masu

Alphabetic transcription of Kato’s poem

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Ramification of Local Fields with Imperfect Residue Fields II

Dedicated to Kazuya Kato on the occasion of his 50th birthday

Ahmed Abbes and Takeshi Saito

Received: October 10, 2002 Revised: August 6, 2003

Abstract. In [1], a filtration by ramification groups and its loga- rithmic version are defined on the absolute Galois group of a complete discrete valuation field without assuming that the residue field is per- fect. In this paper, we study the graded pieces of these filtrations and show that they are abelian except possibly in the absolutely unrami- fied and non-logarithmic case.

2000 Mathematics Subject Classification: primary 11S15, secondary 14G22

Keywords and Phrases: local fields, wild ramification, log structure, affinoid variety.

In the previous paper [1], a filtration by ramification groups and its logarithmic version are defined on the absolute Galois group GK of a complete discrete valuation field K without assuming that the residue field is perfect. In this paper, we study the graded pieces of these filtrations and show that they are abelian except possibly in the absolutely unramified and non-logarithmic case.

Let GjK (j > 0,∈ Q) denote the decreasing filtration by ramification groups and GjK,log (j >0,∈Q) be its logarithmic variant. We put Gj+K =S

j0>jGjK0 and Gj+K,log = S

j0>jGjK,log0 . In [1], we show that the wild inertia subgroup P ⊂GK is equal toG1+K =G0+K,log. The main result is the following.

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Theorem 1 Let K be a complete discrete valuation field.

1. (see Theorem 2.15) Assume either K has equal characteristicsp >0 or K has mixed characteristic and p is not a prime element. Then, for a rational numberj >1, the graded pieceGrjGK=GjK/Gj+K is abelian and is a subgroup of the center of the pro-p-groupG1+K /Gj+K.

2. (see Theorem 5.12)For a rational numberj >0, the graded pieceGrlogj GK = GjK,log/Gj+K,log is abelian and is a subgroup of the center of the pro-p-group G0+K,log/Gj+K,log.

The idea of the proof of 1 is the following. Under some finiteness assumption, denoted by (F), we define a functor ¯Xj from the category of finite ´etale K- algebras with ramification bounded byj+ to the category of finite ´etale schemes over a certain tangent space Θj with continuous semi-linear action ofGK. For a finite Galois extension Lof K with ramification bounded by j+, the image X¯j(L) has two mutually commuting actions of G= Gal(L/K) and GK. The arithmetic action of GK comes from the definition of the functor ¯Xj and the geometric action ofG is defined by functoriality. Using these two commuting actions, we prove the assertion. The assumption thatpis not a prime element is necessary in the construction of the functor ¯Xj.

In Section 1, for a rational numberj >0 and a smooth embedding of a finite flatOK-algebra, we define itsj-th tubular neighborhood as an affinoid variety.

We also define itsj-th twisted reduced normal cone.

We recall the definition of the filtration by ramification groups in Section 2.1 using the notions introduced in Section 1. In the equal characteristic case, under the assumption (F), we define a functor ¯Xj mentioned above in Section 2.2 usingj-th tubular neighborhoods. In the mixed characteristic case, we give a similar but subtler construction using the twisted normal cones, assuming further that the residue characteristicpis not a prime element ofKin Section 2.3. Then, we prove Theorem 2.15 in Section 2.4. We also define a canonical surjectionπ1abj)→GrjGK under the assumption (F).

After some preparations on generalities of log structures in Section 3, we study a logarithmic analogue in Sections 4 and 5. We define a canonical surjection πab1jlog) → GrjlogGK under the assumption (F) and prove the logarithmic part, Theorem 5.12, of the main result in Section 5.2. Among other results, we compare the construction with the logarithmic construction given in [1]

in Lemma 4.10. We also prove in Corollary 4.12 a logarithmic version of [1]

Theorem 7.2 (see also Corollary 1.16).

In Section 6, assuming the residue field is perfect, we show that the surjection πab1jlog)→GrjlogGK induces an isomorphismπab,gp1jlog)→GrjlogGK where πab,gp1jlog) denotes the quotient classifying the ´etale isogenies to Θjlogregarded as an algebraic group.

When one of the authors (T.S.) started studing mathematics, Kazuya Kato, who was his adviser, suggested to read [13] and to study how to generalize it when the residue field is no longer assumed perfect. This paper is a partial

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answer to his suggestion. The authors are very happy to dedicate this paper to him for his 51st anniversary.

Notation. LetKbe a complete discrete valuation field, OK be its valuation ring and F be its residue field of characteristic p >0. Let ¯K be a separable closure of K, OK¯ be the integral closure of OK in ¯K, ¯F be the residue field of OK¯, and GK = Gal( ¯K/K) be the Galois group of ¯K over K. Let π be a uniformizer ofOK and ord be the valuation ofKnormalized by ordπ= 1. We denote also by ord the unique extension of ord to ¯K.

1 Tubular neighborhoods for finite flat algebras

For a semi-local ringR, letmR denote the radical of R. We say that an OK- algebra R is formally of finite type over OK if R is semi-local, mR-adically complete, Noetherian and the quotientR/mRis finite overF. AnOK-algebra Rformally of finite type overOK is formally smooth overOK if and only if its factors are formally smooth. We say that anOK-algebraR is topologically of finite type over OK if R is π-adically complete, Noetherian and the quotient R/πRis of finite type overF. For anOK-algebraRformally of finite type over OK, we put ˆΩR/OK = lim

←−n(R/mnR)/OK. For an OK-algebra R topologically of finite type over OK, we put ˆΩR/OK = lim

←−n(R/πnR)/OK. Here and in the following, Ω denotes the module of differential 1-forms. For a surjection R → R0 of rings, its formal completion is defined to be the projective limit R= lim

←−nR/(Ker(R→R0))n.

In this section,Awill denote a finite flatOK-algebra.

1.1 Embeddings of finite flat algebras

Definition 1.1 1. LetA be a finite flatOK-algebra andAbe an OK-algebra formally of finite type and formally smooth overOK. We say that a surjection A→AofOK-algebras is an embedding if it induces an isomorphismA/mA → A/mA.

2. We define EmbOK to be the category whose objects and morphisms are as follows. An object ofEmbOK is a triple (A→A)where:

• Ais a finite flatOK-algebra.

• Ais anOK-algebra formally of finite type and formally smooth overOK.

• A→Ais an embedding.

A morphism (f,f) : (A → A) → (B → B) of EmbOK is a pair of OK- homomorphismsf :A→B andf :A→B such that the diagram

A −−−−→ A

f



y yf B −−−−→ B

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is commutative.

3. For a finite flatOK-algebraA, letEmbOK(A)be the subcategory ofEmbOK

whose objects are of the form(A→A)and morphisms are of the form(idA,f).

4. We say that a morphism (f,f) : (A → A) → (B → B) of EmbOK is finite flat if f :A → B is finite and flat and if the map B⊗AA →B is an isomorphism.

If (A→A) is an embedding, theA-module ˆΩA/OKis locally free of finite rank.

Lemma 1.2 1. For a finite flatOK-algebraA, the categoryEmbOK(A)is non- empty.

2. For a morphism f : A →B of finite flat OK-algebras and for embeddings (A→A) and(B→B), there exists a morphism(f,f) : (A→A)→(B→B) liftingf.

3. For a morphism f :A→B of finite flat OK-algebras, the following condi- tions are equivalent.

(1) The map f :A→B is flat and locally of complete intersection.

(2) Their exists a finite flat morphism (f,f) : (A → A) → (B → B) of embeddings.

Proof. 1. Take a finite system of generators t1, . . . , tn of A over OK and define a surjection OK[T1, . . . , Tn] → A by Ti 7→ ti. Then the formal completion A → A of OK[T1, . . . , Tn] → A, where A = lim←−mOK[T1, . . . , Tn]/(Ker(OK[T1, . . . , Tn]→A))m, is an embedding.

2. SinceAis formally smooth overOKandB= lim

←−nB/InwhereI= Ker(B→ B), the assertion follows.

3. (1)⇒(2). We may assume A and B are local. By 1 and 2, there exists a morphism (f,f) : (A →A)→(B → B) lifting f. Replacing B →B by the projective limit lim

←−n(A/mnAOKB/mnB)→B/mnB of the formal completion (A/mnAOKB/mnB)→B/mnBof the surjectionsA/mnAOKB/mnB→B/mnB, we may assume that the map A → B is formally smooth. Since A → B is locally of complete intersection, the kernel of the surjection B⊗AA →B is generated by a regular sequence (t1, . . . , tn). Take a lifting (˜t1, . . .˜tn) inBand define a map A[[T1, . . . , Tn]] → B by Ti 7→ ti. We consider an embedding A[[T1, . . . , Tn]] → A defined by the composition A[[T1, . . . , Tn]] → A → A sending Ti to 0. Replacing A by A[[T1, . . . , Tn]], we obtain a map (A → A) → (B → B) such that the map B⊗AA → B is an isomorphism and dimA= dimB. By Nakayama’s lemma, the mapA→Bis finite. Hence the mapA→Bis flat by EGA Chap 0IV Corollaire (17.3.5) (ii).

(2)⇒(1). SinceAandBare regular,Bis locally of complete intersection over A. Since B is flat over A, B is also flat and locally of complete intersection

overA. 2

The base change of an embedding by an extension of complete discrete valuation fields is defined as follows.

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Lemma 1.3 Let K0 be a complete discrete valuation field and K → K0 be a morphism of fields inducing a local homomorphism OK → OK0. Let (A → A) be an object of EmbOK. We define A⊗ˆOKOK0 to be the projective limit lim←−n(A/mnAOKOK0). Then theOK0-algebraA⊗ˆOKOK0 is formally of finite type and formally smooth over OK0. The natural surjection A⊗ˆOKOK0 → A⊗ˆOKOK0 defines an object(A⊗ˆOKOK0 →A⊗OKOK0)ofEmbOK0.

Proof. TheOK-algebraAis finite over the power series ringOK[[T1, . . . , Tn]] for some n≥0. Hence theOK0-algebra A⊗ˆOKOK0 is finite overOK0[[T1, . . . , Tn]]

and is formally of finite type over OK0. The formal smoothness is clear from

the definition. The rest is clear. 2.

For an object (A→A) ofEmbOK, we let the object (A⊗ˆOKOK0 →A⊗OKOK0) of EmbOK0 defined in Lemma 1.3 denoted by (A→ A) ˆ⊗OKOK0. By sending (A → A) to (A → A) ˆ⊗OKOK0, we obtain a functor ˆ⊗OKOK0 : EmbOK → EmbOK0. IfK0 is a finite extension ofK, we haveA⊗ˆOKOK0 =A⊗OKOK0. 1.2 Tubular neighborhoods for embbedings

Let (A → A) be an object of EmbOK and I be the kernel of the surjection A → A. Mimicing [3] Chapter 7, for a pair of positive integers m, n > 0, we define an OK-algebra Am/n topologically of finite type as follows. Let A[Inm] be the subring ofA⊗OKK generated byAand the elementsf /πm for f ∈ In and letAm/n be its π-adic completion. For two pairs of positive integersm, nandm0, n0, ifm0 is a multiple ofmand ifm0/n0 ≤m/n, we have an inclusionA[In0m0]⊂A[Inm]. It induces a continuous homomorphism Am0/n0 → Am/n. Then we have the following.

Lemma 1.4 Let (A → A) be an object of EmbOK and m, n > 0 be a pair of positive integers. Then,

1. The OK-algebra Am/n is topologically of finite type over OK. The tensor productAm/nK =Am/nOKK is an affinoid algebra overK.

2. The map A→ Am/n is continuous with respect to themA-adic topology on A and theπ-adic topology on Am/n.

3. Letm0, n0 be another pair of positive integers and assume that m0 is a mul- tiple of mandj0=m0/n0≤j=m/n. Then, by the mapXm/n= Sp Am/nK → Xm0/n0 = Sp AmK0/n0 induced by the inclusion A[In0m0] ⊂ A[Inm], the affinoid varietyXm/n is identified with a rational subdomain ofXm0/n0. 4. The affinoid varietyXm/n= SpAm/nK depends only on the ratioj=m/n.

The proof is similar to that of [3] Lemma 7.1.2.

Proof. 1. Since the OK-algebra Am/n is π-adically complete, it is sufficient to show that the quotient A[Inm]/(π) is of finite type over F. Since it is finitely generated over A/(π, In) and A/(π, I) =A/(π) is finite over F, the assertion follows.

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2. SinceA/π=A/(π, I) is of finite length, a power ofmAis in (πm, In). Since the image of (πm, In) inAm/n is inπmAm/n, the assertion follows.

3. Take a system of generators f1, . . . , fN of In and define a surjection A[In0m0][T1, . . . , TN]/(πmTi−fi)→A[Inm] by sendingTitofim. Since it induces an isomorphism after tensoring with K, its kernel is annihilated by a power of π. Hence it induces an isomorphismAmK0/n0hT1, . . . , TNi/(πmTi− fi, i= 1, . . . , N)→ Am/nK .

4. Further assume m/n = m0/n0 and put k = m0/m. Let f1, . . . , fN ∈ In be a system of generators of In as above. Then A[Inm] is generated by (f1m)k1· · ·(fNm)kN,0 ≤ ki < k as an A[In0m0]-module. Hence the cokernel of the inclusion Am0/n0 → Am/n is annihilated by a power of πand

the assertion follows. 2

If A = OK[[T1, . . . , TN]] and I = (T1, . . . , TN), the ring Am/1 is isomor- phic to the π-adic completion of OK[T1m, . . . , TNm] and is denoted by OKhT1m, . . . , TNmi. By Lemma 1.4.4, the integral closureAj ofAm/n in the affinoid algebraAm/nOKK depends only onj=m/n.

Definition 1.5 Let (A→A) be an object ofEmbOK andj >0 be a rational number. We define Aj to be the integral closure ofAm/n forj =m/nin the affinoid algebraAm/nOKKand define thej-th tubular neighborhoodXj(A→ A) to be the affinoid varietySpAjK.

In the case A = OK[[T1, . . . , Tn]] and the map A → A =OK is defined by sendingTi to 0, the affinoid varietyXj(A→A) is then-dimensional polydisk D(0, πj)nof center 0 and of radiusπj. For each positive rational numberj >0, the construction attaching thej-th tubular neighboorhoodXj(A→A) to an object (A→A) ofEmbOK defines a functor

Xj:EmbOK →(Affinoid/K)

to the category of affinoid varieties over K. For j0 ≤ j, we have a natural morphismXj→Xj0 of functors. A finite flat morphism of embeddings induces a finite flat morphism of affinoid varieties.

Lemma 1.6 Letj >0be a positive rational number and(A→A)→(B→B) be a finite and flat morphism inEmbOK. Then, the induced map fj:Xj(B→ B)→Xj(A→A) is a finite flat map of affinoid varieties.

Proof. LetI andJ =IBbe the kernels of the surjectionsA→AandB→B.

Since the map A → B is flat, it induces isomorphisms B⊗A A[Inm] → B[Jnm] and B⊗AAjK→ BKj . The assertion follows from this immediately.

2

For an extensionK0 of complete discrete valuation fieldK, the construction of j-th tubular neighborhoods commutes with the base change. More precisely, we have the following. LetK0 be a complete discrete valuation field andK→K0 be a morphism of fields inducing a local homomorphismOK →OK0. Then by

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sending an affinoid variety SpAK overK to the affinoid variety SpAK⊗ˆKK0 over K0, we obtain a functor ˆ⊗KK0 : (Affinoid/K)→ (Affinoid/K0) (see [2]

9.3.6). Letebe the ramification indexeK0/K andj >0 be a positive rational number. Then the canonical map A → A⊗ˆOKOK0 induces an isomorphism Xj(A→A) ˆ⊗KK0 →Xej((A→A) ˆ⊗OKOK0) of affinoid varieties overK0. In other words, we have a commutative diagram of functors

Xj:EmbOK −−−−→ (Affinoid/K)

ˆOKOK0



y yˆKK0 Xej:EmbOK −−−−→ (Affinoid/K0).

Lemma 1.7 For a rational number j > 0, the affinoid algebra AjK is smooth overK.

Proof. By the commutative diagram above, it is sufficient to show that there is a finite separable extension K0 of K such that the base change Xj(A → A)⊗KK0=Xj(A⊗OKOK0 →A⊗OKOK0) is smooth overK0. ReplacingK byK0 and separating the factors ofA, we may assume A/mA=F. Then we also haveA/mA=Fand an isomorphismOK[[T1, . . . , Tn]]→A. We define an object (A→OK) ofEmbOK by sendingTi∈Ato 0. LetIandI0be the kernel of A→A and A→OK respectively and put j =m/n. SinceA/(πm, In) is of finite length, there is an integer n0>0 such thatI0n0 ⊂(πm, In). Then we have an inclusionA[I0n0m]→A[Inm] and hence a mapXm/n(A→A)→ Xm/n0(A → OK). By the similar argument as in the proof of Lemma 1.4.3, the affinoid variety Xm/n(A→A) is identified with a rational subdomain of Xm/n0(A → OK). Since the affinoid variety Xm/n0(A→ OK) is a polydisk,

the assertion follows. 2

By Lemma 1.7, thej-th tubular neighborhoods in fact define a functor Xj:EmbOK −−−−→ (smoooth Affinoid/K)

to the category of smooth affinoid varieties over K. Also by Lemma 1.7, ΩˆAj/OK⊗K is a locally freeAjK-module.

An idea behind the definition of the j-th tubular neighborhood is the fol- lowing description of the valued points. Let (A → A) be an object of EmbOK and j > 0 be a rational number. Let AjK be the affinoid alge- bra defining the affinoid variety Xj(A → A) and let Xj(A → A)( ¯K) be the set of ¯K-valued points. Since a continuous homomorphism AjK → K¯ is determined by the induced map A → OK¯, we have a natural injection Xj(A→A)( ¯K)→Homcont.OK-alg(A, OK¯). The surjectionA→Ainduces an injection

(1.8.0) HomOK-alg(A, OK¯) −−−−→ Xj(A→A)( ¯K).

For a rational numberj >0, let mj denote the idealmj={x∈K; ordx¯ ≥j}. We naturally identify the setHomOK-alg(A, OK¯/mj) ofOK-algebra homomor-

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phisms with a subset of the setHomcont.OK-alg(A, OK¯/mj) of continuous OK- algebra homomorphisms.

Lemma 1.8 Let (A → A) be an object of EmbOK and j > 0 be a rational number. Then by the injection Xj(A → A)( ¯K) → Homcont.OK-alg(A, OK¯) above, the set Xj(A → A)( ¯K) is identified with the inverse image of the subset HomOK-alg(A, OK¯/mj) by the projection Homcont.OK-alg(A, OK¯) → Homcont.OK-alg(A, OK¯/mj). In other words, we have a cartesian diagram

(1.8.1)

Xj(A→A)( ¯K) −−−−→ Homcont.OK-alg(A, OK¯)



y y

HomOK-alg(A, OK¯/mj) −−−−→ Homcont.OK-alg(A, OK¯/mj).

The arrows are compatible with the naturalGK-action.

Proof. Let j = m/n. By the definition of Am/n, a continuous morphism A→OK¯ is extended toAjK →K, if and only if the image of¯ In is contained in the ideal (πm). Hence the assertion follows. 2 For an affinoid varietyX overK, letπ0(XK¯) denote the set lim

←−K0/Kπ0(XK0) of geometric connected components, whereK0 runs over finite extensions ofK in ¯K. The setπ0(XK¯) is finite and carries a natural continuous right action of the absolute Galois groupGK. To get a left action, we letσ∈GK act onXK¯

byσ−1. The natural map Xj(A→A)( ¯K)→π0(XK¯) is compatible with this left GK-action. LetGK-(Finite Sets) denote the category of finite sets with a continuous left action ofGK and let (Finite Flat/OK) be the category of finite flatOK-algebras. Then, for a rational numberj >0, we obtain a sequence of functors

(Finite Flat/OK) ←−−−− EmbOK

Xj

−−−−→

(smooth Affinoid/K) −−−−−−−−→X7→π0(XK¯) GK-(Finite Sets).

We show that the compositionEmbOK →GK-(Finite Sets) induces a functor (Finite Flat/OK)→GK-(Finite Sets).

Lemma 1.9 Letj >0 be a positive rational number.

1. Let (A → A) be an embedding. Then, the map Xj(A → A)( ¯K) → HomOK-alg(A, OK¯/mj) (1.8.1) induces a surjection

(1.9.1) HomOK-alg(A, OK¯/mj) −−−−→ π0(Xj(A→A)K¯).

2. Let (A → A) and (A0 → A) be embeddings. Then, there exists a unique bijectionπ0(Xj(A→A)K¯)→π0(Xj(A0 →A)K¯) such that the diagram

(1.9.2)

HomOK-alg(A, OK¯/mj) −−−−→ π0(Xj(A→A)K¯)

°°

° y

HomOK-alg(A, OK¯/mj) −−−−→ π0(Xj(A0→A)K¯)

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is commutative.

3. Let (f,f) : (A → A) → (B → B) be a morphism of EmbOK. Then, the induced map π0(Xj(B→B)K¯)→π0(Xj(A→A)K¯)depends only on f. 4. Let (f,f) : (A → OK) → (B → B) be a finite flat morphism of EmbOK. Then the map (1.8.0) induces a surjection

(1.9.3) HomOK-alg(B, OK¯) −−−−→ π0(Xj(B→B)K¯).

Proof. 1. The fibers of the map Homcont.OK-alg(A, OK¯) → Homcont.OK-alg(A, OK¯/mj) are ¯K-valued points of polydisks. Hence the surjectionXj(A→A)( ¯K)→Homcont.OK-alg(A, OK¯/mj) induces a surjection Homcont.OK-alg(A, OK¯/mj)→π0(Xj(A→A)K¯) by Lemma 1.8.

2. By 1 and Lemma 1.2.2, there exists a unique surjectionπ0(Xj(A→A)K¯)→ π0(Xj(A0 → A)K¯) such that the diagram (1.9.2) is commutative. Switching A→A andA0→A, we obtain the assertion.

3. In the commutative diagram

Homcont.OK-alg(B, OK¯/mj) −−−−→ π0(Xj(B→B)K¯)

f



y y

Homcont.OK-alg(A, OK¯/mj) −−−−→ π0(Xj(A→A)K¯), the horizontal arrows are surjective by 1. Hence the assertion follows.

4. The mapfj :Xj(B→B)→Xj(A→OK) is finite and flat by Lemma 1.6.

Let y : Xj(A →OK)( ¯K) be the point corresponding to the map A→ OK. Then the fiber (fj)1(y) is identified with the set HomOK-alg(B, OK¯). Since Xj(A→OK)K¯ is isomorphic to a disk and is connected, the assertion follows.

2

For a rational number j >0 and a finite flatOK-algebraA, we put Ψj(A) = lim

(AA)∈E←−mbOK(A)

π0(Xj(A→A)K¯).

By Lemmas 1.2.1 and 1.9.2, the projective system in the right is constant.

Further by Lemma 1.9.3, we obtain a functor

Ψj : (Finite Flat/OK) −−−−→ GK-(Finite Sets)

sending a finite flat OK-algebra A to Ψj(A). Let Ψ : (Finite Flat/OK) → GK-(Finite Sets) be the functor defined by Ψ(A) =HomOK-alg(A,K). Then,¯ the map (1.9.1) induces a map Ψ→Ψj of functors.

1.3 Stable normalized integral models and their closed fibers We briefly recall the stable normalized integral model of an affinoid variety and its closed fiber (cf. [1] Section 4). It is based on the finiteness theorem of Grauert-Remmert.

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Theorem 1.10 (Finiteness theorem of Grauert-Remmert, [1] Theorem 4.2) Let A be an OK-algebra topologically of finite type. Assume that the generic fiberAK =A ⊗OKK is geometrically reduced. Then,

1. There exists a finite separable extension K0 of K such that the geometric closed fiber AOK0OK0 F¯ of the integral closure AOK0 of A in A ⊗OKK0 is reduced.

2. Assume further that A is flat over OK and that the geometric closed fiber A⊗OKF¯ is reduced. LetK0be an extension of complete discrete valuation field over K and π0 be a prime element ofK0. Then theπ0-adic completion of the base change A ⊗OKOK0 is integrally closed inA ⊗OKK0.

LetAbe anOK-algebra topologically of finite type such thatAK is smooth. If a finite separable extensionK0satisfies the condition in Theorem 1.10.1, we say that the integral closureAOK0 ofAinAK0 is a stable normalized integral model of the affinoid variety XK = Sp AK and that the stable normalized integral model is defined overK0. The geometric closed fiber ¯X = SpecAOK0OK0F¯of a stable normalized integral model is independent of the choice of an extension K0over which a stable normalized integral model is defined, by Theorem 1.10.2.

Hence, the scheme ¯X carries a natural continuous action of the absolute Galois group GK = Gal( ¯K/K) compatible with its action on ¯F.

The construction above defines a functor as follows. Let GK-(Aff/F¯) denote the category of affine schemes of finite type over ¯Fwith a semi-linear continuous action of the absolute Galois groupGK. More precisely, an object is an affine scheme Y over ¯F with an action ofGK compatible with the action ofGK on F¯ satisfying the following property: There exist a finite Galois extensionK0 ofK in ¯K, an affine schemeYK0 of finite type over the residue fieldF0 ofK0, an action of Gal(K0/K) onYK0 compatible with the action of Gal(K0/K) on F0 and a GK-equivariant isomorphism YK0F0F¯ →Y. Then Theorem 1.10 implies that the geometric closed fiber of a stable normalized integral model defines a functor

(smooth Affinoid/K)→GK-(Aff/F¯) :X7→X.¯

Corollary 1.11 LetAbe anOK-algebra topologically of finite type such that the generic fiber AK is geometrically reduced as in Theorem 1.10. Let XK = Sp AK be the affinoid variety and XF¯ be the geometric closed fiber of the stable normalized integral model. Then the natural map π0(XF¯)→π0(XK¯) is a bijection.

Proof. Replacing A by its image in AK, we may assume A is flat over OK. LetK0 be a finite separable extension ofKin ¯K such that the stable normal- ized integral model AOK0 is defined over K0. Then since AOK0 is π-adically complete, the canonical maps π0(SpecAOK0) → π0(Spec(AOK0OK0 F0)) is bijective. Since the idempotents of AK0 are in AOK0, the canonical maps π0(SpecAOK0)→π0(SpecAK0) is also bijective. By taking the limit, we obtain

the assertion. 2

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By Corollary 1.11, the functor (smooth Affinoid/K)→GK-(Finite Sets) send- ing a smooth affinoid variety X toπ0(XK¯) may be also regarded as the com- position of the functors

(smooth Affinoid/K) −−−−→X7→X¯ GK-(Aff/F¯) −−−−→π0 GK-(Finite Sets). Lemma 1.12 Letj >0be a positive rational number and(f,f) : (A→OK)→ (B → B) be a finite flat morphism of EmbOK. Let fj : Xj(B → B) → Xj(A → OK) be the induced map and f¯j : ¯Xj(B → B) → X¯j(A → OK) be its reduction. Let y ∈ Xj(A → OK)( ¯K) be the point corresponding to A → A = OK → K¯ and y¯ ∈ X¯j(A → OK) be its specialization. Then the surjections (fj)1(y) =HomOK-alg(B, OK¯)→π0(Xj0(B→B)K¯) (1.9.3) and the specialization map(fj)1(y)→( ¯fj)1(¯y)induces a bijection

(1.12.1) lim

−→j0>jπ0(Xj0(B→B)K¯) −−−−→ ( ¯fj)1(¯y).

Proof. The map (fj)1(y)→ π0(Xj0(B →B)K¯) is a surjection of finite sets by Lemma 1.9.4. Hence there exists a rational number j0 > j such that the surjection π0(Xj0(B → B)K¯) → lim

−→j00>jπ0(Xj00(B → B)K¯) is a bijection.

Let K0 be a finite separable extension such that the surjection π0(Xj0(B → B)K¯) → π0(Xj0(B → B)K0) is a bijection and that the stable normalized integral modelsBjOK0 ofXj(B→B) is defined overK0. EnlargingK0further if necessary, we assume thate0jis an integer wheree0=eK0/Kis the ramification index. Then the integral modelAjOK0 ofXj(A→OK) is also defined overK0. If OK[[T1, . . . , Tn]] →A is an isomorphism such that the kernel of A→ OK

is generated by T1, . . . , Tn and π0 is a prime element of K0, it induces an isomorphism OK0hT10e0j, . . . , Tn0e0ji → AjOK0. Let AjOK0 → OK0 be the map induced by A → OK and AjO

K0 be the formal completion respect to the surjection AjOK0 → OK0. If OK[[T1, . . . , Tn]] → A is an isomorphism as above, it induces an isomorphism OK0[[T10e0j, . . . , Tn0e0j]] → AjO

K0. We putBjO

K0 =BjOK0AjO

K0

AjO

K0. The ringBjO

K0 is finite overAjO

K0 sinceBjOK0

is finite over AjOK0. Enlarging K0 further if necessary, we assume that the canonical map ( ¯fj)1(¯y)→π0(SpecBjO

K0) is a bijection.

We show that the surjection π0(Xj0(B →B)K0)→π0(Spec BjO

K0) is a bijec- tion. For a rational number j0 > 0, let AjK0 and BjK0 denote the affinoid K- algebras definingXj0(A→OK) and Xj0(B→B). We have BKj0 =B⊗AAjK0. Since π0(Xj0(B→B)K¯)→lim

−→j00>jπ0(Xj00(B→B)K¯) is a bijection, the in- jection BjK¯00 → BjK¯0 induce a bijection of idempotents forj < j00 < j0. Since π0(Xj0(B → B)K¯) → π0(Xj0(B → B)K0) is a bijection, the idempotents of BjK¯0 are in BjK00. Hence, for j < j00 < j0, the map BKj000 → BjK00 induces a bi- jection of idempotents for j < j00 < j0. Therefore, the map BjO

K0 → BjK00

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induces a bijection of idempotents by [3] 7.3.6 Proposition. Thus, the map π0(Xj0(B→B)K0)→π0(SpecBjO

K0) is a bijection as required. 2 For later use in the proof of the commutativity in the logarithmic case, we give a more formal description of the functor (smooth Affinoid/K)→GK-(Aff/F¯) : X 7→ X. For this purpose, we introduce a category lim¯

−→K0/K(Aff/F0) and an equivalence lim

−→K0/K(Aff/F0) → GK-(Aff/F) of categories. More gener-¯ ally, we define a category lim

−→K0/KV(K0) in the following setting. Suppose we are given a category V(K0) for each finite separable extension K0 of K and a functor f : V(K00) → V(K0) for each morphism f : K0 → K00 of finite separable extension of K satisfying (f ◦g) = g ◦f and idK0 = idV(K0). In the application here, we will take V(K0) to be (Aff/F0) for the residue field F0. In Section 4, we will take V(K0) to be EmbOK0. We say that a full subcategory C of the category (Ext/K) of finite separable extensions in K¯ is cofinal if C is non empty and a finite extension K00 of an extension K0 in C is also in C. We define lim

−→K0/KV(K0) to be the category whose ob- jects and morphisms are as follows. An object of lim

−→K0/KV(K0) is a sys- tem ((XK0)K0ob(C),(ϕf)f:K0K00mor(C)) where C is some cofinal full sub- category of (Ext/K), XK0 is an object of V(K0) for each object K0 in C and ϕf : XK00 → f(XK0) is an isomorphism in V(K00) for each morphism f :K0 →K00inCsatisfyingϕf◦f0=ff0)◦ϕf for morphismsf0 :K0→K00 andf :K00→K000 inC. For objectsX = ((XK0)K0ob(C),(ϕf)f:K0K00mor(C)) andY = ((YK0)K0ob(C0),(ψf)f:K0K00mor(C0)) of the category lim

−→K0/KV(K0), a morphism g :X → Y is a system (gK0)K0ob(C00), where C00 is some cofinal full subcategory of C ∩ C0 andgK0 :XK0 →YK0 is a morphism inV(K0) such that the diagram

XK00 gK0

−−−−→ YK00 ϕf



y yψf fXK0

gK00

−−−−→ fYK0

is commutative for each morphismf :K0→K00 inC00.

Applying the general construction above, we define a category lim−→K0/K(Aff/F0). An equivalence lim

−→K0/K(Aff/F0) → GK-(Aff/F¯) of categories is defined as follows. Let X = ((XK0)K0∈ob(C), (f)f:K0→K00∈mor(C)) be an object of lim

−→K0/K(Aff/F0). LetCK¯ be the category of finite extensions of K in ¯K which are in C. Then, XK¯ = lim

←−K0∈CK¯ XK0

is an affine scheme over ¯F and has a natural continuous semi-linear ac- tion of the Galois group GK. By sending X to XK¯, we obtain a functor lim−→K0/K(Aff/F0)→GK-(Aff/F¯). We can easily verify that this functor gives an equivalence of categories.

The reduced geometic closed fiber defines a functor (smooth Affinoid/K) → lim−→K0/K(Aff/F0) as follows. LetXbe a smooth affinoid variety overK. LetCX

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be the full subcategory of (Ext/K) consisting of finite extensionsK0such that a stable normalized integral modelAOK0 is defined overK0. By Theorem 1.10.1, the subcategory CX is cofinal. Further, by Theorem 1.10.2, the system ¯X = (Spec AOK0OK0 F0)K0obCX defines an object of lim

−→K0/K(Aff/F0). Thus, by sendingXto ¯X, we obtain a functor (smooth Affinoid/K)→lim

−→K0/K(Aff/F¯0).

By taking the composition with the equivalence of categories, we recover the functor (smooth Affinoid/K)→GK-(Aff/F).¯

1.4 Twisted normal cones

Let (A→A) be an object inEmbOK andj >0 be a positive rational number.

We define ¯Xj(A→A) to be the geometric closed fiber of the stable normalized integral model of Xj(A → A). We will also define a twisted normal cone C¯j(A → A) as a scheme over AF ,red¯ = (A⊗OK F¯)red and a canonical map X¯j(A→A)→C¯j(A→A).

Let I be the kernel of the surjection A → A. Then the normal cone CA/A

of Spec A in Spec A is defined to be the spectrum of the graded A-algebra L

n=0In/In+1. We say that a surjectionR→R0of Noetherian rings is regular if the immersion SpecR0 → SpecR is a regular immersion. If the surjection A → A is regular, the conormal sheaf NA/A = I/I2 is locally free and the normal coneCA/Ais equal to the normal bundle, namely the covariant vector bundle over SpecAdefined by the locally freeA-moduleHomA(NA/A, A).

For a rational numberj, letmjbe the fractional idealmj={x∈OK¯; ord(x)≥ j} and putNj=mjOK¯ F.¯

Definition 1.13 Let(A→A)be an object ofEmbOK andj >0be a rational number. We define the j-th twisted normal coneC¯j(A→A)to be the reduced

part Ã

Spec M n=0

(In/In+1OKNjn)

!

red

of the spectrum of the A⊗OKF¯-algebraL

n=0(In/In+1OKNjn).

It is a reduced affine scheme over Spec AF ,red¯ non-canonically isomorphic to the reduced part of the base changeCA/AOKF. It has a natural continuous¯ semi-linear action of GK via Njn. The restriction to the wild inertia sub- group P is trivial and the GK-action induces an action of the tame quotient GtameK =GK/P. If the surjection A→A is regular, the scheme ¯Cj(A→A) is the covariant vector bundle over Spec AF ,red¯ defined by theAF ,red¯ -module (HomA(I/I2, A)⊗OKNj)⊗A⊗OKF¯AF ,red¯ .

A canonical map ¯Xj(A→A)→C¯j(A→A) is defined as follows. LetK0be a finite separable extension ofK such that the stable normalized integral model AjOK0 is defined over K0 and that the product jewith the ramification index

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