Université Paris-Sud

Ecole Doctorale de Mathématiques(ED n 142)

Discipline : Mathématiques

Thèse de doctoratSoutenue le 24/09/2014 par

Haoyu HU

Ramification et cycles proches pourles faisceaux `-adiques sur unschéma au-dessus d’un trait

Directeur de thèse : M. Ahmed ABBES (IHES)Co-directeur de thèse : M. Lei FU (Université de Nankai)

Composition du jury :Directeur de thèse : M. Ahmed ABBES (IHES)Co-directeur de thèse : M. Lei FU (Université de Nankai)Rapporteur : M. Takeshi SAITO (Université de Tokyo)Examinateurs : M. Gérard LAUMON (Université Paris-Sud)

M. Fabrice ORGOGOZO (École Polytechnique)M. Yichao TIAN (Centre Morningside)

Invité : M. Luc ILLUSIE (Université Paris-Sud)

Rapporteur absent le jour de la soutenance :M. Martin OLSSON (Université Berkeley)

Thèse préparée auDépartement de Mathématiques d’OrsayLaboratoire de Mathématiques (UMR 8628), Bât. 425Université Paris-Sud 1191 405 Orsay CEDEX

Résumé

Dans cette thèse, on étude le complexe des cycles proches d’un faisceau `-adique sur unschéma au-dessus d’un trait en utilisant la théorie de ramification d’Abbes et Saito. La pre-mière partie est consacrée à une nouvelle preuve d’une formule de Deligne et Kato qui calculela dimension du complexe des cycles proches d’un faisceau `-adique sur une courbe relativelisse au-dessus d’un trait strictement local. Deligne a considéré le cas où le faisceau n’a pas deramification verticale, et Kato a traité le cas général. Notre approche est basée sur une notionlocale de cycle caractéristique définie grâce au conducteur de Swan raffiné d’Abbes et Saito.Dans la deuxième partie, on démontre une formule qui calcule le conducteur de Swan de la co-homologie du complexe des cycles proches d’un faisceau `-adique sur une variété lisse au-dessusd’un trait d’égale caractéristique, vérifiant une certaine condition de ramification. Tsushima aintroduit la classe caractéristique raffinée du faisceau et il a démontré qu’elle calcule le con-ducteur de Swan de la cohomologie du complexe de ses cycles proches par une formule du typeLefschetz-Verdier. On calcule la classe caractéristique raffinée comme un produit d’intersectionsur le fibré cotangent logarithmique de la variété faisant apparaître le cycle caractéristique dufaisceau défini par Abbes et Saito et la section nulle.

Mots-clefs : Cycles proches, Théorie de la ramification, Conducteur de Swan raffiné, Formuledu conducteur, Cycle caractéristique, Class caractéristique (raffinée).

Ramification and nearby cycles for `-adic sheaves on a scheme over atrait

Abstract

In this thesis, we study the nearby cycle complex of an `-adic sheaf on a scheme over atrait, using ramification theory of Abbes and Saito. The first part is devoted to a new proofof a formula of Deligne and Kato that computes the dimension of the stalks of the nearbycycle complex of an `-adic sheaf on a smooth relative curve over a strictly local trait. Deligneconsidered the case where the sheaf has no vertical ramification and Kato extended the formulato the general case. Our approach is based on a local notion of characteristic cycle definedusing the refined Swan conductor of Abbes and Saito. In the second part, we prove a formulathat computes the Swan conductor of the cohomology of the nearby cycle complex of an `-adicsheaf on a smooth variety over a trait of equal characteristic, satisfying a certain ramificationcondition. Tsushima introduced the refined characteristic class of the sheaf and he proved thatit computes the Swan conductor of the cohomology of its nearby cycle complex by a Lefschetz-Verdier type formula. We compute the refined characteristic class as an intersection producton the logarithmic cotangent bundle of the variety, involving the characteristic cycle of thesheaf defined by Abbes and Saito and the zero section.

Keywords : Nearby cycles, Ramification theory, Refined Swan conductor, Conductor formula,Characteristic cycle, (Refined) characteristic class.

Remerciements

Je tiens tout d’abord à exprimer ma gratitude la plus profonde envers mes directeursde thèse, Ahmed Abbes et Lei Fu. Ahmed Abbes a consacré beaucoup de temps à dis-cuter avec moi, à lire, à vérifier, et à corriger mes textes mathématiques et non math-ématiques, avec beaucoup de gentillesse et une grande patience. Je souhaite égalementle remercier vivement pour ses encouragements constants pendant la préparation de mathèse. Lei Fu m’a guidé dans un domaine mathématique très intéressant pendant mestrois ans d’études à l’Université de Nankai. Avec sa recommandation, j’ai eu la chanceprécieuse d’étudier les mathématiques à Paris.

Je souhaite remercier chaleureusement Martin Olsson et Takeshi Saito, qui m’ontfait l’honneur d’accepter d’être rapporteurs de cette thèse. Ils ont fait des corrections etm’ont donné des conseils stimulants. Je remercie sincèrement Gérard Laumon, FabriceOrgogozo, et Yichao Tian pour leur participation à mon jury de thèse. Je remercie aussiLuc Illusie pour sa présence à ma soutenance.

Je remercie tous les members de l’Institut des Hautes Études Scientifiques et duDépartement de Mathématiques d’Orsay où j’ai effectué la majeure partie de mon travailde thèse. En particulier, je suis reconnaissant à Christophe Breuil pour sa sollicitude.J’ai été soutenu par le Fonds Chern et la Fondation Mathématique Jacques Hadamardpendant mon séjour en France. Je les remercie cordialement. Je remercie aussi l’Institutde Mathématiques Chern pour son soutien pendant mes études à l’Université de Nankai.

J’adresse des remerciements sincères à mes amis en France, pour leur chaleureuseprésence : Yang Cao, Yiwen Ding, Lie Fu, Ziyang Gao, Yunlong Jiao, Yang Lan, YongqiLiang, Chunhui Liu, Shinan Liu, Ye Lu, Zhe Sun, Junyi Xie, Songyan Xie, Daxin Xu,Cong Xue, Yue Yu et Yeping Zhang, en particulier, Yang Cao, Chunhui Liu et YepingZhang pour leur aide mathématique et non mathématique.

J’exprime ma reconnaissance du fond du cœur à mes parents et à mes grands-parentspour leur soutien constant. Enfin, je remercie Xiaoling Liu pour tout ce qu’elle est pourmoi.

Table des matières

Introduction 31. Théorie de la ramification d’Abbes-Saito 32. Ramification et cycles proches pour les faisceaux `-adiques sur

une courbe relative 73. Classe caractéristique raffinée et formule du conducteur 9

Réferences 13

I. Ramification and nearby cycles for `-adic sheaves on relative curves 151. Introduction 152. Notation 183. Kato’s Swan conductors with differential values 184. Abbes-Saito’s ramification theory 225. Ramification of extensions of type (II) 266. Tubular neighborhoods and normalized integral models 297. Isogenies associated to extensions of type (II): the equal

characteristic case 308. Isogenies associated to extensions of type (II): the unequal

characteristic case 359. The refined Swan conductor of an extension of type (II) 4210. Comparison of Kato’s and Abbes-Saito’s characteristic cycles 4411. Nearby cycles of `-adic sheaves on relative curves 47

Réferences 51

II. Refined characteristic class and conductor formula 531. Introduction 532. Notation 553. Preliminaries on étale cohomology 564. Cohomological correspondences 605. Ramification of `-adic sheaves 646. Clean `-adic sheaves and characteristic cycles 687. Tsushima’s refined characteristic class 738. The conductor formula 77

Réferences 85

1

Introduction

Le complexe des cycles proches d’un faisceau `-adique sur un schéma au-dessus d’untrait (strictement local) a été défini par Grothendieck comme un complexe sur la fibrespéciale muni d’une action du groupe de Galois du corps des fonctions du trait [6, XIII].On s’intéresse à deux invariants de ce complexe : sa caractéristique d’Euler-Poincaré etla somme alternée des conducteurs de Swan de ses groupes de cohomologie. La premièrepartie de cette thèse [Hu1] est consacrée à une nouvelle démonstration de la formulede Deligne-Kato qui calcule la dimension du complexe des cycles proches d’un faisceau`-adique sur une courbe relative. Dans la deuxième partie [Hu2], on montre une formuledu conducteur qui calcule la somme alternée des conducteurs de Swan des groupes decohomologie du complexe des cycles proches d’un faisceau `-adique vérifiant une certainecondition de ramification.

Après un bref rappel de la théorie de la ramification d’Abbes-Saito [2, 3, 5] quijoue un rôle important dans ce travail, on présente les principaux résultats de [Hu1] et[Hu2].

1. Théorie de la ramification d’Abbes-Saito

1.1. Soient K un corps de valuation discrète complet, OK son anneau d’entiers, mK

l’idéal maximal de OK et F le corps résiduel de OK . On suppose que F est de typefini sur un corps parfait de caractéristique p > 0. Soient K une clôture séparable deK, OK la clôture intégrale de OK dans K, F le corps résiduel de OK , v la valuation deK normalisée par v(K×) = Z et GK le groupe de Galois de K/K. Soient ` un nombrepremier inversible dans F et Λ une Z`-algèbre artinnienne locale. On fixe un caractèreadditif non-trivial ψ : Fp → Λ×.

1.2. Abbes et Saito ont défini une filtration décroissante GrK,log (r ∈ Q>0) de GK ,

appelée la filtration de ramification logarithmique [2]. Pour tout nombre rationnel r > 0,on pose Gr+

K,log =⋃b>rG

bK,log. Alors, P = G0+

K,log est le sous-groupe d’inertie sauvage deGK [2, 3.15]. Pour tout nombre rationnel r > 0, le quotient

Grrlog GK = GrK,log

/Gr+K,log

est abélien, annulé par p ([18, 1.24], [19, Theorem 2]) et est contenu dans le centre deP/Gr+

K,log [3].

1.3. Soit L une extension finie séparable de K. Pour tout nombre rationnel r > 0,on dit que la ramification logarithmique est bornée par r (resp. par r+) si Gr

K,log

(resp. Gr+K,log) agit trivialement sur HomK(L,K) via son action sur K. Le conducteur

logarithmique c de L/K est défini comme la borne inférieure des nombres rationnelsr > 0 tels que la ramification logarithmique de L/K soit bornée by r. Alors, c est unnombre rationnel et la ramification logarithmique de L/K est bornée par c+ [2, 9.5]. Sic > 0, la ramification logarithmique de L/K n’est pas bornée par c.

3

4 1. THÉORIE DE LA RAMIFICATION D’ABBES-SAITO

1.4. Soit M un Λ-module sur lequel P = G0+K,log agit Λ-linéairement par un quo-

tient fini discret. Notons l’action ρ : P → AutΛ(M). Alors, M admet une uniquedécomposition en somme directe

M =⊕

r∈Q>0

M (r) (1.4.1)

dont les composantes M (r) sont P -stables, telles que M (0) = MP et pour tout r > 0,

(M (r))GrK,log = 0 et (M (r))G

r+K,log = M (r).

Si r > 0, M (r) = 0 sauf pour un nombre fini de nombres rationnels r pour lesquelsρ(Gr+

K,log) 6= ρ(GrK,log) [16, 1.1]. La décomposition (1.4.1) est appelée la décomposition

en pentes de M . Les valeurs r > 0 pour lesquelles M (r) 6= 0 sont appelées les pentes deM . On dit que M est isocline s’il n’a qu’une seule pente.

1.5. Soit M un Λ-module sur lequel P = G0+K,log agit Λ-linéairement par un quotient

fini discret, qui est isocline de pente r > 0. L’action de P sur M se factorise donc àtravers le groupe P/Gr+

K,log. Soit X(r) l’ensemble des classes d’isomorphisme des carac-tères finis χ : GrrlogGK → Λ×χ tels que Λχ soit une Λ-algèbre finie étale, engendrée parl’image de χ et ayant un spectre connexe. Alors, M admet une unique décomposition[5, 6.7]

M =⊕

χ∈X(r)

Mχ. (1.5.1)

Chaque Mχ est un Λ-sous-module P -stable sur lequel Λ[GrK,log] agit à travers Λχ. De

plus, il y n’a qu’un nombre fini de caractères χ ∈ X(r) pour lesquels Mχ 6= 0. Ladécomposition (1.5.1) est appelée la décomposition en caractères centraux de M .

1.6. On définit le F -espace vectoriel Ω1F (log) par

Ω1F (log) = (Ω1

F ⊕ (F ⊗Z K×))/(da− a⊗ a ; a ∈ O×K),

où a est la classe résiduelle de a ∈ OK dans F . On a alors une suite exacte canoniquede F -espaces vectoriels de dimensions finies

0→ Ω1F → Ω1

F (log)res−→ F → 0, (1.6.1)

où res((0, a ⊗ b)) = a · v(b) pour a ∈ F et b ∈ K×. Pour tout nombre rationnel r, onpose

mrK

= x ∈ K ; v(x) > r et mr+

K= x ∈ K ; v(x) > r.

Pour tout nombre rationnel r > 0, on a une injection canonique ([18, 1.24], [19, Theo-rem 2])

rsw : Hom(GrrlogGK ,Fp)→ HomF (mrK/mr+

K,Ω1

F (log)⊗ F ), (1.6.2)appelée conducteur de Swan raffiné.

1.7. Soient k un corps parfait de caractéristique p > 0, X un k-schéma connexe etlisse de dimension d, D un diviseur à croisements normaux simples sur X, Dii∈I lescomposantes irréductibles de D, U = X − D et j : U → X l’injection canonique. Onnote (X ×k X)′ l’éclatement de X ×k X le long de Di ×k Dii∈I et X∗kX l’ouvertcommplémentaire des transformées strictes de D ×k X et X ×k D dans (X ×k X)′ [18,§2.3]. D’après la propriété universelle des éclatements, l’application diagonale δ : X →X ×k X se relève en une immersion fermée δ : X → X∗kX. On considère X∗kXcomme un X-schéma par la deuxième projection, qui est en fait lisse (loc. cit.).

INTRODUCTION 5

1.8. Un diviseur rationnel effectif sur X à support dans D est un élément R =∑i∈I riDi, où ri ∈ Q>0 pour tout i ∈ I. On appelle points génériques de R les points

génériques des Di tels que ri 6= 0. On note bnRc le diviseur∑

i∈IbnricDi sur X, oubnric est la partie entière de nri. Soit R un diviseur rationnel effectif sur X à supportdans D. On note j : pr−1

2 (U) = U ×k U → X∗kX l’injection canonique et IX lefaisceau d’idéaux de OX∗kX associé à δ. On appelle dilatation de X∗kX le long de δd’épaisseur R, et l’on note (X∗kX)(R), le schéma affine au-dessus de X∗kX, défini parla (OX∗kX)-sous-algèbre quasi-cohérente de j∗(OU×kU)

∑

n>0

pr∗2(OX(bnRc)) ·I nX .

On a une section canonique δ(R) : X → (X∗kX)(R) relevant δ : X → X∗kX [5, 5.26].La dilatation s’insère dans un diagramme canonique cartésien

U

δU //

j

U ×k Uj(R)

Xδ(R)// (X∗kX)(R)

(1.8.1)

où j(R) est une immersion ouvert et δU est l’application diagonale.

r

X

D

r ?D X

XX ×k X

rδ(X)

X∗kX

δ(R)(X)

(X∗kX)(R)

1.9. Soient F un faisceau localement constant et constructible en Λ-modules sur U ,R un diviseur rationnel effectif sur X à support dans D et x un point géométrique deX. On pose H = Hom(pr∗2F , pr∗1F ) sur U ×k U . Alors, le morphisme de changementde base

α : δ(R)∗j(R)∗ (H )→ j∗δ

∗U(H ) = j∗(E nd(F )) (1.9.1)

relativement au diagramme cartésien (1.8.1) est injectif [5, 8.2]. On dit que la ramifica-tion of F en x est bornée par R+ si F satisfait aux conditions équivalantes suivantes(loc. cit.) :

(i) la fibre αx du morphisme α en x est un isomorphisme;(ii) l’image de idF dans j∗(E nd(F ))x est contenue dans l’image de αx.

On dit que la ramification de F le long de D est bornée par R+ si la ramification deF en x est bornée par R+ pour tout point géométrique x ∈ X [5, 8.3].

1.10. Soient F un faisceau localement constant et constructible en Λ-modules surU , R un diviseur rationnel effectif sur X à support dans D, ξ un point générique de D,ξ un point géométrique localisé en ξ, X(ξ) la localisation stricte de X en ξ, η son pointgénérique, η un point géometrique localisé en η et Gη le groupe de Galois de η sur η.

Le conducteur de F en ξ est la borne inférieure de l’ensemble des nombres rationnelsr > 0 tels que F |η soit trivialisé par un revêtement étale fini η′ de η et la ramification

6 1. THÉORIE DE LA RAMIFICATION D’ABBES-SAITO

logarithmique de η′/η soit bornée par r+ (1.3). Le conducteur de F relativement à Xest le diviseur rationnel effectif sur X à support dans D dont la multiplicité en chaquepoint générique ξ de D est le conducteur de F en ξ.

On dit que F est isocline en ξ si la représentation Fη de Gη est isocline (1.4) et queF est isocline le long de D s’il est isocline en tout point générique de D [5, 8.22].

1.11. Soit F un faisceau localement constant et constructible en Λ-modules sur Uqui est isocline le long de D. Abbes et Saito ont introduit la condition d’être ”clean”∗pour F , qui est une condition forte sur sa ramification ([18, §3.2] et [5, 8.23]). Dire queF est clean le long de D revient à dire que sa ramification le long de D est contrôléepar sa ramification en les points génériques de D. Sous cette condition, ils ont défini laclasse caractéristique comme suit.

1.12. On noteT∗X(logD) = Spec(Sym(Ω1

X/k(logD)∨)) (1.12.1)

le fibré cotangent logarithmique de X, σ : X → T∗X(logD) la section nulle, pour i ∈ I,ξi le point générique de Di, Fi le corps résiduel de OX,ξi , Si = Spec(OKi) la hensélisationde X en ξi, ηi = Spec(Ki) le point générique de Si, Ki une clôture séparable de Ki

et Gi le groupe de Galois de Ki sur Ki. Soient F un faisceau localement constant etconstructible en Λ-modules libres sur X et R le conducteur de F . On suppose que Fest isocline et clean le long de D. On note Mi le Λ[Gi]-module associé à F |ηi . CommeF est isocline le long de D, Mi n’a qu’une seule pente ri. On pose Iw = i ∈ I; ri > 0et S =

∑i∈Iw Di. Pour i ∈ Iw, soitMi = ⊕χMi,χ la décomposition en caratères centraux

de Mi (1.5). Notons que Mi,χ est un Λ-module libre de type fini pour tout χ. ÉtendantΛ, on peut supposer que Λχ = Λ pour tout caractère central χ de Mi. Chaque χ sefactorise uniquement en GrrilogGi → Fp

ψ−→ Λ×, où ψ est le caractère additif non-trivialfixé dans 1.1. On note encore χ : GrrilogGi → Fp le caractère ainsi défini et

rsw(χ) : mriKi/mri+

Ki→ Ω1

Fi(log)⊗Fi F i

son conducteur de Swan raffiné (1.6.2). Soit Fχ le corps de définition de rsw(χ), quiest une extension finie de Fi contenue dans F i. Le conducteur de Swan raffiné rsw(χ)définit une droite Lχ dans T∗X(logD) ×X Spec(Fχ). Soit Lχ la clôture de l’image deLχ dans T∗X(logD). Pour i ∈ Iw, on pose

CCi(F ) =∑

χ

ri · rkΛ(Mi,χ)

[Fχ : Fi][Lχ], (1.12.2)

qui est un d-cycle sur T∗X(logD) ×X Di. On définit un d-cycle CC∗(F ) surT∗X(logD)×X S par

CC∗(F ) =∑

i∈IwCCi(F ). (1.12.3)

On définit enfin le cycle caractéristique de F , noté CC(F ), comme un d-cycle surT∗X(logD) par ([18, 3.6] et [5, 1.12])

CC(F ) = (−1)d (rkΛ(F )[σ(X)] + CC∗(F )) . (1.12.4)

∗Faute d’une bonne traduction française, nous utilisons l’adjectif anglais.

INTRODUCTION 7

2. Ramification et cycles proches pour les faisceaux `-adiques sur unecourbe relative

2.1. Soient R un anneau de valuation discrète strictement hensélien et excellent decaractéristique résiduelle p > 0, S = Spec(R), s (resp. η, resp. η) le point fermé (resp.le point générique, resp. un point géométrique générique) de S, X une courbe relativelisse au-dessus de S, x un point fermé de la fibre spéciale Xs, X la localisation stricte deX en x, U un sous-schéma ouvert non-vide de Xη et u : U → Xη l’injection canonique.Soient Λ un corps fini de caractéristique ` 6= p et F un faisceau localement constant etconstructible en Λ-modules sur U . Les espaces des cycles proches de F

Ψix(u!F ) = Hi

ét(Xη, u!F ) (i > 0)

s’annulent lorsque i > 2 ([6, XIII], [9, 9.2.2]) et la dimension de Ψ0x(u!F ) peut être

calculée facilement.

2.2. Soient p le point générique de la fibre spéciale Xs et κ(p) le corps résiduel deX en p, qui est le corps des fractions d’un anneau de valuation discrète strictementhensélien. On suppose que F peut s’étendre comme un faisceau localement constantet constructible F sur un ouvert U de X contenant p. Alors, Deligne a calculé ladimension de Ψ1

x(u!F ). Soient swx(F ) le conducteur de Swan de l’image réciproque deF sur Spec(κ(p)) en x et

ϕ(s) = swx(F ) + rkΛ(F ).

Par ailleurs, pour chaque t ∈ Xη −Uη, soient swt(F ) le conducteur de Swan de l’imageréciproque de F sur Spec(OXη ,t)×X U et

ϕ(η) =∑

t∈Xη−Uη(swt(F ) + rkΛ(F )).

La formule de Deligne est ([17, 5.1.1])

dimΛ Ψ0x(u!F )− dimΛ Ψ1

x(u!F ) = ϕ(s)− ϕ(η). (2.2.1)

2.3. Kato a généralisé la formule de Deligne pour tout F . Sa formule a la mêmeforme que (2.2.1). La définition de l’invariant ϕ(η) est la même que ci-dessus, mais ϕ(s)ne peut pas être défini de la même façon. Kato a donné deux définitions de ϕ(s). L’uneutilise une théorie de la ramification des anneaux de valuation de rang deux [13], quil’a développée pour ce but. L’autre utilise la notion du conducteur de Swan à valeursdifférentielles [14], qui lui est aussi due. Dans la première partie de cette thèse [Hu1],on définit l’invariant ϕ(s) en utilisant la théorie de la ramification d’Abbes-Saito (§1) eton donne une nouvelle démonstration de la formule de Deligne-Kato. Le cas où F estde rang 1 est dû à Abbes et Saito [4, Appendix A].

2.4. Soient K un anneau de valuation discrète complet, OK son anneau d’entiers,F le corps résiduel de OK que l’on suppose de type fini au-dessus d’un corps parfait decaractéristique p > 0, K une clôture séparable de K et GK le groupe de Galois de K surK. On fixe un caractère additif non-trivial ψ : Fp → Λ×. Soit M un Λ-espace vectorielde dimension finie sur lequel P = G0+

K,log agit à travers un quotient discret fini (1.2),

M = ⊕r∈Q>0M (r)

la décomposition en pentes de M (1.4.1), et pour tout nombre rationnel r > 0,

M (r) = ⊕χM (r)χ

82. RAMIFICATION ET CYCLES PROCHES POUR LES FAISCEAUX `-ADIQUES SUR

UNE COURBE RELATIVE

la décomposition en caractères centraux de M (1.5.1). Étendant Λ, on peut supposerque pour tout nombre rationnel r > 0 et pour tout caractère central χ de M (r), Λ = Λχ.Comme Grrlog GK est abélien et annulé par p, χ se factorise uniquement en Grrlog GK →Fp

ψ−→ Λ×. On note encore χ : Grrlog GK → Fp le caractère ainsi défini. On fixe uneuniformisante π de OK . On définit le cycle caractéristique d’Abbes-Saito de M , notéCCψ(M), comme la section suivante [Hu1, 4.12.1]

CCψ(M) =⊗

r∈Q>0

⊗

χ∈X(r)

(rsw(χ)⊗ πr)dimΛM(r)χ ∈ (Ω1

F (log)⊗F F )⊗ dimAM/M(0)

.

C’est bien défini grâce au théorème de Hasse-Arf ([18, 1.26] et [19, 4.4]), malgré qu’onutilise la notation formalisée πr. C’est une version locale du cycle caractéristique d’unfaisceau `-adique défini dans (1.12.4).

2.5. On suppose que p n’est pas une uniformisante de K (c’est-à-dire, soit K estde caractéristique p, soit K est de caractéristique 0 et p n’est pas une uniformisante deOK). Soit L une extension galoisienne finie de K de groupe G. On suppose que l’indicede ramification de L/K est 1 et que l’extension des corps résiduels est non-triviale,purement radicielle et monogène. Une telle extension est dite de type (II) [14, 1.5]. Onvoit que l’ordre de G est une puissance de p.

Proposition 2.6. ([Hu1, 5.7 and 5.10]). Soit M un Λ-espace vectoriel de dimen-sion finie muni d’une action Λ-linéaire et non-triviale de G. Alors, pour tout nombrerationnel r > 0 et pour tout caractère central χ : Grrlog GK → Fp de M (r), on a

rsw(χ) ∈ Ω1F ⊗F m−r

K/m−r+

K⊂ Ω1

F (log)⊗F m−rK/m−r+

K.

En particulier, on aCCψ(M) ∈ (Ω1

F ⊗F F )⊗m,

où m = dimAM/M (0).

2.7. Soit M un Λ-espace vectoriel de dimension finie muni d’une action Λ-linéaireet non-triviale de G. Utilisant la théorie du conducteur de Swan à valeurs différentiellesde Kato, on peut définir le cycle caractéristique de Kato KCCψ(1)(M) ∈ (Ω1

F )m de M ,où m = dimAM/M (0) [Hu1, 3.7.1]. Notre principal résultat dans [Hu1] est le suivant.

Théorème 2.8. ([Hu1, 10.4]). Soit M un Λ-espace vectoriel de dimension finiemuni d’une action Λ-linéaire et non-triviale de G. Alors, on a

CCψ(M) = KCCψ(1)(M) ∈ (Ω1F )m,

où m = dimAM/M (0).

C’est un théorème de type Hasse-Arf pour les cycles caractéristiques d’Abbes-Saito.Saito [20, 3.10] et Xiao [24] ont montré indépendamment des résultats analogues pourdes variétés lisses au-dessus d’un corps parfait.

2.9. On donne une nouvelle définition de ϕ(s) pour tout F utilisant les cyclescaractéristiques d’Abbes-Saito (2.4). D’abord, en vertu d’un résultat d’Epp [8], on peutse réduire au cas où F est trivialisé par un revêtement galoisienne connexe étale fini U ′de U tel que la fibre spéciale de la normalisation X ′ de X dans U ′ soit réduit. On noteOX,p le complété de OX,p, Kp le corps des fractions de OX,p et Fp la représentation deGal(Ksep

p /Kp) associée à l’image réciproque de F sur Spec(OX,p)×XU . Cette dernière sefactorise à travers le groupe de Galois d’une extension galoisienne finie Lp of Kp, qui estde type (II) au-dessus d’une extension ramifiée de Kp. On fixe une uniformisante π de

INTRODUCTION 9

R. On a toujours CCψ(Fp) ∈ (Ω1κ(p))

⊗m [Hu1, 10.7]. On note ordp la valuation de κ(p)

normalisée par ordp(κ(p)×) = Z et encore ordp : Ω1κ(p) − 0 → Z l’application définie

par ordp(αdβ) = ordp(α), si α, β ∈ κ(p)× et ordp(β) = 1. Cette dernière peut s’étendrede façon unique à (Ω1

κ(p))⊗r − 0 pour tout entier r > 1. On note F p la restriction

à Spec(κ(p)) de l’image directe de Fp par l’application Spec(Kp) → Spec(OX,p). Onobtient ainsi une représentation de Gal(κ(p)/κ(p)) notée encore F p. L’invariant ϕ(s)est défini par

ϕ(s) = − ordp(CCψ(Fp)) + swx(F p) + rkΛ(F p). (2.9.1)Cet invariant est stable par tout changement de base par un morphisme fini de traitsS ′ → S. En fait, la définition ϕ(s) de Kato utilisant la théorie du conducteur de Swanà valeurs différentielles [14, 4.4] est obtenue en remplaçant CCψ(Fp) par KCCψ(1)(Fp)dans (2.9.1). On en déduit alors que la formule de Deligne-Kato (2.2.1) est valable pournotre définition [Hu1, 11.9].

3. Classe caractéristique raffinée et formule du conducteur

3.1. Soient k un corps parfait de caractéristique p > 0, f : X → Y un morphismepropre et plat de k-schémas connexes et lisses et d la dimension de X. Soient y unpoint fermé de Y , y un point géométrique localisé en y, Y(y) la localisation stricte deY en y et η un point géométrique de Y(y). On pose W = Y − y, V = f−1(W ) andQ = f−1(y). On suppose que dimY = 1, que la projection fV : V → W est lisse etque Q est un diviseur à croisements normaux de X. Soit D un diviseur à croisementsnormaux simples de X contenant S = Qred tel que D∩V soit un diviseur à croisementsnormaux simples relativement à W . On pose U = X −D et soit j : U → X l’injectioncanonique. On considère le diagramme

Uν //

fU

V

//

fV

X

f

Qoo

fQ

W // Y yoo

(3.1.1)

où ν est l’injection canonique et fU = fV ν. On fixe un nombre premier ` inversibledans k et une Z`-algèbre Λ locale artinienne. Soit F un faisceau localement constantet constructible en Λ-modules libres satisfaisant aux conditions suivantes :

(i) F est modérément ramifié le long du diviseur D ∩ V relativement à V ;(ii) le conducteur R de F est effectif à support dans S (1.10) et F est isocline et

clean le long de D [5, 8.23].La condition (i) implique que fV est universellement localement acyclique relativementà ν!(F ) ([7, Appendice de Th. Finitude], [20, 3.14]). Comme fV est propre, tous lesgroupes de cohomologie de RfU !(F ) sont localement constants et constructibles sur W .On pose [7, Rapport 4.4]

rkΛ(RΓc(Uη,F |Uη)) = Tr(id; RΓc(Uη,F |Uη)), (3.1.2)

swy(RΓc(Uη,F |Uη)) =∑

q∈Z(−1)qswy(R

qΓc(Uη,F |Uη)), (3.1.3)

dimtoty(RΓc(Uη,F |Uη)) = rkΛ(RΓc(Uη,F |Uη)) + swy(RΓc(Uη,F |Uη)), (3.1.4)où swy(R

qΓc(Uη,F |Uη)) est le conducteur de Swan de RqΓc(Uη,F |Uη) en y.Comme F satisfait à la condition (ii), la partie verticale CC∗(F ) du cycle carac-

téristique de F , est bien définie comme un d-cycle sur T∗X(logD)×X S (1.12).

10 3. CLASSE CARACTÉRISTIQUE RAFFINÉE ET FORMULE DU CONDUCTEUR

Théorème 3.2. ([Hu2, 1.3]). Conservons les notations et hypothèses de 3.1 etsupposons que S = D (i.e., U = V ) ou que rkΛ(F ) = 1. Alors, pour toute sections : X → T∗X(logD), on a l’égalité suivante dans Λ

dimtoty(RΓc(Uη,F |Uη))−rkΛ(F )·dimtoty(RΓc(Uη,Λ)) = (−1)d+1 deg(CC∗(F )∩[s(X)]).(3.2.1)

Le cas où rkΛ(F ) = 1 est dû à Tsushima [22, 5.9]. On suit la même méthode pourles faisceaux de rangs supérieurs, mais la situation est techniquement plus compliquée.Notre approche requiert l’hypothèse S = D.

3.3. Pour montrer 3.2, on suit la stratégie de Saito pour la démonstration d’uneformule d’indice pour les faisceaux `-adiques sur des variétés propres et lisses [18]. Cettedernière est schématiquement divisée en deux étapes. la première étape utilise la théoriedes correspondances cohomologiques due à Grothendieck et Verdier pour associer uneclasse de cohomologie au faisceau `-adique, appelée la classe caractéristique, qui calculesa caractéristique d’Euler-Poincaré par la formule de Lefschetz-Verdier [10, III]. Ladeuxième étape est plus géométrique. Elle consiste à calculer la classe caractéristiquepar une formule d’intersection utilisant la théorie de la ramification d’Abbes-Saito (§1).

3.4. L’approche analogue pour la démonstration de la formule du conducteur (3.2.1)a été initiée par Tsushima [22]. Il a raffiné la classe caractéristique du faisceau `-adiqueen une classe de cohomologie à support dans le lieu sauvage, appelée dans cette thèse laclasse caractéristique raffinée. Il a prouvé une formule de Lefschetz-Verdier pour cetteclasse [22, 5.4], qui revient à dire qu’elle commute avec l’image directe propre. Sur unecourbe lisse, la classe caractéristique raffinée donne le conducteur de Swan [22, 4.1]. Lebut principal de la deuxième partie de cette thèse [Hu2] est de prouver une formuled’intersection calculant la classe caractéristique raffinée.

3.5. Plus précisément, avec les notations et hypothèses de 3.1, la classe caractéris-tique raffinée CS(j!(F )) de j!(F ) est définie dans H0

S(X,KX). La formule de Lefschetz-Verdier implique la relation suivante

swy(RΓc(Uη,F |Uη))−rkΛ(F )·swy(RΓc(Uη,Λ)) = −f∗(CS(j!(F ))−rkΛ(F )·CS(j!(ΛU)))(3.5.1)

dans H0y(Y,KY )

∼−→ Λ, où f∗ est l’image directe propre H0S(XKX) → H0

y(Y,KY )

[Hu2, 7.12].

Théorème 3.6. ([Hu2, 8.2]). Conservons les notations et hypothèses de 3.1 etsupposons que D = S ou que rkΛ(F ) = 1. Alors, on a

CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU)) (3.6.1)

= (−1)d rkΛ(F ) · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(logD))XS∩ [X] ∈ H0

S(X,KX),

où cd(−)XS est une classe bivariante qui provient des classes de Chern localisées [Hu2,2.4] et le terme de droite est considéré comme un élément de H0

S(X,KX) par l’applicationde cycle.

Proposition 3.7. (cf. [Hu2, 8.23]). Avec les notations et hypothèses de 3.1, ona l’égalité des classes de cycles suivante :

CC∗(F )∩[s(X)] = rkΛ(F )·cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(logD))XS∩[X] ∈ CH0(S).

INTRODUCTION 11

3.8. Conservons les notations et hypothèses de 3.1 et supposons que D = S ou querkΛ(F ) = 1. En vertu de 3.6 et 3.7, on obtient la formule d’intersection pour les classescaractéristiques raffinées

CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU)) = (−1)dCC∗(F ) ∩ [s(X)] ∈ H0S(X,KX). (3.8.1)

C’est évident que l’application composée

CH0(S)cl−→ H0

S(X,KX)→ H0y(Y,KY )

∼−→ Λ,

où la deuxiéme flèche est l’image directe proper, est le degré des 0-cycles. Alors,f∗(CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU))) = (−1)d deg(CC∗(F ) ∩ [s(X)]). (3.8.2)

Comme F est modérément ramifié le long de D ∩ V relativement à V , on a [11, 2.7]rkΛ(RΓc(Uη,F |Uη)) = rkΛ(F ) · rkΛ(RΓc(Uη,Λ)). (3.8.3)

La formule du conducteur (3.2.1) résulte de (3.5.1), (3.8.2) et (3.8.3).

3.9. Conservons les notations et hypothèses de 3.1 et suppose que D = Q (i.e.,U = V et Q est réduit). En observant que X ×Y Y(y) est semi-stable au-dessus du traitY(y), le groupe de cohomologie H∗(Uη,Λ) = H∗c(Uη,Λ) est modéré [12, 3.3]. Par suite,swy(RΓc(Uη,Λ)) = 0. En vertu de (3.5.1) et (3.8.2), on a

swy(RΓc(Uη,F |Uη)) = (−1)d+1 deg(CC∗(F ) ∩ [s(X)]).

3.10. On termine cette introduction générale par un bref rappel de l’histoire de laformule du conducteur. Comme mentionné plus haut, l’idée principale de notre approcheest due à Tsushima, qui a traité le cas de rang 1 [22, 5.9]. Abbes a donné une formuledu conducteur pour un faisceau `-adique sur une surface arithmétique sous l’hypothèseque le faisceau n’a pas de ramification féroce [1]. Vidal a montré que la somme alternéedes conducteurs de Swan des groupes de cohomologie à support compact d’un faisceau`-adique sur un schéma normal au-dessus d’un corps local ne dépend que du rang et dela ramification sauvage [23]. Pour un faisceau `-adique sur un schéma lisse au-dessusd’un corps local de caractéristique mixte, Kato et Saito ont défini sa classe de Swan,une classe de 0-cycle à support dans le lieu sauvage, qui calcule le conducteur de Swandes groupes de cohomologie à support compact [15]. Dans un récent travail [21], Saitoa défini le cycle caractéristique d’un faisceau `-adique sur une surface lisse sur le fibrécotangent sans condition ”clean”. Lorsque la surface est fibrée au-dessus d’une courbelisse, il a montré une formule du conducteur conjecturée par Deligne [21, 3.16].

Réferences

[Hu1] H. Hu, Ramification and nearby cycles for `-adic sheaves on relative curves. 2013.[Hu2] H. Hu, Refined characteristic class and conductor formula. 2014.[1] A. Abbes, The Grothendieck-Ogg-Shafarevich formula for arithmetic surfaces. J.

Algebr. Geom. 9 (2000), 529–576.[2] A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields.

Amer. J. Math. 124 (2002), 879–920.[3] A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields II.

Doc. Math. Extra Volume Kato (2003), 5–72.[4] A. Abbes and T. Saito, Local fourier transform and epsilon factors. Compos. Math.

146 (2010), 1507–1551.[5] A. Abbes and T. Saito, Ramification and cleanliness. Tohoku Math. J. Centennial

Issue, 63 No.4, (2011), 775–853.[6] P. Deligne and N. Katz, Groupes de monodromie en géométrie algébriques. II. Sémi-

naire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notesof Mathematices 340, springer-verlag, New York, Heidelberg, Berlin, 1973.

[7] P. Deligne et al, Cohomologie étale. Séminaire de Géométrie Algébrique du Bois-Marie SGA 41

2. Par P. Deligne avec la collaboration de J.F. Boutot, A. Grothendieck,

L. Illusie et J.L. Verdier. Lecture Notes in Mathematics 569, Sringer-verlag, Berlin-New York, (1987).

[8] H. Epp, Eliminating wild ramification. Invent. Math. 19 (1973), 235–249.[9] L. Fu, Etale cohomology theory. Nankai Tracts Math. World Scientific Publishing

Co. Pte. Ltd., Hackensack, NJ, 2011.[10] A. Grothendieck et al, Cohomologie `-adique et fonctions L. Séminaire de Géométrie

Algébrique du Bois-Marie 1965–1966 (SGA 5). dirigé par A. Grothendieck avec lacollaboration de I. Bucur, C. Houzel, L. Illusie, J.-P. Jouanolou et J-P. Serre. LectureNotes in Mathematics 589, Springer-verlag, Berlin-New York, (1977).

[11] L. Illusie, Théorie de Brauer et caractéristique d’Euler-Poincaré, d’après P. Deligne.Caractéristique d’Euler-Poincaré, Séminaire ENS 78–79, Astérisque 82–83 (1981),161-172.

[12] L. Illusie, Autour du théorème de monodromie locale. Astérisque, 223 (1994), 9–57.[13] K. Kato, Vanishing cycles, ramification of valuations, and class field theory. Duke

Math. J. 55 (1987), No.3, 629–659.[14] K. Kato, Swan conductors with differential values. Adv. Stud. Pure Math. 12 (1987),

315–342.

13

14

[15] K. Kato, T. Saito, Ramification theory for varieties over a local field. Publ. Math.Inst. Hautes Études Sci. 117 (2013), 1–178.

[16] N. Katz, Gauss sum, Kloosterman sums, and monodromy groups. Annals of Math-ematics Studies 116, Princeton University Press, Princeton, NJ, 1988.

[17] G. Laumon, Semi-continuité du conducteur de Swan (d’après P. Deligne). In theEuler-Poincaré characteristic, Astérisque, 82–83 (1981), 173–219.

[18] T. Saito, Wild ramification and the characteristic cycle of an `-adic sheaf. J. Inst.Math. Jussieu 8 (2009), 769–829.

[19] T. Saito, Ramification of local fields with imperfect residue fields III. Math. Ann.352 (2012), No. 3, 567–580.

[20] T. Saito, Wild Ramification and the cotangent bundle. 2013, arXiv:1301.4632v4.[21] T. Saito, Characteristic cycle and the Euler number of a constructible sheaf on a

surface. 2014, http://arxiv.org/abs/1402.5720.[22] T. Tsushima, On localizations of the characteristic classes of `-adic sheaves and

conductor formula in characteristic p > 0. Math. Z. 269, (2011), 411–447.[23] I. Vidal, Théorie de Brauer et conducteur de Swan. J. Algebr. Geom. 13 (2004),

349–391.[24] L. Xiao, On ramification filtrations and p-adic differential equations, I: equal char-

acteristic case. Algebra Number Theory 4 (2010), 969–1027.

I

Ramification and nearby cycles for `-adic sheaves on relativecurves

1. Introduction

1.1. Let R be an excellent strictly henselian discrete valuation ring of residue char-acteristic p > 0, S = Spec(R), s (resp. η, resp. η) the closed point (resp. the genericpoint, resp. a geometric generic point) of S. Let X be a smooth relative curve overS, x a closed point of the special fiber Xs, X the strict henselization of X at x, U anon-empty open sub-scheme of Xη, and u : U → Xη the canonical injection. Let Λ bea finite field of characteristic ` 6= p, and F a locally constant constructible étale sheafof Λ-modules on U . The spaces of nearby cycles of F

Ψix(u!F ) = Hi

ét(Xη, u!F ) (i > 0)

vanish when i > 2 ([8, XIII], [10, 9.2.2]) and the dimension of Ψ0x(u!F ) is easy to

compute. The aim of this article is to reprove a Deligne-Kato’s formula that computesthe dimension of Ψ1

x(u!F ) [18, 14, 15] using Abbes-Saito’s ramification theory [2, 3].

1.2. Let p be the generic point of the special fiber Xs. We denote by κ(p) theresidue field of p, which is the fraction field of a strictly henselian discrete valuationring. Assume first that F can be extended to a locally constant constructible sheaf Fon an open sub-scheme U of X containing p. Then Deligne computes the dimension ofΨ1x(u!F ). Let swp(F ) be the Swan conductor of the pull-back of F on Spec(κ(p)) and

letϕ(s) = swp(F ) + rank(F ).

On the other hand, for any t ∈ Xη − Uη, let swt(F ) be the Swan conductor of thepull-back of F on Spec(OXη ,t)×X U , and let

ϕ(η) =∑

t∈Xη−Uη(swt(F ) + rank(F )).

Then, Deligne’s formula is ([18, 5.1.1])

dimΛ Ψ0x(u!F )− dimΛ Ψ1

x(u!F ) = ϕ(s)− ϕ(η). (1.2.1)

1.3. Kato generalized Deligne’s formula for any F . His formula has the same formas (1.2.1). The definition of the invariant ϕ(η) is the same as above, but ϕ(s) cannot bedefined by the same method. He provided two definitions of ϕ(s). The first one uses aramification theory for valuation rings of rank two, which he developed for this purpose[14]. The second one uses his notion of Swan conductors with differential values [15].Both methods rely on Epp’s partial semi-stable reduction theorem [9]. In this article,we define the invariant ϕ(s) in terms of ramification theory of Abbes and Saito [2, 3].The case when F has rank 1 is due to Abbes and Saito ([5, Appendix A]).

This chapter reproduces an article that has been accepted for publication at Tohoku Journal ofMathematics.

15

16 1. INTRODUCTION

1.4. Let K be a complete discrete valuation field, OK its integer ring, mK themaximal ideal of OK and F the residue field of OK . We assume that F is of finitetype over a perfect field F0 of characteristic p. We denote by K a separable closureof K, by OK the integral closure of OK in K, by F the residue field of OK , by v thevaluation of K normalized by v(K×) = Z and by GK the Galois group of K/K. Abbesand Saito defined a decreasing filtration Gr

K,log (r ∈ Q>0) of GK , called the logarithmicramification filtration. For any rational number r > 0, we put Gr+

K,log =⋃b>rG

bK,log.

Then P = G0+K,log is the wild inertia subgroup of GK ([2, 3.15]). For any rational number

r > 0, the graded pieceGrrlog GK = Gr

K,log

/Gr+K,log

is abelian and killed by p ([22, 1.24], [23, Theorem 2]).For any r ∈ Q, we denote by mr

K(resp. mr+

K) the set of elements of K such that

v(x) > r (resp. v(x) > r). Let Ω1F (log) be the F -vector space

Ω1F (log) = (Ω1

F/F0⊕ (F ⊗Z K

×))/(da− a⊗ a ; a ∈ O×K),

where a is the residue class of a in F . We have a canonical exact sequence of finitedimensional F -vector spaces

0→ Ω1F → Ω1

F (log)→ F → 0.

For any rational number r > 0, there exists a canonical injective homomorphism ([22,1.24], [23, Theorem 2]), called the refined Swan conductor,

rsw : HomFp(Grrlog GK ,Fp)→ Ω1F (log)⊗F m−r

K/m−r+

K.

Let M be a finite dimensional Λ-vector space on which P acts through a finitediscrete quotient,

M = ⊕r∈Q>0M (r)

the slope decomposition of M (cf. Lemma 4.5), and for any rational number r > 0,

M (r) = ⊕χM (r)χ

the central character decomposition of M (r), where the sum runs over finitely manycharacters χ : Grrlog GK → Λ×χ such that Λχ is a finite extension of Λ (cf. Lemma 4.7).Enlarging Λ, we may assume that for all rational number r > 0 and for all centralcharacters χ of M (r), Λ = Λχ. We fix a non-trivial character ψ0 : Fp → Λ×. SinceGrrlog GK is abelian and killed by p, χ factors uniquely through Grrlog GK → Fp

ψ0−→ Λ×.We denote abusively by χ : Grrlog GK → Fp the induced character. We fix a uniformizerπ of OK . We define Abbes-Saito’s characteristic cycle of M and denote by CCψ0(M)the following section (4.12.1)

CCψ0(M) =⊗

r∈Q>0

⊗

χ∈X(r)

(rsw(χ)⊗ πr)dimΛ M(r)χ ∈ (Ω1

F (log)⊗F F )⊗dimAM/M(0)

.

1.5. In the following, we assume that p is not a uniformizer of K (i.e. either K hascharacteristic p or K has characteristic zero and p is not a uniformizer of OK). Let Lbe a finite Galois extension of K of group G. We assume that L/K has ramificationindex one and that the residue field extension is non-trivial, purely inseparable andmonogenic ; we say that the extension L/K is of type (II) (cf. Subsection 3.3). LetM be a finite Λ-vector space on which GK acts through G. We prove that, for anyrational number r > 0, and any central character χ : Grrlog GK → Fp of M (r), we have(Proposition 5.7)

rsw(χ) ∈ Ω1F ⊗F m−r

K/m−r+

K.

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES17

Hence, we have CCψ0(M) ∈ (Ω1F ⊗F F )⊗m, where m = dimΛM/M (0) (Corollary 5.10).

On the other hand, using Kato’s theory of Swan conductors with differential values, wecan define Kato’s characteristic cycle KCCψ0(1)(M) (3.17.1). Our main result (10.7.4)is the following equality

CCψ0(M) = KCCψ0(1)(M). (1.5.1)

Using Kato’s theory, we deduce a Hasse-Arf type theorem (Corollary 10.5)

CCψ0(M) ∈ (Ω1F )m ⊂ (Ω1

F ⊗F F )m,

and an induction formula (10.6.1) for Abbes-Saito’s characteristic cycle.

1.6. Under the assumptions of subsection 1.1, we can now give a new definition ofϕ(s). Firstly, by Epp’s results [9], we can reduce to the case where F is trivialized by aGalois étale connected covering U ′ of U such that the special fiber of the normalizationX ′ of X in U ′ is reduced. We denote by OX,p the completion of OX,p, by Kp thefraction field of OX,p and by Fp the representation of Gal(Ksep

p /Kp) corresponding tothe pull-back of F on Spec(OX,p) ×X U . The latter factors through the Galois groupof a finite Galois extension Lp of Kp, which is of type (II) over an unramified extensionof Kp. We fix a uniformizer π of R and a non-trivial character ψ0 : Fp → Λ×. Westill have CCψ0(Fp) ∈ (Ω1

κ(p))⊗m (cf. Remark 10.7). We denote by ordp the valuation

of κ(p) normalized by ordp(κ(p)×) = Z and abusively by ordp : Ω1κ(p) − 0 → Z the

map defined by ordp(αdβ) = ordp(α), if α, β ∈ κ(p)× and ordp(β) = 1. The latter canbe uniquely extended to (Ω1

κ(p))⊗r − 0 for any integer r > 1. We denote by F p the

restriction to Spec(κ(p)) of the direct image of Fp by the map Spec(Kp)→ Spec(OX,p).It corresponds to a representation of Gal(κ(p)/κ(p)). The invariant ϕ(s) is defined by

ϕ(s) = − ordp(CCψ0(Fp)) + sws(F p) + rank(F p). (1.6.1)

This invariant is stable by all base changes by a finite morphism of traits S ′ → S (11.7).In fact, Kato’s second definition of ϕ(s) ([15, 4.4]) is obtained by replacing CCψ0(Fp)by KCCψ0(1)(Fp) in (1.6.1). Hence, from (1.5.1), we deduce that Deligne-Kato’s formula(1.2.1) holds true with our definition (cf. Theorem 11.9).

1.7. Deligne-Kato’s formula has already had important applications. For instance,Deligne’s formula could be used in Laumon’s work on local Fourier transform ([19,2.4.3]) and Kato’s formula was recently used in the work of Obus and Wewers on locallifting problem [20]. We would like to mention that Laumon’s formula of the rank ofthe local Fourier transform is a direct application of the formulation of Deligne-Kato’sformula using (1.6.1). Indeed, it was reproved in ([5, Appendix B]) by reducing to therank 1 case by Brauer theorem.

1.8. This article is organized as follows. We briefly introduce Kato’s Swan conduc-tors with differential values and Abbes-Saito’s ramification theory in §3 and §4, respec-tively. We study in §5 the ramification of extensions of type (II). We recall tubularneighborhoods and normalized integral models in §6. We study the isogeny associatedto an extension of type (II) in §7 in the equal character case and in §8 in the unequalcharacteristic case. Using the results of these two sections, we prove the main theorem5.9 in §9. In §10, the heart of this article, we compare Kato’s characteristic cycle andAbbes-Saito’s characteristic cycle. The last section is devoted to Deligne-Kato’s formulaby using Abbes-Saito’s characteristic cycle.

18 3. KATO’S SWAN CONDUCTORS WITH DIFFERENTIAL VALUES

Acknowledgement. This article is a part of the author’s thesis at Université Paris-Sud and Nankai University. The author would like to express his deepest gratitude tohis supervisors Ahmed Abbes and Lei Fu for leading him to this area and for patientlyguiding him in solving this problem. The author is also grateful to Fonds Chern andFondation Mathématiques Jacques Hadamard for their support during his stay in France.

2. Notation

2.1. In this article, K denotes a complete discrete valuation field, OK its integerring, mK the maximal ideal of OK and F the residue field of OK . We assume that thecharacteristic of F is p > 0. We fix a uniformizer π of OK . Let K be a separable closureof K, GK the Galois group of K over K, OK the integral closure of OK in K, F theresidue field of OK and v the valuation of K normalized by v(K×) = Z. We denote byFÉ/K the category of finite étale K-algebras. For any object K ′ of FÉ/K , we denote byOK′ the integer ring of K ′ and by mK′ the radical of OK′ .

2.2. For a field k and one dimensional k-vector spaces V1, . . . , Vm, we denote byk〈V1, . . . , Vm〉 the k-algebra

⊕

(i1,...,im)∈ZmV ⊗i11 ⊗ · · · ⊗ V ⊗imm ,

and by (k〈V1, . . . , Vm〉)× its group of units. An element of (k〈V1, . . . , Vm〉)× is containedin some vector space V ⊗i11 ⊗ · · · ⊗ V ⊗imm . Such an element x will be denoted by [x] andwe adopt the additive notation, i.e. [x] + [y] = [x · y] and −[x] = [x−1]. If for each1 6 i 6 m, ei is a non-zero element of Vi, we have an isomorphism

k〈V1, . . . , Vm〉 ∼−→ k[X1, . . . , Xm, X−11 , . . . , X−1

m ], ei 7→ Xi,

and hence an isomorphism

(k〈V1, . . . , Vm〉)× ∼−→ k× ⊕ Zm. (2.2.1)

3. Kato’s Swan conductors with differential values

3.1. In this section, we fix a finite separable extension L of K of ramification indexe contained in K. We denote by OL its integer ring and by E the residue field of OL.

3.2. We denote the group (F 〈mK/m2K〉)× by RK and the group (E〈mL/m

2L〉)× by

RL (cf. Subsection 2.2). The canonical isomorphisms

E ⊗F (mK/m2K)

∼−→ meL/m

e+1L , (3.2.1)

(mL/m2L)⊗e

∼−→ meL/m

e+1L , (3.2.2)

induce an injective homomorphism of F -algebras

F 〈mK/m2K〉 → E〈mL/m

2L〉

and hence an injective homomorphism RK → RL.

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES19

3.3. Kato’s theory applies if the extension L/K is of one of the following types ([15,1.5]):

(I) L/K is totally ramified (i.e., F = E) ;(II) the ramification index of L/K is 1 and the residue field extension E/F is purely

inseparable and monogenic.

Observe that in both cases, OL is monogenic over OK . These two cases do not cover allfinite separable extensions.

In the remaining part of this section, we assume that L/K is of type (II). We denoteby pn the degree of the residue extension E/F . We choose an element h ∈ OL such thatits reduction h ∈ E is the generator of E/F and a lifting a ∈ OK of a = hp

n ∈ F .

Lemma 3.4. Let V be the kernel of the canonical morphism Ω1F → Ω1

E. Denote by% the morphism E → F, b 7→ bp

n, by φ the morphism F → F, b 7→ bpn, and by ϕ the

morphism E → E, b 7→ bpn.

(i) The F -vector space V is of dimension 1, generated by da.(ii) The E-vector space Ω1

E/F is of dimension 1, generated by dh.(iii) The canonical morphism F ⊗%,E Ω1

E/F → Ω1F/φ(F ) = Ω1

F associated to F → E%−→

F is injective with image V .(iv) For any 1-dimensional E vector space W , the morphism

E ⊗ϕ,E W → W⊗pn , y ⊗ z 7→ yz⊗pn

is an isomorphism.(v) There exists a canonical E-linear isomorphism

E ⊗F V ∼−→ (Ω1E/F )⊗p

n

, (3.4.1)

that maps y ⊗ da to y(dh)⊗pn.

Proof. (i), (ii), (iv) are obvious. We have two canonical exact sequences of differ-ential modules corresponding to the extensions φ : F → E

%−→ F and ϕ : E%−→ F → E,

F ⊗%,E Ω1E/F

β−→ Ω1F → Ω1

F/%(E) → 0,

E ⊗F Ω1F/%(E) → Ω1

E → Ω1E/F → 0.

Since the canonical morphism Ω1F → Ω1

E factors as

Ω1F → Ω1

F/%(E) → E ⊗F Ω1F/%(E) → Ω1

E,

the image of F ⊗%,E Ω1E/F in Ω1

E is 0. Hence the image of β lies in V . Since the kernelof Ω1

F → Ω1F/%(E) is not zero (as it contains da) and since F ⊗%,E Ω1

E/F is of dimension1, β is injective. Hence β induces an isomorphism

β : F ⊗%,E Ω1E/F

∼−→ V.

From (ii) and (iv), we obtain an isomorphism

β′ : E ⊗ϕ,E Ω1E/F → (Ω1

E/F )⊗pn

, y ⊗ zdh 7→ yzpn

(dh)⊗pn

.

We take for (3.4.1) the isomorphism β′ (idE ⊗ β)−1.

20 3. KATO’S SWAN CONDUCTORS WITH DIFFERENTIAL VALUES

3.5. Let V be the kernel of the canonical morphism Ω1F → Ω1

E (Lemma 3.4). Weput (Subsection 2.2)

SK,L = (F 〈mK/m2K , V 〉)× and SL/K = (E〈mL/m

2L,Ω

1E/F 〉)×.

From (3.2.1) and (3.4.1), we obtain an injective homomorphism of F -algebras

F 〈mK/m2K , V 〉 → E〈mL/m

2L,Ω

1E/F 〉,

which induces an injective homomorphism

SK,L → SL/K . (3.5.1)

3.6. Let L′ be a subfield of L containing K, OL′ its integer ring and E ′ its residuefield. When L′ 6= L (resp. L′ 6= K), the extension L/L′ (resp. L′/K) is of type (II) ; weconsider SL′,L (resp. SL′/K) as a subgroup of SL/K containing SK,L, by functoriality. IfK 6= L′ 6= L, the following canonical maps

ker(Ω1F → Ω1

E′)→ ker(Ω1F → Ω1

E),

Ω1E/F → Ω1

E/E′ ,

ker(Ω1E′ → Ω1

E)→ Ω1E′/F

are isomorphisms by considering dimensions, which give the following relations:

SK,L = SK,L′ ⊂ SL′/K = SL′,L ⊂ SL/L′ = SL/K .

3.7. Let i be the maximal integer such that TrL/K(miL) = OK . The surjective

homomorphism TrL/K : miL/m

i+1L → OK/mK = F induces an E-isomorphism

miL/m

i+1L

∼−→ HomF (E,F ), b 7→ (a 7→ TrL/K(ab)),

and hence a basis of (mL/m2L)⊗(−i)⊗E HomF (E,F ), that we call Kato’s different of L/K

and denote by D(L/K) ([15, 2.1]).

3.8. Following Kato ([15, 2.3]), there is an F -linear map TrE/F : Ω1E → Ω1

F charac-terized by

TrE/F

(dx

x

)=

dxpn

xpn, TrE/F

(xi

dx

x

)= 0,

for any x ∈ E× and 1 6 i 6 pn − 1. Its image is V (Lemma 3.4) and it induces anisomorphism

Ω1E/F

∼−→ HomF (E, V ), ω 7→ (a 7→ TrE/F (aω)). (3.8.1)

Hence we obtain a sequence of isomorphisms

HomF (E,F )(3.8.1)−−−→ Ω1

E/F ⊗F V ⊗(−1) (3.4.1)−−−→ Ω1E/F ⊗E (Ω1

E/F )⊗(−pn) = (Ω1E/F )⊗(1−pn),

(3.8.2)by which E〈mL/m

2L〉 ⊗E HomF (E,F ) is a sub-E〈mL/m

2L〉-module of E〈mL/m

2L,Ω

1E/F 〉.

Hence we may consider D(L/K) (Subsection 3.7) as an element of SL/K .

Proposition 3.9. ([15, 2.2]). Let L′ be a subfield of L containing K. If L = L′

(resp. L′ = K), we put D(L/L′) = [1] (resp. D(L′/K) = [1]). Then, we have

D(L/K) = D(L/L′) + D(L′/K) ∈ SL/K . (3.9.1)

We consider D(L′/K) ∈ SL′/K ⊆ SL/K .

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES21

3.10. In the rest of this section, we assume that the extension L/K is Galois withgroup G. For any σ ∈ G− 1, we put

sG(σ) = [dh]− [h− σ(h)] ∈ SL/K ,where the term [dh] corresponds to the element dh in Ω1

E/F and the term [h − σ(σ)]

corresponds abusively to the class of h − σ(h) ∈ (mL/m2L)⊗v(h−σ(h)). The definition of

sG(σ) is independent of the choice of the generator h ([15, 1.8]). We also put

sG(1) = −∑

σ∈G−1sG(σ) ∈ SL/K . (3.10.1)

We have ([15, (2.4)])sG(1) = D(L/K). (3.10.2)

Proposition 3.11. ([15, Proposition 1.9]). Let H be a normal subgroup of G.Then for any element τ ∈ G/H − 1, we have

sG/H(τ) =∑

σ∈Gσ 7→τ

sG(σ).

3.12. In the following of this section, let C be an algebraically closed field of char-acteristic zero, ξ a primitive p-th root of 1 in C and Z the integral closure of Z in C. Forany finite group H, we denote by RC(H) the Grothendieck group of finitely generatedC[H]-modules. For an element χ ∈ R(H), let 〈χ, 1〉 = 1

]H

∑σ∈H trχ(σ).

3.13. For an element χ ∈ RC(G), we put

sG(χ) =∑

σ∈GsG(σ)⊗ trχ(σ) ∈ SL/K ⊗Z Z,

ε(ξ) =∑

r∈F×p ⊆E×[r]⊗ ξr ∈ SL/K ⊗Z Z.

Kato defined the Swan conductor with differential values of χ as

swξ(χ) = sG(χ) + (dimχ− 〈χ, 1〉)ε(ξ) ∈ SL/K ⊗ Z. (3.13.1)

For any r ∈ F×p , we have swξr(χ) = swξ(χ) + (dimχ− 〈χ, 1〉)[r].Proposition 3.14. ([15, 3.3(1)]). Let H be a normal subgroup of G, ϑ an element

in RC(G/H) and ϑ′ the image of ϑ under the canonical map RC(G/H)→ RC(G). Then,we have

sG(ϑ′) = sG/H(ϑ) and swξ(ϑ′) = swξ(ϑ).

Proposition 3.15. ([15, 3.3(2)]). Let H be a subgroup of G. For any θ ∈ RC(H),we have

sG(IndGH θ) = [G : H](sH(θ) + dim θ ·D(LH/K)

)

swξ(IndGH θ) = [G : H](swξ(θ) + (dim θ − 〈θ, 1〉) ·D(LH/K)

). (3.15.1)

By (3.10.1), (3.10.2) and (3.9.1), equation (3.15.1) can be written as

swξ(IndGH θ) = [G : H]

(swξ(θ)− (dim θ − 〈θ, 1〉)

( ∑

σ∈G−H([dh]− [h− σ(h)])

)).

(3.15.2)

Theorem 3.16. ([15, 3.4]). For any χ ∈ RC(G), we have

swξ(χ) ∈ SK,L ⊂ SL/K ⊗Z Z.

22 4. ABBES-SAITO’S RAMIFICATION THEORY

This is a generalization of Hasse-Arf’s theorem. It can be reduced to the case whereG is cyclic of rank ps and χ is 1-dimensional by the induction formula (3.15.1) andBrauer theorem. Then the proof relies on the higher dimensional class field theory ofKato ([15, 3.6, 3.7]).

3.17. For an element χ ∈ RC(G), the Swan conductor with differential values swξ(χ)is given by

swξ(χ) = −]G(dimC χ− 〈χ, 1〉)[dh] + ∆,

where

∆ =∑

σ∈G−1[h− σ(h)]⊗ (dimC χ− trχ(σ)) + (dimC χ− 〈χ, 1〉)ε(ξ) ∈ RL ⊗Z Z.

From (3.4.1) and theorem 3.16, we have ]G[dh] = [da] and ∆ ∈ RK . Hence, we get

swξ(χ) = [πc] + [∆′]−m[da] ∈ SK,L,

where π is the uniformizer of OK fixed in subsection 2.1, c is an integer, m = dimC χ−〈χ, 1〉 and ∆′ ∈ F such that [πc∆′] = ∆. We define Kato’s characteristic cycle of χ anddenote by KCCξ(χ) the element

KCCξ(χ) = ∆′(da)m ∈ (Ω1F )⊗m. (3.17.1)

Remark 3.18. ([15, 3.15]). If the extension L/K is not of type (II), but thereexists a subfield K ′ of L containing K such that K ′/K is an unramified extension andL/K ′ is of type (II), we define

swξ(χ) = swξ(ResGGal(L/K′) χ).

Denote by OK′ the integer ring of K, mK′ the maximal ideal of OK′ and F ′ the residuefield of OK′ . Observe that swξ(χ) is fixed by Gal(K ′/K) and that the Gal(K ′/K)-invariant part of F ′〈mK′/m

2K′ , ker(Ω1

F ′ → Ω1E)〉 is F 〈mK/m

2K , ker(Ω1

F → Ω1E)〉. Thus

swξ(χ) is still contained in SK,L.

Remark 3.19. ([15, 3.16]). Let A be an algebraically closed field of characteristic` /∈ 0, p. We denote by A′ an algebraic closure of the fraction field of the ring ofWitt vectors W (A). Let χ be an element of RA(G) and let χ be a pre-image of χ inRA′(G) ([26, 16.1 Theorem 33]). We denote by ξ the p-th root of unity in A′ lifting ofa primitive p-th root of unity ξ in A. Then we put

swξ(χ) = swξ(χ).

This definition is independent of the choice of χ because of ([26, 18.2 Theorem 42]) and(3.13.1).

4. Abbes-Saito’s ramification theory

4.1. Abbes and Saito defined two decreasing filtrations GrK and Gr

K,log (r ∈ Q>0)of GK by closed normal subgroups called the ramification filtration and the logarithmicramification filtration, respectively ([2, 3.1, 3.2]).

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES23

4.2. We denote by G0K the group GK . For any r ∈ Q>0, we put

Gr+K =

⋃

s∈Q>rGsK and GrrGK = Gr

K/Gr+K .

Let L be a finite separable extension of K. For a rational number r > 0, we say thatthe ramification of L/K is bounded by r (resp. by r+) if Gr

K (resp. Gr+K ) acts trivially

on HomK(L,K) via its action on K. We define the conductor c of L/K as the infimumof rational numbers r > 0 such that the ramification of L/K is bounded by r. Then cis a rational number and L/K is bounded by c+ ([2, 6.4]). If c > 0, the ramification ofL/K is not bounded by c.

4.3. We denote by G0K,log the inertia subgroup of GK . For any r ∈ Q>0, we put

Gr+K,log =

⋃

s∈Q>rGsK,log and GrrlogGK = Gr

K,log

/Gr+K,log.

By ([2, 3.15]), P = G0+K,log is the wild inertia subgroup of GK , i.e., the p-Sylow subgroup

of G0K,log. Let L be a finite separable extension ofK. For a rational number r > 0, we say

that the logarithmic ramification of L/K is bounded by r (resp. by r+) if GrK,log (resp.

Gr+K,log) acts trivially on HomK(L,K) via its action on K. We define the logarithmic

conductor c of L/K as the infimum of rational numbers r > 0 such that the ramificationof L/K is bounded by r. Then c is a rational number and L/K is bounded by c+ ([2,9.5]). If c > 0, the ramification of L/K is not bounded by c.

Theorem 4.4. ([3, Theorem 1]). For every rational number r > 0, the groupGrrlogGK is abelian and is contained in the center of P/Gr

K,log.

Lemma 4.5. ([17, 1.1]). Let M be a Z[1p]-module on which P = G0+

K,log acts througha finite discrete quotient, say by ρ : P → AutZ(M). Then,

(i) The module M has a unique direct sum decomposition

M =⊕

r∈Q>0

M (r) (4.5.1)

into P -stable submodules M (r), such that M (0) = MP and for every r > 0,

(M (r))GrK,log = 0 and (M (r))G

r+K,log = M (r).

(ii) If r > 0, then M (r) = 0 for all but the finitely many values of r for whichρ(Gr+

K,log) 6= ρ(GrK,log).

(iii) For any r > 0, the functor M 7→M (r) is exact.(iv) For M , N as above, we have HomP−mod(M (r), N (r′)) = 0 if r 6= r′.

The decomposition (4.5.1) is called the slope decomposition of M . The values r > 0for which M (r) 6= 0 are called the slopes of M . We say that M is isoclinic if it has onlyone slope.

4.6. In the following of this section, we fix a prime number ` different from p, alocal Z`-algebra Λ which is of finite type as a Z`-module and a non-trivial characterψ0 : Fp → Λ×.

Lemma 4.7. ([6, 6.7]). Let M be a Λ-module on which P acts Λ-linearly through afinite discrete quotient, which is isoclinic of slope r > 0. So the P action on M factorsthrough the group P/Gr+

K,log.

24 4. ABBES-SAITO’S RAMIFICATION THEORY

(i) Let X(r) be the set of isomorphism classes of finite characters χ : GrrlogGK →Λ×χ such that Λχ is a finite étale Λ-algebra, generated by the image of χ, andhaving a connected spectrum. Then M has a unique direct sum decomposition

M =⊕

χ∈X(r)

Mχ. (4.7.1)

Each Mχ is a P stable sub-Λ-module such that Λ[GrK,log] acts on Mχ through

Λχ.(ii) There are finitely many characters χ ∈ X(r) for which Mχ 6= 0.(iii) For a fixed χ ∈ X(r), the functor M →Mχ is exact.(iv) For M , N as above, we have HomΛ(Mχ, Nχ′) = 0 if χ 6= χ′.

The decomposition (4.7.1) is called the central character decomposition of M . Thecharacters χ : GrrlogGK → Λ×χ for which Mχ 6= 0 are called the central characters of M([6, 6.8]).

Let P0 be a finite discrete quotient of P/Gr+K,log through which P acts on M and let

C0 be the image of GrrlogGK in P0. By theorem 4.4, we know that C0 is contained in thecenter of P0. The connected components of Spec(Λ[C0]) correspond to the isomorphismclasses of characters χ : C0 → Λ×χ , where Λχ is finite étale Λ-algebra, generated by theimage of χ, and having a connected spectrum. If pnC = 0, and Λ contains a primitivepn-th root of 1, then Λχ = Λ for every χ such that Mχ 6= 0.

Lemma 4.8. ([17, 1.4], [6, 6.10]). Let A be a Λ-algebra and M a left A-module onwhich P acts A-linearly through a finite discrete quotient. Then,

(i) In the slope decomposition M =⊕

rM(r), each M (r) is a sub-A-module of M .

For any A-algebra B, the decomposition of B ⊗A M is given by B ⊗A M =⊕r(B ⊗AM(r)).

(ii) If M is isoclinic, then in the central character decomposition M =⊕

χMχ,each Mχ is a sub-A-module of M . For any A-algebra B, the central characterdecomposition of B ⊗AM is given by B ⊗AM =

⊕χ(B ⊗AMχ).

4.9. Let V be a finite dimensional F -vector space and we denote by V ∗ its dualspace. We consider V as a smooth abelian algebraic group over F , i.e. Spec(Sym(V ∗)).Let πalg

1 (V ) be the quotient of πab1 (V ) classifying étale isogenies. Then πalg

1 (V ) is aprofinite group killed by p and the group Hom(πalg

1 (V ),Fp) is canonically identified withthe dual space V ∗ by pulling-back the Lang’s isogeny A1 → A1 : t 7→ tp − t by linearforms (cf. [24, 1.19]).

4.10. For the rest of this section, we assume that F is of finite type over a perfectsubfield F0. We define the F -vector space Ω1

F (log) by

Ω1F (log) = (Ω1

F/F0⊕ (F ⊗Z K

×))/(da− a⊗ a ; a ∈ O×K).

Then we have an exact sequence of finite dimensional F -vector spaces

0 −→ Ω1F −→ Ω1

F (log)res−→ F −→ 0, (4.10.1)

where res((0, a⊗ b)) = a · v(b) for a ∈ F and b ∈ K×. If K has characteristic p, we put

Ω1OK/F0

= lim←−n

Ω1(OK/mnK)/F0

.

We have an exact sequence of F -vector spaces

0→ mK/m2K → Ω1

OK/F0⊗OK F → Ω1

F → 0. (4.10.2)

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES25

If K has characteristic zero and p is not a uniformizer of OK , we denote by OK0 thering of Witt vectors W (F0) regarded as a sub-algebra of OK . Then, we put

Ω1OK/OK0

= lim←−n

Ω1(OK/mnK)/OK0

.

We have an exact sequence of F -vector spaces

0→ mK/m2K → Ω1

OK/OK0⊗OK F → Ω1

F → 0. (4.10.3)

For any rational number r, we put

mrK

= x ∈ K ; v(x) > r, mr+

K= x ∈ K ; v(x) > r,

Θ(r)

F ,log= HomF

(Ω1F (log),mr

K/m

(r+)

K

),

Ξ(r)

F= HomF

(Ω1F ,m

rK/m

(r+)

K

). (4.10.4)

When K has characteristic p (resp. characteristic zero and p is not a uniformizer ofOK), for any rational number r > 0, we denote by Θ

(r)

Fthe F -vector space

Θ(r)

F= HomF

(Ω1OK/F0

⊗OK F,mrK/m

(r+)

K

)(4.10.5)

(resp. Θ

(r)

F= HomF

(Ω1OK/OK0

⊗OK F,mrK/m

(r+)

K

) ).

By (4.10.1), (4.10.2) and (4.10.3), when p is not a uniformizer of K, we have homomor-phisms

Θ(r)

F ,log→ Ξ

(r)

F→ Θ

(r)

F.

By ([3, 5.12]), we have a canonical surjection

πab1 (Θ

(r)

F ,log)→ GrrlogGK . (4.10.6)

Theorem 4.11. ([22, 1.24], [23, Theorem 2]). For every rational number r > 0,the canonical surjection (4.10.6) factors through the quotient πalg

1 (Θ(r)

F ,log). In particular,

the abelian group GrrlogGK is killed by p and the surjection (4.10.6) induces an injectivehomomorphism

rsw : Hom(GrrlogGK ,Fp)→ HomF (mrK/mr+

K,Ω1

F (log)⊗ F ). (4.11.1)

The morphism (4.11.1) is called the refined Swan conductor.

4.12. Let M be a free Λ-module of finite type on which P acts Λ-linearly througha finite discrete quotient. Let

M =⊕

r∈Q>0

M (r)

be the slope decomposition of M and for each rational number r > 0, let

M (r) =⊕

χ∈X(r)

M (r)χ

be the central character decomposition of M (r). We notice that each M (r)χ is a free Λ-

module. Enlarging Λ, we may assume that for all rational number r > 0 and χ ∈ X(r),Λ = Λχ (Lemma 4.7). Each χ factors uniquely through ψ0 (Subsection 4.6)

Grrlog GK → Fpψ0−→ Λ×.

26 5. RAMIFICATION OF EXTENSIONS OF TYPE (II)

We denote abusively by χ the induced character Grrlog GK → Fp. We define the Abbes-Saito characteristic cycle CCψ0(M) of M by

CCψ0(M) =⊗

r∈Q>0

⊗

χ∈X(r)

(rsw(χ)⊗ πr)dimΛ M(r)χ ∈ (Ω1

F (log)⊗F F )⊗ dimΛM/M(0)

. (4.12.1)

Thanks to the Hasse-Arf theorem ([22, 1.26] and [23, 4.4]), it is well defined althoughwe use the formal notation πr.

5. Ramification of extensions of type (II)

5.1. In this section, we assume that the residue field F of OK is of finite type overa perfect field F0 of characteristic p. Let L be a finite Galois extension of K of groupG and type (II) (Subsection 3.3), OL the integer ring of L and E the residue field ofOL. We denote by pn the degree of the residue extension E/F . We choose an elementh ∈ OL such that its residue class h ∈ E is a generator of E/F . We have OL = OK [h].Let f(T ) ∈ OK [T ] be the minimal polynomial of h:

f(T ) = T pn

+ apn−1Tpn−1 + · · ·+ a0. (5.1.1)

Notice that a0 = −hpn ∈ F . We put

c = supσ∈G−1

v(h− σ(h)) +∑

σ∈G−1v(h− σ(h)), (5.1.2)

which is an integer > pn.For any rational number r > 0, we denote by Gr (resp. Gr

log) the image of GrK

(resp. GrK,log) in G ([2, 3.1]). Using the monogenic presentation OL = OK [T ]/(f(T )),

we obtain that, for any rational number r > 1, Gr = Grlog([2, 3.1, 3.2]) and that the

conductor of L/K is c ([2, 6.6]). By theorem 4.11, the normal subgroup Gc of G iscommutative and killed by p. In the following, we put ]Gc = ps.

5.2. For any integer j > 1, we denote by Dj the j-dimensional closed poly-disc ofradius one over K and by Dj the j-dimensional open disc of radius one over K. Fora rational number r > 0, the j-dimensional closed poly-disc of radius r is denoted byDj,(r) = (x1, . . . , xj) ∈ Dj ; v(xi) > r. Let

f : D1 → D1, x 7→ f(x),

be the morphism induced by f . For any rational number r > 0, it is easy to see thatf−1(D1,(r)) is a disjoint union of closed discs with the same radius, i.e. there exists arational number ρ(r) > 0 such that

f−1(D1,(r)) =∐

16j6i

(xj +D1,(ρ(r))

),

where the xj’s are zeros of f . The function ρ : Q>0 → Q>0 is called the Herbrandfunction of the extension L/K. By ([3, 6.6]), we have ρ(c) = supσ∈G−1 v(h−σ(h)) and

Gc = σ ∈ G; v(h− σ(h)) > ρ(c). (5.2.1)

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES27

5.3. We denote by u the map

u : G→ E, σ 7→uσ =

(h−σ(h)

πv(h−σ(h))

), if σ 6= 1,

uσ = 0, if σ = 1.(5.3.1)

The restriction u|Gc : Gc → E of u to Gc is an injective homomorphism of groups.Indeed, for any σ ∈ Gc − 1, we have v(h − σ(h)) = ρ(c). Hence, for σ1, σ2 ∈ Gc, wehave

uσ1σ2 =

(h− σ1σ2(h)

πρ(c)

)=

(h− σ1(h) + σ1(h− σ2(h))

πρ(c)

)= uσ1 + uσ2 .

Proposition 5.4. The polynomial fc(T ) = f(πρ(c)T + h)/πc ∈ L[T ] has integralcoefficients. Its reduction fc ∈ E[T ] is an additive polynomial of degree ps = ]Gc with anon-zero linear term.

Proof. We have

fc(T ) = T∏

σ∈G−1

πρ(c)T + h− σ(h)

πv(h−σ(h))∈ OL[T ].

Hence

fc(T ) = T∏

σ∈G−1

(πρ(c)T + h− σ(h)

πv(h−σ(h))

)=

∏

σ∈G−Gcuσ ·

∏

σ∈Gc(T + uσ) . (5.4.1)

Choose an Fp-basis τ1, . . . , τs of Gc, we get∏

σ∈Gc(T + uσ) =

∏

(j1,...,js)∈Fsp

(T + j1uτ1 + · · ·+ jsuτs).

We conclude by the lemma below.

Lemma 5.5. Let C be a field of characteristic p. For any integer r > 0, let x1, . . . , xrbe r elements of C such that for any (j1, . . . , jr) ∈ Frp−0, j1x1 + · · ·+ jrxr 6= 0. Thenwe have

∏

(j1,...,jr)∈Frp

(T + j1x1 + · · ·+ jrxr) = T pr

+ λr−1Tpr−1

+ · · ·+ λ1Tp + λ0T ∈ C[T ], (5.5.1)

whereλ0 =

∏

(j1,...,jr)∈Frp−0(j1x1 + · · ·+ jrxr) 6= 0.

Proof. We proceed by induction on r. If r = 1,∏

j1∈Frp

(T + j1x1) = T p − xp−11 T,

which satisfies (5.5.1). Assume that (5.5.1) holds for (r − 1)-tuples where r > 2, let(x1, . . . , xr) ∈ Cr be as in the lemma. We put

gr−1(T ) =∏

(j1,...,jr−1)∈Fr−1p

(T + j1x1 + · · ·+ jr−1xr−1).

28 5. RAMIFICATION OF EXTENSIONS OF TYPE (II)

Then, we have∏

(j1,...,jr)∈Frp

(T + j1x1 + · · ·+ jrxr) =∏

jr∈Fp(gr−1(T + jrxr))

=∏

jr∈Fp(gr−1(T ) + jrgr−1(xr))

= gpr−1(T )− gp−1r−1(xr)gr−1(T ),

which satisfies (5.5.1) since gr−1 does.

In the following of this section, we assume that p is not a uniformizer of K.

Lemma 5.6. Suppose c > 2. Then, for any 1 6 i 6 pn − 1, we have v(ai) > 2(5.1.1).

Proof. From the equation f(T ) =∏

σ∈G(T − σ(h)), for any 1 6 i 6 pn − 1, weobtain

ai = (−1)(pn−i)∑

σ1,...,σpn−i⊆Gσ1(h)σ2(h) · · ·σpn−i(h) (5.6.1)

= (−1)(pn−i)∑

σ1,...,σpn−i⊆G(σ1(h)− h+ h) · · · (σpn−i(h)− h+ h)

= (−1)(pn−i)((

pn

i

)hp

n−i +

(pn − 1

i

)hp

n−i−1∑

σ∈G(σ(h)− h) + ∆

),

where v(∆) > 2. Since the integer(pn

i

)is divisible by p, v(

(pn

i

)hp

n) > 2. Hence it is

sufficient to show that

v

(∑

σ∈G(σ(h)− h)

)> 2.

Assume first that for any σ ∈ G − 1, v(h − σ(h)) = ρ(c), i.e. G = Gc. It sufficesto treat the case where ρ(c) = 1. In this case, ]G = c > 2 (5.1.2). From subsection 5.1,G is an Fp-vector space of dimension n and we choose an Fp-basis τ1, . . . , τn of G. Bysubsection 5.3, we have

∑

σ∈Guσ =

∑

(j1,...,jn)∈Fnp

(j1uτ1 + · · ·+ jnuτn)

=pn(p− 1)

2(uτ1 + · · ·+ uτn) = 0,

which means that v(∑

σ∈G−1(σ(h)− h)) > ρ(c) + 1 = 2.Assume next that for σ ∈ G − 1, the v(h − σ(h))’s are not equal. Let c′ be the

smallest jump of the ramification filtration of G and let ](Gc′+) = pn′ for some integer

n′ < n. Let ς1 = 1, ς2, . . . , ςpn−n′ be liftings of all the elements of G/Gc′+ in G. Observethat for any ς ∈ G−Gc′+ and σ ∈ Gc′+, we have uςσ = uς . Hence

∑

ς∈G−Gc′+uς =

pn−n′

∑

j=2

∑

σ∈Gc′+uςj = pn

′pn−n

′∑

j=2

uςj = 0.

Hence v(∑

σ∈G−Gc′+(σ(h) − h)) > 2. Meanwhile, v(∑

σ∈Gc′+(σ(h) − h)) > 2, hence weobtain the inequality v(

∑σ∈G(σ(h)− h)) > 2.

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES29

Proposition 5.7. The composition of the canonical homomorphisms (Theorem4.11)

πalg1 (Θ

(c)

F ,log)→ Grclog GK → Gc

factors through πalg1 (Ξ

(c)

F) (4.10.4). In particular, for any non-trivial character χ : Gc →

Fp, we have rsw(χ) ∈ Ω1F ⊗F m−c

K/m−c+

K.

The proof of this proposition is given in subsection 9.3.

5.8. For a non-trivial character χ : Gc → Fp, we denote by fc,χ(T ) the polynomial(Subsection 5.3)

fc,χ(T ) =∏

σ∈kerχ

(T + uσ) ∈ F [T ], (5.8.1)

and by τ ∈ Gc a lifting of 1 ∈ Fp. Recall that fc,χ is an additive polynomial with anon-zero linear term (Lemma 5.5), and that fc,χ(uτ ) is independent of the choice of τ .

Theorem 5.9. For any non-trivial character χ : Gc → Fp, the refined Swan con-ductor rsw(χ) is given by

rsw(χ) = −da0 ⊗π−c(∏

σ∈G−Gc uσ)fpc,χ(uτ )

∈ Ω1F ⊗F m−c

K/m−c+

K.

The proof of this theorem is given in subsection 9.4.

Corollary 5.10. Let M be a finite dimensional Λ-vector space with a non-triviallinear G-action. Then, with the notation of subsection 4.6, we have (4.12.1)

CCψ0(M) ∈ (Ω1F ⊗F F )⊗r,

where r = dimΛM/M (0) (Lemma 4.5).

6. Tubular neighborhoods and normalized integral models

6.1. Let R be an OK-algebra. Following ([3, 1]), we say that R is formally of finitetype over OK if it is semi-local with radical mR, mR-adically complete, Noetherian andif the quotient R/mR is of finite type over F . We say that R is topologically of finitetype over OK if it is π-adically complete, Noetherian and if the quotient R/πR is offinite type over F .

6.2. We denote by AFSOK the category of affine Noetherian adic formal schemesX over Spf(OK) such that the closed sub-scheme Xred defined by the largest ideal ofdefinition of X, is a scheme of finite type over Spec(F ). Let A be a finite flat algebraover OK , and i : Spf(A) → X a closed immersion in AFSOK . For any rational numberr > 0, following ([13, 7.1] and [1, 2.1]), we associate to i a K-affinoid variety Xr, calledthe tubular neighborhood of i of thickening r, as follows. Let X = Spf(A), I be the idealof A which defines the immersion i and t, s > 0 be two integer such that r = t/s. LetA〈Is/πt〉 be the π-adic completion of the subalgebra of A⊗OK K generated by A andf/πt for f ∈ Is. Then A〈Is/πt〉 ⊗OK K is a K-affinoid algebra which depends only onr. We denote by Xr the K-affinoid variety Sp(A〈Is/πt〉 ⊗OK K). For rational numbersr′ > r > 0, there exists a canonical morphism Xr′ → Xr which makes Xr′ a rationalsub-domain of Xr. The admissible union of the affinoid spaces Xr for r ∈ Q>0 is aquasi-separated rigid variety over K.

307. ISOGENIES ASSOCIATED TO EXTENSIONS OF TYPE (II): THE EQUAL

CHARACTERISTIC CASE

Proposition 6.3. (Finiteness theorem of Grauert-Remmert, [7, 6.4.1/3], [2, 4.2]).Let R be a geometrically reduced K-affinoid algebra. Then, there exists a finite separableextension K ′ of K such that the supremum norm unit ball ([7, 3.8.1])

ROK′ = f ∈ R⊗K K ′; |f |sup 6 1 ⊆ R⊗K K ′ (6.3.1)

has a reduced geometric closed fiber ROK′ ⊗OK′ F . Moreover, the formation of ROK′commutes with any finite extension of K ′.

6.4. LetR be a geometrically reducedK-affinoid algebra. We consider the collectionof OK′-formal scheme Spf(ROK′ ), where K ′ and ROK′ are as in proposition 6.3, as aunique model of Sp(R) over OK . We call it the normalized integral model over OK . Wesay that the normalized integral model of Sp(R) is defined overK ′ if the supremum normunit ball ROK′ has a reduced geometric special fiber. We call this reduced geometricspecial fiber over F the special fiber of the normalized integral model of Sp(R) over OK .

Proposition 6.5. ([2, 4.4]). Let X be a geometrically reduced affinoid varietyover K, X its normalized integral model over OK and X the special fiber of X. Then theset of geometric connected components of X and X are isomorphic.

6.6. Let X be a geometrically reduced affinoid variety over K, X its normalizedintegral model over OK and X the special fiber of X. If X is defined over a finiteGalois extension K ′ of K, we denote by XOK′ the normalized integral model of X overOK′ . The natural K ′-semi-linear action of GK on X ⊗K K ′ extends to an OK′-semi-linear action of GK on XOK′ . If K ′′ is another finite Galois extension of K containingK ′, then X′OK′′ = XOK′ ⊗OK′ OK′′ and the semi-linear action of GK on both sides arecompatible. Hence, it induces an F -semi-linear action of GK on the special fiber X,called the geometric monodromy ([2, 4.5]).

7. Isogenies associated to extensions of type (II): the equal characteristiccase

7.1. In this section, we assume that K has characteristic p and that the residue fieldF of OK is of finite type over a perfect field F0. For an object L of FÉ/K and an integerr > 1, we denote by (OL/mr

L)⊗F0OK the completion of (OL/mrL) ⊗F0 OK relatively to

the kernel of the homomorphism

(OL/mrL)⊗F0 OK → OL/mr

L, a⊗ b 7→ ab, (7.1.1)

and by OL ⊗F0OK the projective limit

lim←−r

(OL/mrL)⊗F0OK .

We will always consider OL ⊗F0OK as an OK-algebra by the homomorphism

OK → OL ⊗F0OK , u 7→ 1⊗ u, (7.1.2)

(in the following, we always abbreviate 1 ⊗ u by u) and we will consider it as an OL-algebra by

OL → OL ⊗F0OK , v 7→ v ⊗ 1.

There is a canonical surjective homomorphism

OL ⊗F0OK → OL (7.1.3)

induced by the surjections (7.1.1). We denote by IL its kernel.

Proposition 7.2. ([3, 2.3]). Let L be an object of FÉ/K.

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES31

(i) The OK-algebra OL ⊗F0OK is formally of finite type and formally smooth overOK and the morphism (OL ⊗F0OK)/mOL ⊗F0

OK → OL/mOL (7.1.3) is an iso-morphism.

(ii) Any morphism L→ L′ of FÉ/K induces an isomorphism

OL′ ⊗OL (OL ⊗F0OK)∼−→ OL′ ⊗F0OK . (7.2.1)

7.3. Let L be an object of FÉ/K . By proposition 7.2, Spf(OL ⊗F0OK) is an objectof AFSOK (Subsection 6.2). For any rational number r > 0 and integer numbers s, t > 0such that r = t/s, we denote by Rr

L the K-affinoid algebra

RrL = (OL ⊗F0OK)〈IsL/πt〉 ⊗OK K, (7.3.1)

by XrL = Sp(Rr

L) the tubular neighborhood of thickening r of the closed immersionSpf(OL)→ Spf(OL ⊗F0OK) (7.1.3), (Subsection 6.2), which is smooth over K ([3, 1.7]).By proposition 6.3, there exists a finite separable extension K ′ of K such that thenormalized integral model of Xr

L over OK is defined over K ′ (Subsection 6.4). We denoteby Rr

L,OK′ the supremum norm unit ball of RrL ⊗K K ′ (6.3.1), by Xr

L the normalizedintegral model of Xr

L over OK and by Xr

L the special fiber of XrL (Subsection 6.4).

7.4. Let m be the dimension of the F -vector space Ω1F , which is finite. By ([3,

1.14.3]), there is an isomorphism of OK-algebrasOK [[T0, . . . , Tm]]

∼−→ OK ⊗F0OK , (7.4.1)

such that the composition of it and (7.1.3) OK [[T0, . . . , Tm]]∼−→ OK ⊗F0OK → OK maps

Ti to 0. Here the OK-algebra structure of OK ⊗F0OK is as in (7.1.2). If r is an integer> 1, we have an isomorphism of K-affinoid algebras

K〈T0/πr, . . . , Tm/π

r〉 ∼−→ RrK . (7.4.2)

The normalized integral model XrK is defined over OK , and we have an isomorphism

OK〈T0/πr, . . . , Tm/π

r〉 ∼−→ (OK ⊗F0OK)〈IK/πr〉 = RrK,OK . (7.4.3)

Hence the closed fiber Xr

K is isomorphic to the affine scheme

SpecF [T0/πr, . . . , Tm/π

r].

In general, for any rational number r > 0, the K-affinoid variety XrK is isomorphic to

Dm+1,(r) and the rigid space XK = ∪r>0XrK is isomorphic to Dm+1 (Subsection 5.2).

By ([3, 1.13, 2.4]), for any rational number r > 0, there exists a canonical isomor-phism X

r

K∼−→ Θ

(r)

F(4.10.5) which is compatible with the geometric monodromy on X

r

K

and the natural GK-action on Θ(r)

F(via its action on mr

K/mr+

K). If r is an integer, it is

constructed as follows. Firstly, we have a natural ring isomorphism∞⊕

i=0

I iK/Ii+1K ⊗OK m−irK /m−ir+1

K → RrK,OK/mKRr

K,OK , b⊗ c 7→ bc, (7.4.4)

by (7.4.1) and (7.4.3). Extending scalars, we have

Xr

K∼−→ Spec

( ∞⊕

i=0

I iK/Ii+1K ⊗OK m−ir

K/m−ir+

K

). (7.4.5)

Then, from ([3, 1.14.3, 2.4]), we have an isomorphism of free OK-modules

Ω1OK/F0

→ IK/I2K , dt 7→ 1⊗ t− t⊗ 1, (7.4.6)

327. ISOGENIES ASSOCIATED TO EXTENSIONS OF TYPE (II): THE EQUAL

CHARACTERISTIC CASE

which induces the isomorphism Xr

K → Θ(r)

F.

7.5. Let L be a finite Galois extension of K of group G and conductor r > 1. By([2, 7.2]), the natural action of G on OL ⊗F0OK induces an OK-linear action of G on Xr

L

making it an étale G-torsor over XrK . In particular, Xr

L and Xr

L are étale G-torsors ofXrK and X

r

K , respectively. The geometric monodromy action of GK on Xr

L (Subsection6.6) commutes with the action of G. Let X

r

L,0 be a connected component of Xr

L. Thestabilizers of Xr

L,0 via these two actions are Gr and GrK , respectively. Then, we get an

isomorphism Gr ∼−→ Aut(Xr

L,0/Xr

K) and a surjection GrK → Aut(X

r

L,0/Xr

K) which impliesthat Gr is commutative (cf. [3, 2.15.1]). Composing with X

r

K∼−→ Θ

(r)

F, the étale covering

Xr

L,0 → Θ(r)

Finduces a surjective homomorphism ([3, 2.15.1])

πab1 (Θ

(r)

F)→ GrrGK → Gr.

7.6. In the rest of this section, let L/K be a finite Galois extension of type (II) andwe take again the notation and assumptions of subsections 5.1 and 5.2. By (7.2.1) andthe proof of ([3, 1.6]), for any rational number r > 0, we have an isomorphism

RrK ⊗OK ⊗F0

OK (OL ⊗F0OK)∼−→ Rr

L. (7.6.1)

It induces, for any rational numbers r > r′ > 0, an isomorphism

RrK ⊗Rr′K R

r′L∼−→ Rr

L,

which gives a Cartesian diagram of rigid spaces

XrL

// XL

XrK

// XK

(7.6.2)

where XK =⋃r>0X

rK and XL =

⋃r>0X

rL.

We put

f(T ) = T pn

+ (apn−1 ⊗ 1)T pn−1 + · · ·+ (a0 ⊗ 1) ∈ (OK ⊗F0OK)[T ].

From (7.2.1) and (7.6.1), we have a surjection

τL : RrK〈T 〉 → Rr

L, T 7→ h⊗ 1,

which induces an isomorphism that we denote abusively also by

τL : RrK〈T 〉/f(T )

∼−→ RrL. (7.6.3)

In other terms, we have a co-Cartesian diagram of homomorphisms of RrK-algebras

RrL Rr

K〈T 〉τLoo

RrK

OO

RrK〈T 〉

φ

OO

τKoo

(7.6.4)

where φ(T ) = f(T ) and τK(T ) = 0. Hence, taking the union of the K-affinoid varietiesassociated to each of the K-affinoid algebras in (7.6.4) for r ∈ Q>0, we have a Cartesian

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES33

diagram

XL

iL // XK ×D1

f

XKiK // XK ×D1

(7.6.5)

where iL, f and iK are the morphisms induced by τL, φ and τK .

7.7. In the following, for any 0 6 i 6 pn−1, we denote by αi the element ai−ai⊗1 ∈IK (Subsection 7.1). When the conductor c > 2, for each 1 6 i 6 pn − 1, v(ai) > 2(Lemma 5.6). Let a′i = π−2ai ∈ OK . We denote by α′i the element a′i − a′i ⊗ 1 ∈ IK andby β the element π − π ⊗ 1 ∈ IK . Then, we have

αi = (a′i − α′i)(2πβ − β2) + π2α′i.

Since α′i, β ∈ IK ⊂ πcRcK,OK , we have αi ∈ πc+1Rc

K,OK .When c = 2, we have p = 2, ]G = 2, ρ(c) = 1 and a′′1 = π−1a1 ∈ OK . We denote by

α′′1 the element a′′1 − a′′1 ⊗ 1 ∈ IK . Then we have

α1 = (a′′1 − α′′1)β + πα′′1.

Since α′′1, β ∈ πcRcK,OK , we have α1 ∈ πcRc

K,OK , and α1/πc = a′′1β/πc ∈ Rc

K,OK/πRc

K,OK .We put

f0(T ) =∑

06i6pn−1

(αi/πc) · T i ∈ Rc

K,OK [T ].

We havef(T ) = f(T )−

∑

06i6pn−1

αiTi = f(T )− πcf0(T ).

In the rest of this section, we fix an embedding L→ K. Recall that we put ](Gc) = ps

(Subsection 5.1).

Proposition 7.8. The K-affinoid XcL has ](G/Gc) = pn−s geometric connected

components. Let σ1, . . . , σpn−s be liftings of all the elements of G/Gc in G. We have

iL(XcL) ⊆

∐

16j6pn−sXcK × (σj(h) +D1,(ρ(c))) ⊆ XK ×D1, (7.8.1)

and each disc of the disjoint union contains exact one geometric connected componentof Xc

L.

Proof. By the Cartesian diagrams (7.6.2) and (7.6.5), we have

iL(XcL) = f−1(iK(Xc)) ⊆ Xc

K ×D1 ⊆ XK ×D1.

Taking in account the isomorphisms (7.4.2) and (7.4.3), for any point

(t0, . . . , tm, t) ∈ XcK ×D1 −

∐

16k6pn−sXcK × (σk(h) +D1,(ρ(c))),

we have v(f(t)) < c and v((αi/πc)(t0, . . . , tm)ti) > 0. Hence v(f(t) −

πcf0(t0, . . . , tm, t)) < c which means f(t0, . . . , tm, t) = (t1, . . . , tm, f(t0, . . . , tm, t)) 6∈iK(Xc

K). Thus (7.8.1) holds. By the proof of ([3, 2.15]), XcL has exactly pn−s

geometric connected components. Moreover, for any 1 6 j 6 pn−s, f(σj(h)) −πcf0(0, . . . , 0, σj(h)) = 0, hence each disc Xc

K × (σj(h) + D1,(ρ(c))) contains at least onegeometric connected component of Xc

L.

347. ISOGENIES ASSOCIATED TO EXTENSIONS OF TYPE (II): THE EQUAL

CHARACTERISTIC CASE

In the following, we denote by Xc

L,0 the connected component of Xc

L correspondingto the connected component Xc

L,0 of XcL containing (0, . . . , 0, h) ∈ Xc

K×D1 defined overL.

Proposition 7.9. There exists a canonical Cartesian diagram

Xc

L,0

ν // A1F

fc

Θ(c)

F

µ // A1F

(7.9.1)

where fc is defined in (5.4.1), such that if ξ is the canonical coordinate of A1F, we have

µ∗(ξ) =

da0 ⊗ π−c, if c > 2,(a′′1hdπ + da0)⊗ π−2, if c = 2.

Moreover, for any σ ∈ Gc, the following diagram

Xc

L,0

σ

ν // A1F

dσ

Xc

L,0ν // A1

F

(7.9.2)

where d∗σ(ξ) = ξ − uσ (Subsection 5.3), is commutative.

Proof. We consider the K-affinoid algebra RcK (resp. Rc

L) as a sub-ring of theL-affinoid algebra Rc

K ⊗K L (resp. RcL ⊗K L). By proposition 7.8, we have

XcL,0 = i−1

L (XcK × (h+D1,(ρ(c)))) ∩Xc

L.

Hence XcL,0 is presented by the L-affinoid algebra

(RcL ⊗K L)〈T ′〉/(πρ(c)T ′ + h− h⊗ 1). (7.9.3)

By the isomorphism (7.6.3), (7.9.3) is isomorphic to

(RcK ⊗K L)〈T, T ′〉/(f(T ), πρ(c)T ′ + h− T ),

which, after eliminating T by the relation πρ(c)T ′ + h− T = 0, is

(RcK ⊗K L)〈T ′〉/(f(πρ(c)T ′ + h)). (7.9.4)

In both cases, by proposition 5.4 and subsection 7.7, we have

f(πρ(c)T ′ + h)/πc ∈ RcK,OL〈T ′〉,

f(πρ(c)T ′ + h)/πc+1 /∈ RcK,OL〈T ′〉.

Then the image of RcK,OL〈T ′〉 by the canonical surjection

(RcK ⊗K L)〈T ′〉 → (Rc

K ⊗K L)〈T ′〉/(f(πρ(c)T ′ + h))

isRcK,OL〈T ′〉/(f(πρ(c)T ′ + h)/πc). (7.9.5)

Extending the scalars from OL to F , we obtain the following F -algebra:(i) if c > 2,

(RcK,OL ⊗OL F )[T ′]/(fc(T

′)− α0/πc) ; (7.9.6)(ii) if c = 2,

(RcK,OL ⊗OL F )[T ′]/(f2(T ′)− (α0 + a′′1hβ)/π2). (7.9.7)

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES35

From isomorphisms (7.4.4), (7.4.6) and the canonical exact sequence (4.10.2), we knowthat when c > 2 (resp. c = 2), α0/πc (resp. (α0 + a′′1hβ)/π2) is a non-zero linear termin F ⊗OL Rc

K,OL . Hence (7.9.6) and (7.9.7) are all reduced. Then, by ([2, 4.1]),

Spf(RcK,OL〈T ′〉/(f(πρ(c)T ′ + h)/πc))

is the normalized integral model of XcK,0 defined over OL. Hence X

c

L,0 is defined by theF -algebra (7.9.6) (resp. (7.9.7)) when c > 2 (resp. c = 2). We put

ν : Xc

L,0 → A1F

= Spec(F [ξ]), ν∗(ξ) = T ′.

It follows form the isomorphism Xc

K∼−→ Θ

(c)

F(Subsection 7.4) that (7.9.1) is Cartesian.

For any σ ∈ Gc, let yσ(x) be a polynomial brxr + · · ·+ b0 ∈ OK [x], where r 6 pn− 1,such that yσ(h) = (h− σ(h))/πρ(c) ∈ OL. We denote by yσ the polynomial

yσ(x) = (br ⊗ 1)xr + · · ·+ (b0 ⊗ 1) ∈ RcK [x].

The action of σ on RcK〈T 〉/f(T ) (isomorphic to Rc

L (7.6.3)) is given by : T 7→ T −(πρ(c) ⊗ 1)y(T ). Hence the action of σ on (7.9.4) is given by

T ′ 7→ T ′ − yσ(πρ(c)T ′ + h)− ((πρ(c) ⊗ 1− πρ(c))/πρ(c))yσ(πρ(c)T ′ + h)

and the induced action on (7.9.5) is given by the same formula. Since πρ(c)⊗ 1− πρ(c) ∈πcRc

K,OK and c > ρ(c), the reduction of (πρ(c) ⊗ 1− πρ(c))/πρ(c) to the geometric specialfiber is 0. For any 0 6 j 6 r, bj⊗1−bj ∈ πcRc

K,OK . Then, the reduction of yσ(πρ(c)T ′+h)to the geometric special fiber is (Subsection 5.3)

yσ(πρ(c)T ′ + h) = yσ(πρ(c)T ′ + h) = yσ(h) = uσ.

Hence, diagram (7.9.2) is commutative.

8. Isogenies associated to extensions of type (II): the unequal characteristiccase

8.1. In this section, we assume that K has characteristic 0 and that the residuefield F of OK is of finite type over a perfect field F0. Let K0 be the fraction field of thering of Witt vectors W (F0) = OK0 considered as a subfield of K. We denote by m thedimension of the F -vector space Ω1

F , which is finite.

8.2. Let L be an object of FÉ/K . We call an OK0-presentation of Cartier type of OLa pair (AL, j : AL → OL), where AL is a complete semi-local Noetherian OK0-algebraformally smooth of relative dimension m+1 over OK0 and j a surjective homomorphismof OK0-algebra inducing an isomorphism AL/mAL

∼−→ OL/mL such that the kernel of jis generated by a non-zero divisor of AL.

Let L1, L2 be two objects of FÉ/K and (AL1 , j1 : AL1 → OL1), (AL2 , j2 : AL2 → OL2)two OK0-presentations of Cartier type. A morphism (g,g) from (AL1 , j1) to (AL2 , j2)is a pair of OK0-homomorphisms g : OL1 → OL2 and g : AL1 → AL2 such that thediagram

AL1

g

j1 // OL1

g

AL2

j2 // OL2

(8.2.1)

is commutative. We say that (g,g) is finite and flat if g is finite and flat and if thediagram (8.2.1) is co-Cartesian.

Proposition 8.3. ([3, 2.7, 2.8]).

368. ISOGENIES ASSOCIATED TO EXTENSIONS OF TYPE (II): THE UNEQUAL

CHARACTERISTIC CASE

(i) Any object of FÉ/K admits an OK0-presentation of Cartier type.(ii) Let g : L1 → L2 be a morphism of FÉ/K, and (AL1 , j1), (AL2 , j2) two OK0-

presentations of Cartier type. Then there exist a morphism g : AL1 → AL2

such that (g,g) is a morphism of OK0-presentations of Cartier type.(iii) Let g : L1 → L2 be a morphism of FÉ/K and (g,g) a morphism between OK0-

presentations of Cartier type (AL1 , j1) and (AL2 , j2). If a uniformizer π0 of K0

is not a uniformizer of any factor of OL1, then (g,g) is finite and flat.

8.4. Let L be an object of FÉ/K , and (AL, j : AL → OL) an OK0-presentation ofCartier type. We denote by (AL/mr

AL)⊗OK0OK the formal completion of (AL/mr

AL)⊗OK0OK relatively to the kernel of the homomorphism

(AL/mrAL)⊗OK0

OK → OL/mrOL , a⊗ b 7→ ab, (8.4.1)

and by AL ⊗OK0OK the projective limit

AL ⊗OK0OK = lim←−

r

((AL/mrAL)⊗OK0

OK). (8.4.2)

We will always consider AL ⊗OK0OK as an OK-algebra by the homomorphism

OK → AL ⊗OK0OK , u 7→ 1⊗ u,

(in the following, we always abbreviate 1 ⊗ u by u) and we will consider it as an AL-algebra by

AL → AL ⊗OK0OK , v 7→ v ⊗ 1.

There is a canonical surjective homomorphism

AL ⊗OK0OK → OL, (8.4.3)

induced by the surjections (8.4.1). We denote by IL its kernel.

Proposition 8.5. ([3, 2.9]). Let L be an object of FÉ/K, and (AL, j : AL → OL)an OK0-presentation of Cartier type. Then,

(i) The OK-algebra AL ⊗OK0OK is formally of finite type and formally smooth over

OK and the morphism AL ⊗OK0OK/mAL ⊗OK0

OK → OL/mOL (8.4.3) is an iso-

morphism.(ii) Let L′ be another object in FÉ/K and (AL′ , j′ : AL′ → OL′) an OK0-presentation

of Cartier type. If a uniformizer π0 is not a uniformizer of any factor of OL,then, any morphism (AL, j)→ (AL′ , j′) induces an isomorphism

AL′ ⊗AL (AL ⊗OK0OK)

∼−→ AL′ ⊗OK0OK . (8.5.1)

Proof. Part (i) is proved in ([3, 2.9]). For part (ii), we may assume L and L′ arefields. We denote by e the ramification index of the extension L′/L. For any integerr > 1, we have the following canonical commutative diagram

ALpr1 //

(AL/mrAL)⊗OK0

OK

(8.4.1)// OL/mr

L

AL′

pr′1 // (AL′/mrALAL′)⊗OK0

OKgL′ // OL′/mer

L′

such that each square is co-Cartesian. We denote by (AL′/mrALAL′)⊗OK0

OK the formalcompletion of (AL′/mr

ALAL′) ⊗OK0OK relatively to the kernel of gL′ . Since AL is a

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES37

Noetherian local ring, by proposition 8.3(iii) and Nakayama’s lemma, AL′ is a finite freeAL-module. Then, we have

AL′ ⊗AL (AL/mrAL)⊗OK0

OK ∼−→ (AL′/mrALAL′)⊗OK0

OK .After taking projective limit on both sides, we obtain

AL′ ⊗AL (AL ⊗OK0OK)

∼−→ lim←−r

((AL′/mrALAL′)⊗OK0

OK). (8.5.2)

By the proof of ([3, 2.7.3]), we obtain that meAL′ ⊆ mALAL′ ⊆ mAL′ . Hence, for any

integer r > 1, we have two surjections

(AL′/merAL′ )⊗OK0

OK (AL′/mrALAL′)⊗OK0

OK (AL′/mrAL′ )⊗OK0

OK ,which imply

lim←−r

((AL′/mrALAL′)⊗OK0

OK)∼−→ AL′ ⊗OK0

OK . (8.5.3)

Combining (8.5.2) and (8.5.3), we get (ii).

8.6. Let L be an object of FÉ/K , and (AL, j : AL → OL) an OK0-presentation ofCartier type. We will introduce objects analogue of those defined in §7, and denotethem by the same notation. For any rational number r > 0 and integer numbers s, t > 0such that r = t/s, we denote by Rr

L the K-affinoid algebra

RrL = (AL ⊗OK0

OK)〈IsK/πt〉 ⊗OK K,by Xr

L = Sp(RrL) the tubular neighborhood of thickening r of the immersion

Spf(OL)→ Spf(AL ⊗OK0OK),

which is smooth over K ([3, 1.7]). By proposition 6.3, there exists a finite separableextension K ′ of K such that the normalized integral model of Xc

L is defined over K ′(Subsection 6.4). We denote by Rr

L,OK′ the supremum norm unit ball of RrL ⊗K K ′

(6.3.1), by XrL the normalized integral model of Xr

L over OK and by Xr

L the special fiberof Xr

L.

8.7. In the following of this section, we assume that p is not a uniformizer of K. By([3] 1.14.3), there is an isomorphism of OK-algebras

OK [[T0, . . . , Tm]]∼−→ AK ⊗OK0

OK , (8.7.1)

such that the composition of it and (8.4.3) OK [[T0, . . . , Tm]]→ AK ⊗OK0OK → OK maps

Ti to 0. If r is an integer > 1, we have an isomorphism of K-affinoid algebras

K〈T0/πr, . . . , Tm/π

r〉 ∼−→ RrK . (8.7.2)

The normalized integral model XrK is defined over OK , and we have an isomorphism

OK〈T0/πr, . . . , Tm/π

r〉 ∼−→ (AK ⊗OK0OK)〈IK/πr〉 = Rr

K,OK . (8.7.3)

Hence the geometric closed fiber Xr

K is isomorphic to the affine scheme

SpecF [T0/πr, . . . , Tm/π

r].

In general, for any rational number r > 0, the K-affinoid variety XrK is isomorphic to

Dm+1,(r) and the rigid space XK =⋃r>0X

rK is isomorphic to Dm+1 (Subsection 5.2).

By ([3, 2.11.2]), we have an isomorphism

(IK/I2K)⊗OK F → Ω1

OK/OK0⊗OK F, (8.7.4)

388. ISOGENIES ASSOCIATED TO EXTENSIONS OF TYPE (II): THE UNEQUAL

CHARACTERISTIC CASE

such that for any x ∈ OK and x a lifting in AK , the image of (1⊗ x− x⊗ 1)⊗1 is dx⊗1.From ([3, 1.14.3, 2.11.2]), for any rational number r > 0, the inverse of (8.7.4) gives anisomorphism X

r

K∼−→ Θ

(r)

F. When r is an integer, the construction of the isomorphism is

similar to the equal characteristic case (Subsection 7.4).

Remark 8.8. From (8.7.4), we notice that for any element x ∈ ker(AK → OK), theclass (x⊗ 1)⊗1 vanishes in (IK/I

2K)⊗OKF . It is equivalent to say that x⊗1 ∈ I2

K+πIK .

8.9. Let L be a finite Galois extension of K of group G and conductor c > 1. Let(g,g) be a finite and flat morphism from (AK , jK : AK → OK) to (AL, jL : AL → OL)

(Subsection 8.2). By (8.5.1), g induces a finite flat morphism g ⊗ id : AK ⊗OK0OK →

AL ⊗OK0OK . Hence, for any rational number r > 0, it gives a morphism of smooth

K-affinoid varieties XrL → Xr

K ([3, 1.6]) which induces morphisms XrL → Xr

K andXr

K → Xr

L. For any σ ∈ G, there is a morphism gσ making the following diagramcommutative (Proposition 8.3(iii))

ALgσ

jL // OLσ

AL

jL // OL.

(8.9.1)

The pair (σ,gσ) induces automorphisms of XrL, Xr

L and Xr

L. Notice that, gσ is notunique in general and may not be an AK-homomorphism. Hence the automorphismsof Xr

L, XrL and X

r

L induced by all possible gσ may not be uniquely determined byσ ∈ G. Luckily, by ([3, 2.13]), the induced automorphism of Xc

L is Xc

K-invariant andindependent of the choice of gσ. Hence X

c

L → Xc

K is a finite étale G-torsor ([3, 1.16.2]).The geometric monodromy action of GK on X

c

L commutes with the action of G. Let Xc

L,0

be a connected component of Xc

L. The stabilizers of Xc

L,0 via these two actions are Gc

and GcK , respectively ([3, 2.15.1]). Then, we get an isomorphism Gc ∼−→ Aut(X

c

L,0/Xc

K)

and a surjection GcK → Aut(X

c

L,0/Xc

K) which imply that Gc is commutative (cf. [3,2.15.1]). Composing with X

r

K∼−→ Θ

(r)

F, the étale covering Xc

L,0 → Θ(r)

Finduces a surjective

homomorphism ([3, 2.15.1])

πab1 (Θ

(r)

F)→ GrcGK → Gc.

8.10. In the following of this section, we assume that the finite Galois extensionL/K is of type (II) and we take again the notation and assumptions of subsections5.1 and 5.2. Let (g,g) be a finite and flat morphism as in subsection 8.9. Let h be alifting of h ∈ OL in AL. Since AK is a Noetherian local ring, by proposition 8.3(iii) andNakayama’s lemma, we have that AL is a finite free AK-module of rank ]G and thatAL = AK [h]. Let

f(T ) = T pn

+ apn−1Tpn−1 + · · ·+ a0 ∈ AK [T ],

be a lifting of f [T ] ∈ OK [T ] such that h is a zero. We have an isomorphism

AK [T ]/(f(T ))∼−→ AL, T 7→ h. (8.10.1)

By (8.5.1) and the proof of ([3, 1.6]), we have an isomorphism

RrK ⊗AK ⊗OK0

OK (AL ⊗OK0OK)

∼−→ RrL. (8.10.2)

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES39

It induces, for any rational numbers r > r′ > 0, an isomorphism

RrK ⊗Rr′K R

r′L∼−→ Rr

L,

which gives a Cartesian diagram of rigid spaces

XrL

// XL

XrK

// XK

(8.10.3)

where XK =⋃r>0X

rK and XL =

⋃r>0X

rL. We put

f(T ) = T pn

+ (apn−1 ⊗ 1)T pn−1 + · · ·+ (a0 ⊗ 1) ∈ (AK ⊗OK0

OK)[T ].

From (8.5.1) and (8.10.2), we have a surjection

τL : RrK〈T 〉 → Rr

L, T 7→ h⊗ 1,

which induces an isomorphism that we denote abusively also by

τL : RrK〈T 〉/f(T )

∼−→ RrL. (8.10.4)

In other terms, we have a co-Cartesian diagram of homomorphisms of RrK-algebras

RrL Rr

K〈T 〉τLoo

RrK

OO

RrK〈T 〉,

φ

OO

τKoo

(8.10.5)

where φ(T ) = f(T ) and τK(T ) = 0. Hence, taking the union of the K-affinoid varietiesassociated to each of the K-affinoid algebras in (8.10.5) for r ∈ Q>0, we obtain aCartesian diagram

XL

iL // XK ×D1

f

XKiK // XK ×D1,

(8.10.6)

where iL, f and iK are the morphisms induced by τL, φ and τK .

8.11. In the following, for any 0 6 i 6 pn − 1, we denote by αi the element ai −ai ⊗ 1 ∈ IK and fix π ∈ AK a lifting of π ∈ OK . When the conductor c > 2, for each1 6 i 6 pn − 1, v(ai) > 2 (Lemma 5.6). Let a′i = π−2ai ∈ OK and a′i ∈ AK a liftingof a′i. Then we have ai = π2a′i + yi, where yi ∈ ker(AK → OK). We denote by α′i theelement a′i − a′i ⊗ 1 ∈ IK and by β the element π − π ⊗ 1 ∈ IK . Then, we have

αi = (a′i − α′i)(2πβ − β2) + π2α′i + yi ⊗ 1.

Since α′i, β ∈ IK ⊂ πcRcK,OK and yi ⊗ 1 ∈ I2

K + πIK ⊂ πc+1RcK,OK (Remark 8.8), we

have αi ∈ πc+1RcK,OK . When c = 2, we have p = 2, ]G = 2, deg f = 2 and ρ(c) = 1. Let

a′′1 ∈ AK be a lifting of a′′1 = π−1a1. We have α1 = πa′′1 + z1, where z1 ∈ ker(AK → OK).We denote by α′′1 the element a′′1 − a′′1 ⊗ 1 ∈ IK . Then we have

α1 = (a′′1 − α′′1)β + πα′′1 + z1 ⊗ 1.

Since α′′1, β ∈ πcRcK,OK and z1 ⊗ 1 ∈ I2

K + πIK ⊂ πc+1RcK,OK , we have α1 ∈ πcRc

K,OK ,and α1/πc = a′′1β/π

c ∈ RcK,OK

/πRc

K,OK .

408. ISOGENIES ASSOCIATED TO EXTENSIONS OF TYPE (II): THE UNEQUAL

CHARACTERISTIC CASE

Putf0(T ) =

∑

06i6pn−1

(αi/πc) · T i ∈ Rc

K,OK [T ].

We havef(T ) = f(T )−

∑

06i6pn−1

αiTi = f(T )− πcf0(T ).

In the following, we fix an embedding L → K. Recall that we put ](Gc) = ps

(Subsection 5.1).

Proposition 8.12. The K-affinoid XcL has ](G/Gc) = pn−s geometric connected

components. Let σ1, . . . , σpn−s be liftings of all the elements of G/Gc in G. We have

iL(XcL) ⊆

∐

16j6pn−sXcK × (σj(h) +D1,(ρ(c))) ⊆ XK ×D1.

Proof. The proof is the same as in the equal characteristic case (Proposition 7.8).

In the following, we denote by Xc

L,0 the connected component of Xc

L correspondingto the connected component Xc

L,0 of XcL containing (0, . . . , 0, h) ∈ Xc

K×D1 defined overL.

Proposition 8.13. There exists a canonical Cartesian diagram

Xc

L,0

ν // A1F

fc

Θ(c)

F

µ // A1F,

(8.13.1)

where fc is defined in (5.4.1) and if ξ is the canonical coordinate of A1F, we have

µ∗(ξ) =

da0 ⊗ π−c, if c > 2,(a′′1hdπ + da0)⊗ π−2, if c = 2.

Moreover, for any σ ∈ Gc, the following diagram

Xc

L,0

σ

ν // A1F

dσ

Xc

L,0ν // A1

F,

(8.13.2)

where d∗σ(ξ) = ξ − uσ (Subsection 5.3), is commutative.

Proof. We consider the K-affinoid algebra RcK (resp. Rc

L) as a sub-ring of theL-affinoid algebra Rc

K ⊗K L (resp. RcL ⊗K L). By (8.12), we have

XcL,0 = i−1

L (XcK × (h+D1,(ρ(c)))) ∩Xc

L.

Hence XcL,0 is presented by the L-affinoid algebra

(RcL ⊗K L)〈T ′〉/(πρ(c)T ′ + h− h⊗ 1). (8.13.3)

By the isomorphism (8.10.4), (8.13.3) is isomorphic to

(RcK ⊗K L)〈T, T ′〉/(f(T ), πρ(c)T ′ + h− T ),

which, after eliminating T by the relation πρ(c)T ′ + h− T = 0, is

(RcK ⊗K L)〈T ′〉/(f(πρ(c)T ′ + h)). (8.13.4)

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES41

In both cases, by proposition 5.4 and subsection 8.11, we have

f(πρ(c)T ′ + h)/πc ∈ RcK,OL〈T ′〉,

f(πρ(c)T ′ + h)/πc+1 /∈ RcK,OL〈T ′〉.

Then the image of RcK,OL〈T ′〉 in (8.13.4) through the canonical surjective map

(RcK ⊗K L)〈T ′〉 → (Rc

K ⊗K L)〈T ′〉/(f(πρ(c)T ′ + h)),

isRcK,OL〈T ′〉/(f(πρ(c)T ′ + h)/πc). (8.13.5)

Extending scalars from OL to F , we obtain the following F -algebra:(i) if c > 2,

(RcK,OL ⊗OL F )[T ′]/(fc(T

′)− α0/πc). (8.13.6)(ii) if c = 2,

(RcK,OL ⊗OL F )[T ′]/(f2(T ′)− (α0 + a′′1hβ)/π2). (8.13.7)

From the isomorphism (8.7.4) and the canonical exact sequence (4.10.3), we know thatwhen c > 2 (resp. c = 2), α0/πc (resp. (α0 + a′′1hβ)/π2) is a non-zero linear term inRcK,OL ⊗OL F . Hence (8.13.6) and (8.13.7) are all reduced. Then, by ([2, 4.1]),

Spf(RcK,OL〈T ′〉/(f(πρ(c)T ′ + h)/πc))

is the normalized integral model of XcK,0 defined over OL. Hence X

c

L,0 is defined by theF -algebra (8.13.6) (resp. (8.13.7)) when c > 2 (resp. c = 2). We put

ν : Xc

L,0 → A1F

= Spec(F [ξ]), ν∗(ξ) = T ′.

It follows form the isomorphism Xc

K → Θ(c)

Fthat (8.13.1) is Cartesian.

For any σ ∈ Gc, let yσ(x) = brxr + · · · + b0 ∈ OK [x] be a polynomial such that

yσ(h) = (h− σ(h))/πρ(c) ∈ OL. We denote by yσ(x) = brxr + · · · + b0 a lifting of yσ(x)

in AK [x] and by y(x) the polynomial

y(x) = (br ⊗ 1)xr + · · ·+ (b0 ⊗ 1) ∈ (AK ⊗OK0OK)[x].

Let gσ : AL → AL be a homomorphism as in (8.9.1). We denote by gσ the inducedmorphism of gσ on (8.13.5). By (8.10.1), we have

ker(AL → OL) =

pn−1⊕

i=0

ker(AK → OK)hi.

Hence, we have gσ(h) = h− πρ(c)yσ(h) + ε(h), where ε is a polynomial with coefficientsin ker(AK → OK). Then, we have

gσ(T ′) = T ′ − yσ(πρ(c)T ′ + h) + ∆(T ′),

where

∆(T ′) = −((πρ(c) ⊗ 1− πρ(c))/πρ(c))yσ(πρ(c)T ′ + h) + ε(πρ(c)T ′ + h)/πρ(c),

and ε is a polynomials with coefficients in J = x⊗1 ∈ A ⊗OK0OK ; x ∈ ker(AK → OK).

Since J ⊆ πc+1RK,OK (Remark 8.8), πρ(c) ⊗ 1 − πρ(c) ∈ πcRK,OK and c > ρ(c), it iseasy to see that the reduction of ∆(T ′) to X

c

L,0 is zero. For any 0 6 j 6 r, we havebj ⊗ 1− bj ∈ πcRc

K,OK . Then

yσ(πρ(c)T ′ + h) = yσ(πρ(c)T ′ + h) = yσ(h) = uσ.

42 9. THE REFINED SWAN CONDUCTOR OF AN EXTENSION OF TYPE (II)

Hence, by ([3, 2.13]), the diagram (8.13.2) is commutative.

9. The refined Swan conductor of an extension of type (II)

9.1. In this section, we assume either that K has characteristic p or that it has char-acteristic 0 and that p is not a uniformizer of K. Let L be a finitely generated extensionof K of type (II) and we take again the notation and assumptions of subsections 5.1,5.2, 7.9 and 8.13.

Proposition 9.2. The fibre product Xc

L,0 ×Θ(c)

F

Ξ(c)

F(4.10.4) is a connected affine

scheme.

Proof. The image of da0 ⊗ 1 and (a′′1hdπ + da0) ⊗ 1 by the canonical map fromΩ1OK/F0

⊗OK F (resp. Ω1OK/OK0

⊗OK F ) to Ω1F ⊗F F is da0 ⊗ 1, which is a non-zero

element. So we have a Cartesian diagram

Xc

L,0 ×Θ(c)

F

Ξ(c)

F

// A1F

fc

Ξ

(c)

F

µ′ // A1F

(9.2.1)

where µ′∗(ξ) = da0 ⊗ π−c. Since da0 ⊗ π−c is a non-zero linear term in the affine spaceΞ

(c)

F, Xc

L,0 ×Θ(c)

F

Ξ(c)

Fis connected.

9.3. Proof of proposition 5.7. By ([3, 5.13]), both in the equal and unequal charac-teristic case, we have a commutative diagram

πab1 (Θ

(c)

F ,log) //

γ1

πab1 (Θ

(c)

F)

γ2

Gc

log Gc.

(9.3.1)

The surjection γ1 factors through πalg1 (Θ

(c)

F ,log) (Theorem 4.11). By propositions 7.9 and

8.13, γ2 also factors through πalg1 (Θ

(c)

F). Combining (9.3.1) and the following canonical

commutative diagram

πab1 (Θ

(c)

F ,log) //

πab1 (Ξ

(c)

F)

// πab1 (Θ

(c)

F)

πalg1 (Θ

(c)

F ,log) // πalg

1 (Ξ(c)

F) // πalg

1 (Θ(c)

F),

we obtain thatπalg

1 (Θ(c)

F ,log) //

πalg1 (Ξ

(c)

F) // πalg

1 (Θ(c)

F)

Gc

log Gc

(9.3.2)

is commutative. The composition of morphisms πalg1 (Ξ

(c)

F) → πalg

1 (Θ(c)

F) → Gc corre-

sponds to the isogeny Xc

L,0 ×Θ(c)

F

Ξ(c)

F→ Ξ

(c)

F(cf. (9.2.1)). Hence, by (9.3.2), we have a

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES43

commutative diagram

Hom(πalg1 (Ξ

(c)

F),Fp) //

Ω1F ⊗F m−c

K/m−c+

K

Hom(Gc,Fp) //

66

Hom(πalg1 (Θ

(c)

F ,log),Fp) // Ω1

F (log)⊗F m−cK/m−c+

K,

(9.3.3)

which concludes proposition 5.7.

9.4. Proof of theorem 5.9. Since the surjection πalg1 (Ξ

(c)

F) → Gc is obtained by

pulling-back the isogeny fc : A1F→ A1

Fby µ′ (cf. (9.2.1)), it is an étale Gc-torsor with

the action of Gc given by dσ for σ ∈ Gc (7.9.2), (8.13.2). With notation in subsection5.8, we denote by fc,χ(ξ) the polynomial

fc,χ(ξ) =

( ∏

σ∈G−Gcuσ

)(ξp − fp−1

c,χ (uτ )ξ) ∈ F [ξ].

Observe that fc,χ(fc,χ(ξ)) = fc(ξ), hence the isogeny fc is the composition of two isoge-nies

A1F

fc,χ−−→ A1F

fc,χ−−→ A1F.

For any σ ∈ kerχ, f ∗c,χ(ξ − uσ) = f ∗c,χ(ξ), i.e. fc,χdσ = fc,χ. Hence the isogeny fc,χ :

A1F→ A1

Fis an étale (Gc/ kerχ)-torsor. Then, the surjection πalg

1 (Ξ(c)

F) → Gc χ−→ Fp

corresponds to the pull-back of fc,χ by µ′ and the Fp-action on this torsor is given by1∗ : ξ 7→ ξ − fc,χ(uτ ). We have the following Cartesian diagram

Fp

id

φ // A1F

λ2

fc,χ // A1F

λ1

Fp // A1

F

L // A1F

(9.4.1)

where L denotes the Lang’s isogeny defined by L∗(ξ) = ξp − ξ. The morphisms λ1, λ2

and φ are given as follows

λ∗1(ξ) = −ξ/( ∏

σ∈G−Gcuσ

)fpc,χ(uτ ),

λ∗2(ξ) = −ξ/fc,χ(uτ ),

φ(1) = −fc,χ(uτ ).

The sign is chosen in order that, for any σ ∈ Gc, the translation by φ(χ(σ)) is inducedby dσ. Consequently, πalg

1 (Ξ(c)

F) → Gc χ−→ Fp corresponds to the pull-back of L by λ1µ

′.Hence the image of χ ∈ Hom(Gc,Fp) in Ω1

F ⊗F m−cK/m−c+

K(9.3.3) is

−da0 ⊗π−c(∏

σ∈G−Gc uσ)fpc,χ(uτ )

∈ Ω1F ⊗F m−c

K/m−c+

K.

Then the theorem follows from (9.3.3).

44 10. COMPARISON OF KATO’S AND ABBES-SAITO’S CHARACTERISTIC CYCLES

10. Comparison of Kato’s and Abbes-Saito’s characteristic cycles

10.1. In this section, let L be a finite Galois extension of K of type (II) and we takeagain the notation and assumptions of subsections 5.1 and 5.2. Let C be an algebraicallyclosed field of characteristic zero. We fix a non-trivial character ψ0 : Fp → C×. Anycharacter χ : Gc → C× factors uniquely through Gc → Fp

ψ0−→ C×. We denote by χ theinduced character Gc → Fp.

Proposition 10.2. Let χ : G→ C× be a character of G such that its restrictionto Gc is non-trivial. Let τ ∈ Gc be a lifting of 1 ∈ Fp in Gc through χ : Gc → Fp. ThenKato’s Swan conductor with differential values swψ0(1)(χ) is given by (Subsection 5.3),(Subsection 5.8)

swψ0(1)(χ) = [πc] + [−fpc,χ(uτ )] +∑

σ∈G−Gc[uσ]− [da0] ∈ SK,L.

Proof. By definition (3.13.1), we have

swψ0(1)(χ) =∑

σ∈G−1([h− σ(h)]− [dh])⊗ (1− χ(σ)) +

∑

r∈F×p

[r]⊗ ψ0(r)(10.2.1)

=∑

σ∈Gc−1[h− σ(h)]⊗ (1− χ(σ)) +

∑

r∈F×p

[r]⊗ ψ0(r) +

∑

σ∈G−Gc[h− σ(h)]−

∑

σ∈G−Gc[h− σ(h)]⊗ χ(σ)−

∑

σ∈G−1[dh]⊗ (1− χ(σ)).

Choose an Fp-basis τ1 = τ, τ2, . . . , τs of Gc such that χ(τ1) = 1 ∈ Fp and, that for any2 6 j 6 s, χ(τj) = 0. Then, by lemma 5.5, we have

∑

σ∈Gc−1[h− σ(h)]⊗ (1− χ(σ)) +

∑

r∈F×p

[r]⊗ ψ0(r) (10.2.2)

= [πρ(c)]Gc ] +∑

j1,...,js∈Fsp−0[j1uτ1 + · · ·+ jsuτs ]⊗ (1− ψ0(j1)) +

∑

r∈F×p

[r]⊗ ψ0(r)

= [πρ(c)]Gc ] +∑

r∈F×p

[fc,χ(ruτ1)]⊗ (1− ψ0(r)) +∑

r∈F×p

[r]⊗ ψ0(r)

= [πρ(c)]Gc ] +∑

r∈F×p

([fc,χ(uτ1)] + [r])⊗ (1− ψ0(r)) +∑

r∈F×p

[r]⊗ ψ0(r)

= [πρ(c)]Gc ] +∑

r∈F×p

[fc,χ(uτ1)]⊗ (1− ψ0(r)) +∑

r∈F×p

[r]

= [πρ(c)]Gc ] + [−fpc,χ(uτ1)] ∈ SL/K .

Let σ1 = 1, σ2, . . . , σpn−s be liftings of all the elements of G/Gc in G and denote by Jthe set σ2, . . . , σpn−s. Observe that for any ς ∈ J and σ ∈ Gc, we have

[h− ςσ(h)] = [h− ς(h) + ς(h− σ(h))] = [h− ς(h)].

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES45

Hence∑

σ∈G−Gc[h− σ(h)]⊗ χ(σ) =

∑

ς∈J

∑

σ∈Gc[h− ςσ(h)]⊗ χ(ςσ) (10.2.3)

=∑

ς∈J

∑

σ∈Gc[h− ς(h)]⊗ χ(ς)χ(σ) = 0.

Moreover, by the isomorphism (3.4.1), we have∑

σ∈Gc−1[dh]⊗ (1− χ(σ)) = ]G[dh] = [da0] ∈ SK,L. (10.2.4)

Hence, combining (10.2.1), (10.2.2), (10.2.3) and (10.2.4), we obtain that

swψ0(1)(χ) = [πρ(c)]Gc ] + [−fpc,χ(uτ1)] +∑

σ∈G−Gc[h− σ(h)]⊗ 1− ]G[dh]

= [πc] + [−fpc,χ(uτ )] +∑

σ∈G−Gc[uσ]− [da0].

Lemma 10.3. Let M be a finite dimensional C-vector space with an irreduciblelinear action of G. Then, there exist a subgroup H of G satisfying Gc ⊆ H and a1-dimensional representation θ of H, such that M = IndGH θ.

Proof. SinceM is irreducible and G is nilpotent (hence super-solvable), there exista subgroup H of G and a 1-dimensional representation θ of H, such that M = IndGH θ([26, 8.5 Theorem 16]). Let ResGGcM =

⊕iMi be the canonical decomposition of

ResGGcM into isotypic Gc-representations (cf. [26, 2.6]). Since Gc is contained in thecenter of G, any σ ∈ G defines an automorphism of the Gc-representation ResGGcM . Inparticular, for any i, σ induces an automorphism of Mi. On the other hand, since M isirreducible, G permutes transitively the Mi’s. Hence ResGGcM is isotypic. By ([26, 7.3Propsition 22]), we have

ResGGcM = ResGGc IndGH θ =⊕

H\G/GcIndG

c

H∩Gc ResHH∩Gc θ. (10.3.1)

We notice that, if H ∩ Gc 6= Gc, since Gc = H ∩ Gc ⊕ Gc/H ∩ Gc, IndGc

H∩Gc ResHH∩Gc θis isomorphic to the tensor of the regular representation of Gc/H ∩ Gc with ResHH∩Gc θwhich is not isotypic.

Theorem 10.4. Assume that p is not a uniformizer of K. Let M be a finitedimensional C-vector space with a linear action of G. Then,

CCψ0(M) = KCCψ0(1)(M). (10.4.1)

Proof. From the definitions, we may assume that M is irreducible. We denote byc0 the unique slope ofM . By definitions and proposition 3.14, both sides of (10.4.1) willnot change if replacing G by G/Gc0+. Hence we may assume further that the uniqueslope of M is equal to c. By lemma 10.3, M = IndGH θ where H is a subgroup of Gcontaining Gc and θ is a character of H. Since the slope of M is c, the restriction ofθ to Gc is non-trivial (10.3.1). We notice that [G : H] = dimCM . Choose an Fp-basisτ1, . . . , τs of Gc such that θ(τ1) = 1 ∈ Fp and, for any 2 6 j 6 s, θ(τj) = 0. Letc′ = ρ(c) +

∑σ∈H−1 v(h − σ(h)). Since L/LH is still of type (II), we obtain that the

conductor of L/LH is c′, that Hc′ = Gc and, denoting by ρ′ the Herbrand function of

46 10. COMPARISON OF KATO’S AND ABBES-SAITO’S CHARACTERISTIC CYCLES

L/LH , that ρ′(c′) = ρ(c). Using proposition 10.2 for the group H and the representationθ, we have

swψ0(1)(θ) = [πc′] + [−fp

c,θ(uτ1)] +

∑

σ∈H−Hc′

[uσ]− ]H[dh]. (10.4.2)

Meanwhile, we have

−∑

σ∈G−H([dh]− [h− σ(h)]) = (]H − ]G)[dh] + [πc−c

′] +

∑

σ∈G−Huσ. (10.4.3)

Hence, combining (10.4.2), (10.4.3) and the induction formula for Kato’s Swan con-ductors (3.15.2), we have

swψ0(1)(M) = [G : H]

(swψ0(1)(θ)−

∑

σ∈G−H([dh]− [h− σ(h)]

)

= [G : H]

([πc] + [−fp

c,θ(uτ1)]− [da0] +

∑

σ∈G−Gc[uσ]

).

Hence Kato’s characteristic cycle KCCψ0(1)(M) is given by

KCCψ0(1)(M) =(−da0)⊗[G:H]

( (∏σ∈G−Gc uσ

)fpc,θ

(uτ1))[G:H]

∈ (Ω1F )⊗[G:H].

On the other hand, ResGGcM =⊕

G/H ResHGc θ (10.3.1). Hence the Abbes-Saito’s char-acteristic cycle CCψ0(M) is given by

CCψ0(M) =(rsw(ResHGc(θ))⊗ πc

)[G:H]=

(−da0)⊗[G:H]

( (∏σ∈G−Gc uσ

)fpc,θ

(uτ1))[G:H]

∈ (Ω1F⊗FF )⊗[G:H].

(10.4.4)So, we have CCψ0(M) = KCCψ0(1)(M).

Corollary 10.5. Assume that p is not a uniformizer of K. Let M be a finitedimensional Λ-vector space with a linear action of G and r = dimΛM/M (0). Then, wehave

CCψ0(M) ∈ (Ω1F )r ⊆ (Ω1

F ⊗F F )r

It is a Hasse-Arf type result for Abbes-Saito characteristic cycle. We should mentionthat T. Saito ([24, 3.10]) and L. Xiao [27] proved independently analogue results forsmooth varieties of any dimension over perfect fields.

Corollary 10.6. Assume that p is not a uniformizer of K. Let H be a sub-groupof G, and N a finite dimensional C-linear representation of H. We denote by r thedimension of N and by r′ the dimension of N (0). Then, we have

CCψ0(IndGH N) = CCψ0(N)⊗[G:H] ⊗ (da0)⊗([G:H]−1)

(∏σ∈G−H uσ

)[G:H]∈ (Ω1

F )⊗([G:H]r−r′). (10.6.1)

Indeed, (10.6.1) follows from the induction formula for Kato’s Swan conductor withdifferential values (3.15.2) and theorem 10.4.

Remark 10.7. Assume that p is not a uniformizer of K. Let L′ be a finite Galoisextension of K of group G′ which contains a sub-extension K ′ of K such that K ′/K isunramified and L′/K ′ is of type (II). We denote by P ′ the Galois group of the extensionL′/K ′ and by F ′ the residue field of OK′ . Let Λ be a finite field of characteristic` 6= p which contains a primitive (]P ′)-th root of unity and let N be a Λ-vector space

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES47

of finite dimension with a linear-G′ action. We fix a non-trivial character ψ : Fp →Λ×. By remarks 3.18 and 3.19, we can still define KCCψ(1)(N) ∈ (Ω1

F )⊗r, where r =

dimΛN/N(0). On the other hand, the wild inertia subgroup P of GK acts on N through

P ′, we can define CCψ(N) (Subsection 4.12). By ([22, 1.22]) and ([23, 3.1]), we have

CCψ(ResG′

P ′ N) = CCψ(N) ∈ (Ω1F (log)⊗F F )⊗r (10.7.1)

through the canonical isomorphism Ω1F (log)⊗F F ′ ∼−→ Ω1

F ′(log). Moreover, let Λ′ be thealgebraic closure of the fraction field of the ring of Witt vectors W (Λ), N ′ a pre-imageof the class of ResG

′P ′ N in the Grothendieck ring RΛ′(P

′) ([26, 16.1 Theorem 33] ) andψ′ : Fp → Λ′× the unique lifting of ψ : Fp → Λ×. By lemma 4.8, we deduce that

CCψ′(N′) = CCψ(ResG

′P ′ N). (10.7.2)

From theorem 10.4, we have

CCψ′(N′) = KCCψ(1)(N). (10.7.3)

By (10.7.1), (10.7.2) and (10.7.3), we conclude that

CCψ(N) = KCCψ(1)(N) ∈ (Ω1F )⊗r. (10.7.4)

11. Nearby cycles of `-adic sheaves on relative curves

11.1. In this section, we denote by S = Spec(R) an excellent strictly henselian trait.Assume that the residue field of R has characteristic p and that p is not a uniformizer ofR. We denote by s (resp. η, resp. η) the closed point (resp. generic point, a geometricgeneric point) of S. A finite covering of (S, η, s) stands for a trait (S ′, η′, s′) equippedwith a finite morphism S ′ → S. Let Λ be a finite field of characteristic ` 6= p and fix anon-trivial character ψ0 : Fp → Λ×.

11.2. We define a category CS as follows. An object of CS is a normal affine S-scheme H for which there exist a flat S-scheme of relative dimension one X and aclosed point x of Xs, such that X −x is smooth over S and H is S-isomorphic to thehenselization of X at x. A morphism between two objects of CS is a generically étalefinite morphism of S-schemes. Let (S ′, η′, s′) be a finite covering of (S, η, s). Then forany object H of CS, H ×S S ′ is an object of CS′ ([14, 5.4]).

11.3. Let H be an object of CS. We denote by P (H) the set of height 1 points ofH, by

Ps(H) = P (H) ∩Hs, Pη(H) = P (H) ∩Hη.

We have ([14, 5.2], [5, A.6]):

(i) Hη is geometrically regular over η and for any p ∈ Pη(H), the residue field κ(p)of H at p is a finite extension of the fraction field K(S) of S.

(ii) Hs is a reduced henselian noetherian local scheme over s of dimension 1, hencePs(H) is a finite set.

We denote by Hs the normalization of Hs, which is a finite union of strictly henseliantraits. We put

δ(H) = dimk(OHs/OHs).

48 11. NEARBY CYCLES OF `-ADIC SHEAVES ON RELATIVE CURVES

11.4. Let H be an object of CS, U a non-empty open sub-scheme of Hη and F alocally constant constructible étale sheaf of Λ-modules over U . For a triple (H,U,F ) anda finite covering (S ′, η′, s′) of (S, η, s), we denote by (H,U,F )S′ the triple (H ′, U ′,F ′)where H ′ = H ⊗S S ′, U ′ is the inverse image of U in H ′ and F ′ is the inverse imageof F on U ′. We call the triple (H,U,F ) stable if there is an étale connected Galoiscovering U of U such that

(i) The pull-back of F to U is constant.(ii) The normalization H of H in U belongs to CS and the residue field of H at all

points of Hη − Uη are finite separable extensions of κ(η).

Proposition 11.5. ([14, 6.3]). Let (H,U,F ) be a triple as subsection 11.4.(i) If (H,U,F ) is stable, (H,U,F )S′ is stable for any finite covering S ′ of S.(ii) For any triple (H,U,F ), there exist a finite covering (S ′, η′, s′) of (S, η, s) such

that (H,U,F )S′ is stable.

Proposition (i) follows form ([14, 5.4]) and proposition (ii) follows form [9].

11.6. Let (H,U,F ) be a stable triple. For p ∈ P (H), we denote by OH,p thecompletion of the local ring of H at p and by κ(p) its residue field. For p ∈ Ps(H), wedenote by Hs,p the integral closure of Hs in κ(p), which is a strictly henselian trait. Letords,p be the valuation of κ(p) associated to Hs,p normalized by ords,p(κ(p)×) = Z. Wedenote also by ords,p : Ω1

κ(p)−0 → Z the valuation defined by ords,p(αdβ) = ords,p(α),if α, β ∈ κ(p)× and ords,p(β) = 1. It can be canonically extended, for any integer r > 0,to (Ω1

κ(p))⊗r−0. Following ([12, XVI], [18] and [14, 6.4]), we call the total dimension

of F at a point p ∈ P (H), and denote by dimtotp(F ) the integer defined as follows:(i) For p ∈ Pη(H), we put

dimtotp(F ) = [κ(p) : κ(η)](swp(F ) + rank(F )),

where swp(F ) is the Swan conductor of the pull-back of F over Spec(OH,p)×HU .

(ii) For p ∈ Ps(H), we denote by Kp the fraction field of OH,p. Since the triple(H,U,F ) is stable, there exists a finite Galois extension Lp of Kp of ramifica-tion index one, such that the representation Fp of Gal(Ksep

p /Kp) defined by Ffactors through the quotient Gal(Lp/Kp). Notice that Lp/Kp factors througha field K ′p such that K ′p/Kp is unramified and Lp/K

′p is of type (II) (Subsec-

tion 3.3). Fixing a uniformizer π of R (also a uniformizer of Kp), we haveCCψ0(Fp) ∈ (Ω1

κ(p))m (cf. Remark 10.7). We denote by F p the restriction

to Spec(κ(p)) of the direct image of F under Spec(Kp) → Spec(OH,p) andby dimtots,p(F p) the sum of rank(F p) and the Swan conductor of F p overSpec(κ(p)). We put

dimtotp(F ) = − ords,p(CCψ0(Fp)) + dimtots,p(F p). (11.6.1)We notice that ords,p(CCψ0(F )) dose not depend on the choice of ψ0 (10.4.4)and the choice of π.

We put

ϕη(H,U,F ) =∑

p∈Hη−Udimtotp(F ), (11.6.2)

ϕs(H,U,F ) =∑

p∈Ps(H)

dimtotp(F ). (11.6.3)

I. RAMIFICATION AND NEARBY CYCLES FOR `-ADIC SHEAVES ON RELATIVE CURVES49

Lemma 11.7. ([14, 6.5]). Let (H,U,F ) be a stable triple (Subsection 11.4),(S ′, η′, s′) a finite covering of (S, η, s). We put (H ′, U ′,F ′) = (H,U,F )S′.

(i) For any p ∈ Ps(H) and for the unique p′ ∈ Ps(H) above p, we have

dimtotp(F ) = dimtotp′(F′).

(ii) For any p ∈ Hη − U , we have

dimtotp(F ) =∑

p′

dimtotp′(F′),

where p′ runs over the points above p.

11.8. Let (H,U,F ) be a triple (Subsection 11.4). By proposition 11.5, there existsa finite covering (S ′, η′, s′) of (S, η, s) such that (H,U,F )S′ is stable. We put

ϕη(H,U,F ) = ϕη′((H,U,F )S′),

ϕs(H,U,F ) = ϕs′((H,U,F )S′).

By lemma 11.7, they don’t depend on the choice of the covering (S ′, η′, s′).

Theorem 11.9 (Deligne, Kato). Let (H,U,F ) be a triple (Subsection 11.4), x theclosed point of H, u : U → Hη the canonical open immersion. Then we have

dimΛ(Ψ0x(u!F ))− dimΛ(Ψ1

x(u!F )) = ϕs(H,U,F )− ϕη(H,U,F )− 2δ(H) rank(F ).(11.9.1)

Proof. Indeed, for a stable triple (H,U,F ) and any p ∈ Ps(H), dimtotp(F ) is thesame as Kato’s definition in ([15, 4.4]) by (10.7.4).

Remark 11.10. The theorem 11.9 is proved by Deligne if F is unramified at everypoint of Ps(H) ([18, 5.1.1]). In the general case, Kato proved the theorem with twodifferent definitions of the invariant ϕs(H,U,F ) ([14, 6.7], [15, 4.5]). T. Saito giveanother proof with another definition of ϕs(H,U,F ) ([21]) which corresponds to thelatter definition of Kato ([15, 4.5]). If F is of rank 1, Abbes and Saito gave a definitionof ϕs(H,U,F ) ([5, A.10]) using the refined Swan conductor in their ramification theory[4], which coincides with Kato’s latter definition ([16, remark after 6.8]). Here, usingAbbes and Saito’s ramification theory, we give the definition of ϕs(H,U,F ) for any ranksheaf F which is equal to Kato’s latter formula (Theorem 10.4).

Réferences

[1] A. Abbes and A. Mokrane, Sous-groupes canoniques et cycles évanescents p-adiquespour les variétés abéliennes. Inst. Hautes Études Sci. Publ. Math. 99 (2004), 117–162.

[2] A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields.Amer. J. Math. 124 (2002), 879–920.

[3] A. Abbes and T. Saito, Ramification of local fields with imperfect residue fields II.Doc. Math. Extra Volume Kato (2003), 5–72.

[4] A. Abbes and T. Saito, Analyse micro-locale `-adique en caractéristique p > 0 le casd’un trait. Publ. Res. Inst. Math. Sci. 45 (2009), 25–74.

[5] A. Abbes and T. Saito, Local fourier transform and epsilon factors. Compos. Math.146 (2010), 1507–1551.

[6] A. Abbes and T. Saito, Ramification and cleanliness. Tohuku Math. J. CentennialIssue 63 (2011), No. 4, 775–853.

[7] S. Bosch, U. Güntzer and R. Remmert, Non-archimedean analysis. Grundlehrender Mathematischen Wissenschaften 261, Springer-Verlag, New York, Heidelberg,Berlin, 1984.

[8] P. Deligne and N. Katz, Groupes de monodromie en géométrie algébriques. II. Sémi-naire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Lecture Notesof Mathematices 340, springer-verlag, New York, Heidelberg, Berlin, 1973.

[9] H. Epp, Eliminating wild ramification. Invent. Math. 19 (1973), 235–249.[10] L. Fu, Etale cohomology theory. Nankai Tracts Math. World Scientific Publishing

Co. Pte. Ltd., Hackensack, NJ, 2011.[11] A. Grothendieck and J.A. Dieudonné, Éléments de géometrie algébrique. IV. Étude

locale des schémas et des morphisms de shémas. Inst. Hautes Études Sci. Publ.Math. 20 (1961), 24 (1965), 28 (1966), 32 (1967).

[12] A. Grothendieck et al, Groupes de monodromie en géométrie algébriques. I. Sémi-naire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 I). Dirigé par A.Grothendieck. Avec la collaboration de M. Raynaud et D.S. Rim. Lecture Notes inMathematics 288. Sringer-Verlag, Berlin-New York, 1972.

[13] A.J. de Jong, Crystalline Dieudonné module theory via formal and rigid geometry.Inst. Hautes Études Sci. Publ. Math. 82 (1995), 5–96.

[14] K. Kato, Vanishing cycles, ramification of valuations, and class field theory. DukeMath. J. 55 (1987), No.3, 629–659.

[15] K. Kato, Swan conductors with differential values. Adv. Stud. Pure Math. 12 (1987),315–342.

51

52

[16] K. Kato, Swan conductors for characters of degree one in the imperfect residue fieldcase. Contemp. Math. 83 (1989), 101–131.

[17] N. Katz, Gauss sum, Kloosterman sums, and monodromy groups. Annals of Math-ematics Studies 116, Princeton University Press, Princeton, NJ, 1988.

[18] G. Laumon, Semi-continuité du conducteur de Swan (d’après P. Deligne). In theEuler-Poincaré characteristic, Astérisque, 82–83 (1981), 173–219.

[19] G. Laumon, Transformation de Fourier, constantes d’équations fonctionnelles etconjecture de Weil. Inst. Hautes Études Sci. Publ. Math. 65 (1987), 131–210.

[20] A. Obus and S. Wewers, Cyclic extensions and the local lifting problem. 2012,arXiv:1203.5057.

[21] T. Saito, Trace formula for vanishing cycles of curves, Math. Ann. 276 (1987),311–315.

[22] T. Saito, Wild ramification and the characteristic cycle of an `-adic sheaf. J. Inst.Math. Jussieu 8 (2009), 769–829.

[23] T. Saito, Ramification of local fields with imperfect residue fields III. Math. Ann.352 (2012), No. 3, 567–580.

[24] T. Saito, Wild Ramification and the cotangent bundle. 2013, arXiv:1301.4632v4.[25] J.P. Serre, Corps locaux. Deuxième édition, Publications de l’Université de

Nancago, No. VIII. Hermann, Paris, 1968.[26] J.P. Serre, Linear representations of finite groups. Graduate Texts in Mathematics

42. Springer-Verlag, New York-Heidelberg, (1977).[27] L. Xiao, On ramification filtrations and p-adic differential equations, I: equal char-

acteristic case. Algebra Number Theory 4 (2010), 969–1027.

II

Refined characteristic class and conductor formula

1. Introduction

1.1. This article is devoted to the proof of a conductor formula for `-adic sheaves in ageometric situation (1.3.1) which generalizes the classical Grothendieck-Ogg-Shafarevichformula ([11] X 7.1) as well as the index formula of Saito ([18] 3.8). It uses the ramifica-tion theory developed by Abbes and Saito and it relies on a previous work of Tsushima,who proved a special case ([23] 5.9).

1.2. Let k be a perfect field of characteristic p > 0, f : X → Y a proper flatmorphism of smooth connected k-schemes and d the dimension of X. We assume thatdimY = 1 and let y be a closed point of Y , y a geometric point localized at y, Y(y) thestrict localization of Y at y and η a geometric generic point of Y(y). Put W = Y − y,V = f−1(W ) and that Q = f−1(y). We assume that the canonical projection fV : V →W is smooth and S is a divisor with simple normal crossing on X. Let D be a divisorwith simple normal crossing on X containing S = Qred such that D∩V is a divisor withsimple normal crossing relatively to W . We put U = X −D and let j : U → X be thecanonical injection. We consider the diagram

Uν //

fU

V

//

fV

X

f

Qoo

W // Y yoo

where ν is the canonical injection and fU = fV ν. We fix a prime number ` invertiblein k, and an Artinian local Z`-algebra Λ. Let F be a locally constant and constructiblesheaf of free Λ-modules on U such that

(i) F is tamely ramified along the divisor D ∩ V relatively to V ;(ii) the conductor R of F is effective with support contained in S ([5] 8.10) and

F is isoclinic and clean along D ([5] 8.22 and 8.23).Condition (i) implies that fV is universally locally acyclic relatively to ν!(F ) ([6]

Appendice to Th. Finitude, [19] 3.14). Since fV is proper, all cohomology groups ofRfU !(F ) are locally constant and constructible on W . We put ([6] Rapport 4.4)

rkΛ(RΓc(Uη,F |Uη)) = Tr(id; RΓc(Uη,F |Uη)),swy(RΓc(Uη,F |Uη)) =

∑

q∈Z(−1)qswy(R

qΓc(Uη,F |Uη)),

dimtoty(RΓc(Uη,F |Uη)) = rkΛ(RΓc(Uη,F |Uη)) + swy(RΓc(Uη,F |Uη)),where swy(R

qΓc(Uη,F |Uη)) denotes the Swan conductor of RqΓc(Uη,F |Uη) at y.We denote by

T∗X(logD) = V(Ω1X/k(logD)∨)

the logarithmic cotangent bundle over X and by σ : X → T∗X(logD) the zero section.Under the conditions (i) and (ii), Abbes and Saito defined the characteristic cycles of

53

54 1. INTRODUCTION

F , denoted by CC(F ), as a d-cycle on T∗X(logD) ([5] 1.12; [18] 3.6; cf. 6.15). Thevertical part CC∗(F ) of CC(F ) is a d-cycle on T∗X(logD)×X S such that

CC(F ) = (−1)d(rkΛ(F )[σ(X)] + CC∗(F )).

Theorem 1.3. We keep the notation and assumptions of 1.2 and assume moreoverthat S = D (i.e., U = V ) or that rkΛ(F ) = 1. Then, for any section s : X →T∗X(logD), we have the following equality in Λ

dimtoty(RΓc(Uη,F |Uη))−rkΛ(F )·dimtoty(RΓc(Uη,Λ)) = (−1)d+1 deg(CC∗(F )∩[s(X)]).(1.3.1)

The case where rkΛ(F ) = 1 is due to Tsushima ([23] 5.9). Although we follow thesame lines for sheaves of higher ranks, the situation is technically more involved. Ourapproach requires the assumption that S = D.

1.4. To prove 1.3, we follow the strategy of Saito for the proof of an index formulafor `-adic sheaves on proper smooth varieties [18]. The latter can be schematicallydivided into two steps. The first step uses the theory of cohomological correspondencesdue to Grothendieck and Verdier to associate a cohomology class to the `-adic sheaf,called the characteristic class, that computes its Euler-Poincaré characteristic by theLefschetz-Verdier formula ([11] III). The second step is more geometric. It consistsof computing the characteristic class as an intersection product using the ramificationtheory developed by Abbes and Saito [2].

1.5. The analogous approach for the proof of the conductor formula (1.3.1) wasstarted by Tsushima in [23]. He refined the characteristic class of an `-adic sheaf intoa cohomology class with support in the wild locus, called in this article the refinedcharacteristic class. He proved a Lefschetz-Verdier formula for this class ([23] 5.4)which amounts to say that it commutes with proper push-forward. On a smooth curve,the refined characteristic class gives the Swan conductor ([23] 4.1). The main goal ofthis article is to prove an intersection formula that computes the refined characteristicclass.

1.6. More precisely, with the notation and assumptions of 1.2, the refined charac-teristic class CS(j!(F )) of j!(F ) is defined as an element in H0

S(X,KX). The Lefschetz-Verdier formula implies the following relation

swy(RΓc(Uη,F |Uη))−rkΛ(F )·swy(RΓc(Uη,Λ)) = −f∗(CS(j!(F ))−rkΛ(F )·CS(j!(ΛU)))

in H0y(Y,KY )

∼−→ Λ, where f∗ in the left hand side is the proper push-forwardH0S(X,KX) → H0

y(Y,KY ) (cf. 7.12). Assume that D = S or that rkΛ(F ) = 1.Then, our main result is the following formula (8.2)

CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU))

= (−1)d rkΛ(F ) · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(logD))XS∩ [X] ∈ H0

S(X,KX),

where cd(−)XS is a bivariant class built of localized Chern classes (cf. 2.4). The right handside is the image of a zero cycle class in CH0(S), whose degree is (−1)d deg(CC∗(F ) ∩[s(X)]) (cf. 8.24), which implies theorem 1.3.

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 55

1.7. Beyond Tsushima’s work already mentioned, there have been several workson the conductor formula. Abbes gave a conductor formula for an `-adic sheaf on anarithmetic surface, under the condition that the sheaf has no fierce ramification [1].Vidal proved that the alternating sum of the Swan conductor of the cohomology groupswith compact support of an `-adic sheaf on a normal scheme over a local field onlydepends on its rank and its wild ramification [24]. For an `-adic sheaf on a smoothscheme over a local field of mixed characteristic, Kato and Saito defined its Swan class,which is a 0-cycle class supported on the wild locus, that computes the Swan conductorof the cohomology groups with compact support [15]. In a recent work [19], Saitodefined the characteristic cycle of an `-adic sheaf on a smooth surface as a cycle on thecotangent bundle without the cleanliness condition. When the surface is fibered over asmooth curve, he proved a conductor formula conjectured by Deligne ([19] 3.16).

1.8. This article is organized as follows. After preliminaries on étale cohomology,we briefly introduce the cohomological correspondences and the characteristic class ofan `-adic sheaf in §4. We recall Abbes and Saito’s ramification theory in §5 and reviewthe definition of clean sheaves and the characteristic cycle in §6. We give the definitionof Tsushima’s refined characteristic class and introduce the corresponding Lefschetz-Verdier formula in §7. The last section is devoted to the proof of the conductor formula.

Acknowledgement. This article is a part of the author’s thesis at Université Paris-Sud and Nankai University. The author would like to express his deepest gratitude tohis supervisors Abbes and Fu for leading him to this area and for patiently guiding himin solving this problem. This work is developed during a long visit to IHES supported byFonds Chern and Fondation Mathématiques Jacques Hadamard. The author is gratefulto these institutions for their support.

2. Notation

2.1. In this article, k denotes a perfect field of characteristic p > 0. We fix a primenumber ` invertible in k, an Artinian local Z`-algebra Λ and a non-trivial additivecharacter ψ : Fp → Λ×. All k-schemes are assumed to be separated and of finite typeover Spec(k).

2.2. For a k-scheme X, we denote by D(X,Λ) the derived category of complexes ofétale sheaves of Λ-modules on X and by Db

ctf(X,Λ) (resp. D−(X,Λ), resp. D+(X,Λ)and resp. Db

c(X,Λ)) its full subcategory consisting of objects bounded of finite tor-dimension with constructible cohomologies (resp. of objects bounded above, resp. ofobjects bounded below and resp. of objects bounded with constructible cohomologies).We denote by KX the complex Rf !Λ, where f : X → Spec(k) is the structure map andby DX the functor RHom(−,KX) on Db

ctf(X,Λ). For two k-schemes X and Y , andan étale sheaf of Λ-modules F (resp. G ) on X (resp. Y ), F G denotes the sheafpr∗1F ⊗ pr∗2G on X ×k Y .

2.3. Let X be a scheme and E a sheaf of OX-modules of finite type. Following ([9]1.7.8), we denote by V(E ) the vector bundle Spec(SymOX

(E )) over X.

2.4. Let X be a k-scheme of equidimension e, Z a closed subscheme of X, E1 andE0 locally free OX-modules of rank e, f : E1 → E0 an OX-linear map which is anisomorphism on X −Z, and E = [E1

f−→ E0] the complex such that E0 is in degree 0. For

56 3. PRELIMINARIES ON ÉTALE COHOMOLOGY

i > 0, we put ([14] 3.3)

ci(E0 − E1)XZ =

min(e,i−1)∑

j=0

cj(E1) ∩ ci−jXZ (E )

as a bivariant class, where c(E1) denotes the Chern class of E1 ([8] 3.2) and cXZ (E ) thelocalized Chern class of E ([8] 18.1).

3. Preliminaries on étale cohomology

3.1. Let

X ′f ′ //

g′

Y ′

g

X

f // Y

(3.1.1)

be a commutative diagram of k-schemes. We have the base change maps ([10] XVII4.1.5, XVIII 3.1.13.2)

g∗Rf∗ → Rf ′∗g′∗, (3.1.2)

Rf ′! Rg′! → Rg!Rf!. (3.1.3)

Assume that diagram (3.1.1) is Cartesian. There exists a canonical base changeisomorphism (the proper base change theorem) ([10] XVII 5.2.6)

g∗Rf!∼−→ Rf ′! g

′∗. (3.1.4)

There exists a canonical isomorphism of functors ([10] XVIII 3.1.12.3)

Rf ′∗Rg′! ∼−→ Rg!Rf∗. (3.1.5)

There exists a canonical morphism of functors ([10] XVIII 3.1.14.2)

g′∗Rf ! → Rf ′!g∗. (3.1.6)

It is defined as the adjoint of the composed morphisms

Rf ′! g′∗Rf ! ∼−→ g∗Rf!Rf

! → g∗,

where the first arrow is induced by the inverse of the proper base change theorem andthe second arrow is induced by the adjunction map Rf!Rf

! → id.

3.2. Let f : X → Y be a morphism of k-schemes and F (resp. G ) an object ofDb

ctf(X,Λ) (resp. Dbctf(Y,Λ)). There exists a canonical isomorphism (the projection

formula) ([10] XVII 5.2.9)

Rf!(f∗G ⊗L F )

∼−→ G ⊗L Rf!F . (3.2.1)

3.3. For a morphism f : X → Y of k-schemes and two objects F and G ofDb

ctf(Y,Λ), we have a canonical map

f ∗F ⊗L Rf !G → Rf !(F ⊗L G ), (3.3.1)

defined as follows. By the projection formula (3.2.1), we have a canonical isomorphism

Rf!(f∗F ⊗L Rf !G )

∼−→ F ⊗L Rf!(Rf!G ).

Composing with the adjunction map Rf!(Rf!G ) → G , we obtain a map Rf!(f

∗F ⊗LRf !G )→ F ⊗L G , which gives (3.3.1) by adjunction.

If f is a closed immersion, (3.3.1) induces a cup product

Hi(X, f ∗F )× HjX(Y,G )

∪−→ Hi+jX (Y,F ⊗L G ). (3.3.2)

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 57

3.4. Let g : W → X and f : X → Y be closed immersions of k-schemes, and Fand G objects of Dctf(Y,Λ). Then, the following diagram is commutative

f ∗F ⊗L g∗Rg!(Rf !G )

∼ // g∗((fg)∗F ⊗L R(fg)!G )(3.3.1)

// g∗Rg!(Rf !(F ⊗L G ))

f ∗F ⊗L Rf !G

(3.3.1)// Rf !(F ⊗L G )

(3.4.1)where the vertical maps are induced by the adjunction map g∗Rg! → id, and the iso-morphic map is induced by the projection formula (3.2.1). Indeed, it is enough to showthat the following diagram is commutative

f∗(f ∗F ⊗L g∗Rg!(Rf !G ))

∼ // (fg)∗((fg)∗F ⊗L R(fg)!G )∼ // F ⊗L (fg)∗R(fg)!G

f∗(f ∗F ⊗L Rf !G )

∼ // F ⊗L f∗(Rf !G ) // F ⊗L G

where the isomorphic maps are the projection formulae and the other maps are inducedby adjunction. Since the composition of the upper horizontal maps is the projectionformula

f∗(f∗F ⊗L g∗Rg!(Rf !G ))

∼−→ F ⊗L (fg)∗R(fg)!G ,

we are reduced to show the following diagram is commutative

f∗(f ∗F ⊗L (g∗Rg!(Rf !G )))

adj

∼ // F ⊗L f∗(g∗Rg!(Rf !G ))

adj

f∗(f ∗F ⊗L Rf !G )∼ // F ⊗L f∗(Rf !G )

which is obvious.Diagram (3.4.1) induces a commutative diagram

Hi(X, f ∗F )× HjW (Y,G )

∪W //

Hi+jW (Y,F ⊗L G )

Hi(X, f ∗F )× HjX(Y,G )

∪ // Hi+jX (Y,F ⊗L G )

(3.4.2)

where ∪W is defined by the upper horizontal arrows of (3.4.1).

3.5. Let f : X → Y be a morphism of k-schemes, F an object of D−(X,Λ) and Gan object of D+(Y,Λ). We have a canonical isomorphism ([10] XVIII 3.1.10, [7] 8.4)

Rf∗RHom(F , Rf !G )∼−→ RHom(Rf!F ,G ). (3.5.1)

Taking G = KY , we obtain an isomorphism (2.2)

Rf∗(DX(F ))∼−→ DY (Rf!F ). (3.5.2)

3.6. For a morphism f : X → Y of k-schemes and two objects F and G ofDb

ctf(Y,Λ), we recall the definition of the canonical isomorphism ([10] XVIII 3.1.12.2,[7] 8.4.7)

RHom(f ∗F ,Rf !G )∼−→ Rf !RHom(F ,G ). (3.6.1)

By the inverse of the projection formula (3.2.1), we have a canonical isomorphism

F ⊗L Rf!RHom(f ∗F , Rf !G )∼−→ Rf!(f

∗F ⊗L RHom(f ∗F ,Rf !G )).

58 3. PRELIMINARIES ON ÉTALE COHOMOLOGY

Composing with the canonical morphisms

Rf!(f∗F ⊗L RHom(f ∗F ,Rf !G ))→ Rf!Rf

!G → G ,

we obtain a morphism

F ⊗L Rf!RHom(f ∗F ,Rf !G )→ G .

It induces the map (3.6.1) by adjunction. Taking G = KY , we obtain a canonicalisomorphism

DX(f ∗F )∼−→ Rf !(DY (F )). (3.6.2)

3.7. For two k-schemes X and Y and objects F and G of Dbctf(X,Λ) and Db

ctf(Y,Λ),a canonical isomorphism

RHom(pr∗2G ,Rpr!1F )

∼−→ F L DY (G ) (3.7.1)

is defined in ([11] III 3.1.1).Let g : X → X and h : Y → Y be open immersions. By (3.7.1), (3.5.2) and the

Künneth formula, we have a canonical isomorphism on X ×k Y

RHom(pr∗2h!G ,Rpr!1g!F )

∼−→ (g!F )L DY (h!G )∼−→ (g!F )L (Rh∗(DY (G )))(3.7.2)

∼−→ (g × 1)!(R(1× h)∗(F L DY (G )))∼−→ (g × 1)!(R(1× h)∗RHom(pr∗2G ,Rpr!

1F )).

3.8. Let f : X → Y be a flat morphism of k-schemes with fibers of equidimension dand F an object of Db

ctf(Y,Λ). We have a canonical trace map ([10] XVIII 2.9)

Trf : Rf!f∗F (d)[2d]→ F .

Its adjoint

tf : f ∗F (d)[2d]→ Rf !F (3.8.1)

is called the class map ([10] XVIII 3.2.3). If f is smooth, tf (3.8.1) is an isomorphism(Poincaré duality) ([10] XVIII 3.2.5, [7] 8.5.2).

3.9. Let f : X → Y be a morphism of smooth k-schemes with the same equidimen-sion d. For any object F of Db

ctf(Y,Λ), we recall the definition of the canonical map([4] (1.9))

f ∗F → Rf !F . (3.9.1)

The map f is the composition of the graph Γf : X → X ×k Y of f and the projectionpr2 : X ×k Y → Y . Since Γf is a section of the projection pr1 : X ×k Y → X, thereexists a canonical isomorphism Λ→ Rf !Λ defined as the composition

Λ∼−→ RΓ!

fRpr!1Λ

∼−→ RΓ!fΛ(d)[2d]

∼−→ RΓ!fRpr!

2Λ∼−→ Rf !Λ,

where the second and the third arrows are induced by Poincaré duality. Then, thecanonical map (3.3.1) induces (3.9.1).

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 59

3.10. Let V be a k-scheme, Z an integral closed subscheme of V of equidimensiond. The canonical class map Λ(d)[2d]→ KZ induces a morphism

H0(Z,Λ)→ H−2dZ (V,KV (−d)). (3.10.1)

The cycle class [Z] ∈ H−2dZ (V,KV (−d)) is defined as the image of 1 ∈ H0(Z,Λ) by the

map (3.10.1). We obtain a homomorphism Zd(V ) → H−2d(V,KV (−d)), where Zd(V )denotes the free abelian group generated by integral closed subschemes of equidimensiond of V . This map factors through the Chow group CHd(V ), and induces the cycle map

cl : CHd(V )→ H−2d(V,KV (−d)). (3.10.2)

If V is a closed subscheme of a smooth k-scheme X of dimension d+ c, by Poincaréduality, we have

H−2d(V,KV (−d))∼−→ H−2d

V (X,KX(−d))∼−→ H2c

V (X,Λ(c)).

Let Y be another smooth k-scheme of dimension e + c, f : X → Y a k-morphismand W a closed subscheme of Y such that f−1(W ) is a closed subscheme of V . By ([14]2.1.2), we have a commutative diagram

CHe(W )cl //

f !

H2cW (Y,Λ(c))

f∗

CHd(V )

cl // H2cV (X,Λ(c))

(3.10.3)

where the map f ! denotes the refined Gysin homomorphism ([8] 6.6).

3.11. Let X be a k-scheme, Z a closed subscheme of X, V = X − Z the comple-mentary open subscheme of Z in X, U an open subscheme of X, i : Z → X, j : V → X,iU : Z ∩ U → U and jU : U ∩ V → U the canonical injections, and F an objectof Db

ctf(X,Λ). Assume that for any integer q, H q(F )|U is locally constant and con-structible. Then, we have a canonical isomorphism ([7] 6.5.5)

(F |U)⊗L RjU∗Λ∼−→ RjU∗j

∗U(F |U).

Since Ri!URjU∗ = 0 (3.1.5), we have Ri!U((F |U) ⊗L RjU∗Λ) = 0. In particular, for anyinteger q, the canonical map

HqZ−U(X,F ⊗L Rj∗(ΛV ))→ Hq

Z(X,F ⊗L Rj∗(ΛV ))

is an isomorphism ([23] Lemma 3.1).

3.12. Let X be a k-scheme, Z a closed subscheme of X, V = X − Z the comple-mentary open subscheme of Z in X, i : Z → X and j : V → X the canonical injections,and F an object of D−(X,Λ). Applying the functor Ri!(F ⊗L −) to the distinguishedtriangle i∗Ri!ΛX → ΛX → Rj∗ΛV →, we obtain a distinguished triangle ([23] Lemma3.5)

i∗F ⊗L Ri!(ΛX)a−→ Ri!F

b−→ Ri!(F ⊗L Rj∗(ΛV ))→, (3.12.1)

where a is the map (3.3.1). We denote the functor Ri!(− ⊗L Rj∗(ΛV )) : D−(X,Λ) →D−(Z,Λ) by ∆i(−).

60 4. COHOMOLOGICAL CORRESPONDENCES

3.13. Let X be a k-scheme, δ : X → X ×k X the diagonal map and S a closedsubscheme of X such that its complement is dense in X. Consider the composed map

KX → Rδ!δ∗KX → ∆δ(δ∗(KX)), (3.13.1)

where the first arrow is the adjunction map and the second arrow is b in (3.12.1). IfX is an equidimensional smooth k-scheme, the following map induced by (3.13.1) is anisomorphism ([4] 5.2)

H0S(X,KX)

∼−→ H0S(X,∆δ(δ∗(KX))). (3.13.2)

4. Cohomological correspondences

Definition 4.1 ([11] III 3.2, [4],1.2.1). Let X and Y be two k-schemes. A corre-spondence between X and Y is a k-scheme C equipped with k-morphisms c1 : C → Xand c2 : C → Y . Let F and G be objects of Db

ctf(X,Λ) and Dbctf(Y,Λ), respectively. A

cohomological correspondence is a morphism u : c∗2G → Rc!1F from G to F on C.

We switch the factors compared to ([11] III 3.2).

4.2. Let X and Y be two k-schemes and (C, c1 : C → X, c2 : C → Y ) a corre-spondence between X and Y . We denote by c the map (c1, c2) : C → X ×k Y and bypr1 : X ×k Y → X and pr2 : X ×k Y → Y the canonical projections. Let F and Gbe objects of Db

ctf(X,Λ) and Dbctf(Y,Λ), respectively. We have a canonical isomorphism

(3.6.1)

RHom(c∗2G ,Rc!1F )

∼−→ Rc!RHom(pr∗2G ,Rpr!1F ).

Taking global sections on C, we get a canonical isomorphism

Hom(c∗2G ,Rc!1F )

∼−→ H0(C,Rc!RHom(pr∗2G , Rpr!1F )), (4.2.1)

which shows that cohomological correspondences u : c∗2G → Rc!1F are in one to one cor-

respond with morphisms ΛC → Rc!RHom(pr∗2G ,Rpr!1F ), and hence with morphisms

Rc!(ΛC)→ RHom(pr∗2G ,Rpr!1F ) by adjunction.

If c : C → X ×k Y is a closed immersion, and X and Y are smooth k-schemes ofdimension d, we have

H0(C,Rc!RHom(pr∗2G ,Rpr!1F ))

∼−→ H2dC (X ×k Y,RHom(pr∗2G , pr∗1F )(d)).

If we further assume that F and G are sheaves of free Λ-modules and that G is lo-cally constant and constructible. Then the canonical map c∗RHom(pr∗2G , pr∗1F ) →RHom(c∗2G , c

∗1F ) is an isomorphism, and we have

Hom(c∗2G , c∗1F )

∼−→ H0(C, c∗RHom(pr∗2G , pr∗1F )).

Then, the cycle class map CHd(C)→ H2dC (X ×k Y,Λ(d)) induces a pairing

CHd(C)⊗ Hom(c∗2G , c∗1F ) → H2d

C (X ×k Y,Λ(d))⊗ H0(C, c∗RHom(pr∗2G , pr∗1F ))∪−→ H0

C(X ×k Y,RHom(pr∗2G ,Rpr!1F )) = Hom(c∗2G ,Rc

!1F ).

In this case, for a cycle class Γ ∈ CHd(C) and a homomorphism γ : c∗2G → c∗1F , thepair (Γ, γ) induces a cohomological correspondence u(Γ, γ) from F to G on C.

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 61

4.3. We consider a commutative diagram of k-schemes

X

f

Cc2 //c1oo

h

Y

g

X ′ C ′c′2 //

c′1oo Y ′

(4.3.1)

and let F and G be objects ofDbctf(X,Λ) and Db

ctf(Y,Λ), respectively. By (3.5.2), (3.7.1)and the Künneth formula, we have a canonical isomorphism

R(f × g)∗RHom(pr∗2G ,Rpr!1F )

∼−→ RHom(pr∗2Rg!G ,Rpr!1Rf∗F ). (4.3.2)

Diagram (4.3.1) gives a commutative diagram

C

h

c // X ×k Yf×g

C ′c′ // X ′ ×k Y ′

We assume that f , g and h are proper. A cohomology correspondence u : c∗2G → Rc!1F

is identified with a map u : ΛC → Rc!RHom(pr∗2G ,Rpr!1F ) (4.2). It induces a map

ΛC′ → Rh∗Rc!RHom(pr∗2G ,Rpr!

1F ). (4.3.3)

The base change map (3.1.3) gives

Rh∗Rc! = Rh!Rc

! → Rc′!R(f × g)! = Rc′!R(f × g)∗. (4.3.4)

Composing (4.3.2), (4.3.3) and (4.3.4), we obtain a map

ΛC′ → Rc′!R(f × g)∗RHom(pr∗2G , Rpr!1F )

∼−→ Rc′!RHom(pr∗2Rg!G ,Rpr!1Rf∗F ).

By (4.2.1), we obtain a map

c′∗2 Rg!G = c′∗2 Rg∗G → Rc′!1Rf∗F ,

which is a correspondence form Rg∗G to Rf∗F on C ′, that we denote by h∗(u) and callthe push-forward of u by h. The map h∗(u) is equal to the composition of the maps

c′∗2 Rg∗G → Rh∗c∗2G

Rh∗(u)−−−−→ Rh∗Rc!1F → Rc′!1Rf∗F ,

where the left and right arrows are the base change maps.

4.4. We consider a commutative diagram of k-schemes

U

jU

C ′c′2 //

jC

c′1oo V

jV

X Cc2 //c1oo Y

(4.4.1)

where all the vertical arrows are open immersions. Let F and G be objects of Dbctf(X,Λ)

and Dbctf(Y,Λ), respectively, and u : c∗2G → Rc!

1F a cohomological correspondence onC. Denote by FU and GV the restrictions of F and G to U and V , respectively. We haveRj!

C = j∗C and Rj!U = j∗U . Hence, the restriction u′ of u to C ′ defines a cohomological

correspondence

u′ : c′∗2 (GV ) = j∗Cc∗2G

j∗C(u)−−−→ j∗CRc!1F = Rc′!1(FU). (4.4.2)

62 4. COHOMOLOGICAL CORRESPONDENCES

We denote by j the map jU×jV : U×kV → X×kY , by c the map (c1, c2) : C → X×kYand by c′ the map (c′1, c

′2) : C ′ → U ×k V . We have a commutative diagram

C ′c′ //

jC

U ×k Vj

C

c // X ×k Y

(4.4.3)

The base change map (3.1.3) gives a canonical morphism

Rc′!(ΛC′) = Rc′!Rj!C(ΛC)→ Rj!Rc!(ΛC) = j∗Rc!(ΛC). (4.4.4)

Put

H ′ = RHom(pr∗2(GV ),Rpr!1(FU)) on U ×k V, (4.4.5)

H = RHom(pr∗2G ,Rpr!1F ) on X ×k Y. (4.4.6)

By (4.2), we identify a cohomological correspondence u : c∗2G → Rc!1F with a map

u : ΛC → Rc!H and also with the associated map u : Rc!(ΛC) → H . We identify therestriction u′ : c′∗2 (GV ) → Rc′!1(FU) of u with a map u′ : ΛC′ → Rc′!(H ′) also with theassociated map u′ : Rc′!(ΛC′)→ H ′. Since Rj!

U = j∗U and Rj! = j∗, by (3.6.1), we havea canonical isomorphism

j∗H∼−→H ′. (4.4.7)

Lemma 4.5 ([4] Lemma 1.2.2). We take the notation and assumptions of (4.4).Then,

1. The map u′ : ΛC′ → Rc′!(H ′) coincides with the restriction of u : ΛC → Rc!Hto C ′ by the composed isomorphism j∗CRc!H = Rj!

CRc!H → Rc′!Rj!H =Rc′!j∗H = Rc′!(H ′).

2. The following diagram is commutative

j∗Rc!(ΛC)j∗u // j∗H

(4.4.7)

Rc′!(ΛC′)

(4.4.4)

OO

u′ // H ′

(4.5.1)

Lemma 4.6 ([4] Lemma 1.2.3). Consider diagram (4.4.1) again and assume more-over that its right square is Cartesian. Let F ′ and G ′ be objects of Db

ctf(U,Λ) andDb

ctf(V,Λ), and u′ : c′∗2 (G ′) → Rc′!1(F ′) a cohomological correspondence on C ′. Then,there exist a unique cohomological correspondence u : c∗2jV !(G ′)→ Rc!

1jU !(F ′) on C suchthat its restriction to C ′ is u′.

We call u in (4.6) the extension by zero of u′ and denote it by jC!u′.

4.7. Let X, X ′, Y , Y ′ be smooth equidimensional k-schemes such that dimX =dimX ′ and dimY = dimY ′, f : X ′ → X and g : Y ′ → Y morphisms of k-schemes,(C, c1 : C → X, c2 : C → Y ) a correspondence between X and Y , and F and G objectsof Db

ctf(X,Λ) and Dbctf(X,Λ), respectively. We denote by c the map (c1, c2) : C →

X ×k Y . By (3.6.1) and (3.9.1), we have a map

(f × g)∗RHom(pr∗2G ,Rpr!1F ) → R(f × g)!RHom(pr∗2G ,Rpr!

1F ) (4.7.1)→ RHom(pr∗2g

∗G ,Rpr!1Rf !F ).

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 63

Let u : c∗2G → Rc!1F be a cohomological correspondence on C, that we identify with

a map u : Rc!ΛC → RHom(pr∗2G ,Rpr!1F ). We define a correspondence c′ = (c′1, c

′2) :

C ′ → X ′ ×k Y ′ by the Cartesian diagram

C ′

c′ //

h

X ′ ×k Y ′

f×g

Cc // X ×k Y

By the proper base change theorem, the base change map (f × g)∗Rc!ΛC → Rc′!ΛC′ isan isomorphism. The composed map

Rc′!(ΛC′)∼−→ (f × g)∗Rc!(ΛC) → (f × g)∗RHom(pr∗2G ,Rpr!

1F )

→ RHom(pr∗2g∗G ,Rpr!

1Rf !F ),

where the first arrow is the inverse of the base change isomorphism, corresponds to acohomological correspondence

(f × g)∗(u) : c′∗2 g∗G → Rc′!1Rf !F ,

called the pull-back of u by f × g.

4.8. LetX be a k-scheme, F an object ofDbctf(X,Λ). We denote by δ : X → X×kX

the diagonal map and put H = RHom(pr∗2F ,Rpr!1F ). The canonical isomorphism

H∼−→ F L DX(F ) (3.7.1) induces an isomorphism

δ∗H∼−→ F ⊗L DX(F ).

Composed with the evaluation map F ⊗L DX(F )→ KX , we get a map

ev : δ∗H → KX , (4.8.1)

that we also call the evaluation map.Let C be a closed subscheme of X×kX and u a cohomological correspondence of F

on C. We denote by c : C → X ×k X the canonical injection. By (4.2), u correspondsto a section

u ∈ H0(C,Rc!H ) = H0C(X ×k X,H ).

We call the image of u by the following composed maps

H0C(X ×k X,H )

δ∗−→ H0C∩X(X, δ∗H )

ev−→ H0C∩X(X,KX)

the characteristic class of the cohomological correspondence u, and denote it byC(F , C, u) ∈ H0

C∩X(X,KX) ([11] III, [4] 2.1.8). If C = δ(X), and u : F → F isan endomorphism (resp. the identity of F ), we abbreviate the notation of the charac-teristic class of u by C(F , u) ∈ H0(X,KX) (resp. C(F ) ∈ H0(X,KX), and call it thecharacteristic class of F ).

4.9. LetX be a k-scheme, U an open subscheme ofX, and F an object ofDbctf(U,Λ).

We denote by j : U → X the canonical open immersion, and by δ : X → X ×k X andδU : U → U ×k U the diagonal maps. Put

H = RHom(pr∗2F ,Rpr!1F ) on U ×k U,

H = RHom(pr∗2j!F ,Rpr!1j!F ) on X ×k X.

By (3.7.1) and the projection formula for j! (3.2.1), we have

δ∗(H ) ∼= (j!F )⊗L D(j!F ) ∼= j!(F ⊗L D(F )) ∼= j!(δ∗UH ).

64 5. RAMIFICATION OF `-ADIC SHEAVES

Then, the evaluation map ev : δ∗UH → KU (4.8.1) induces a map

ev′ : δ∗(H )→ j!(KU).

Let C be a closed subscheme of U ×k U and u a cohomological correspondence of Fon C. We denote by C the closure of C in X ×k X. We have a commutative diagram

Cc //

jC

U ×k Uj×j

Cc // X ×k X

where j, c and c are the canonical injections. Assume C = (X×kU)∩C. The extensionby zero jC!(u) of u (4.6) corresponds, by (4.2), to a section

jC!(u) ∈ H0(C,Rc!(H )) = H0C

(X ×k X,H ).

We denote by C!(j!F , C, jC!(u)) the image of jC!(u) by the composed map

H0C

(X ×k X,H )δ∗−→ H0

C∩X(X, δ∗(H ))ev′−→ H0

C∩X(X, j!(KU)).

By ([4] 2.1.7), the characteristic class C(jU !F , C, jC!(u)) ∈ H0C∩X(X,KX) is the canon-

ical image of C!(j!F , C, jC!(u)).

4.10. Let X be an equidimensional smooth k-scheme, S a closed subscheme of X,U = X − S the complementary open subscheme of S in X that we assume to be densein X, j : U → X the canonical injection, δ : X → X ×k X the diagonal map, and Fan object of Db

ctf(X,Λ) such that for any integer q, H q(F )|U is locally constant. PutH = RHom(pr∗2F ,Rpr!

1F ) on X ×k X, we have Rδ!H∼−→ RHom(F ,F ) (3.6.1).

Hence idF ∈ End(F ) corresponds to a map ΛX → Rδ!H , and by adjunction to amap δ∗ΛX → H . Let ev′ : H → δ∗KX be the adjoint of the evaluation maps (4.8.1).Applying the functor ∆δ (3.13) to the composition of the two morphisms above, weobtain a map

∆δ(δ∗(ΛX))→ ∆δ(H )∆δ(ev′)−−−−→ ∆δ(δ∗(KX)). (4.10.1)

Since, for each integer q, H q(H )|U×kU is locally constant and constructible, by (3.11),we have

H0S(X,∆δ(H ))

∼−→ H0(X,∆δ(H )). (4.10.2)

Hence, the canonical map ΛX = Rδ!δ∗ΛX → ∆δ(δ∗ΛX), (4.10.1), (4.10.2) and the inverseof (3.13.2) define a map

H0(X,ΛX)→ H0(X,∆δ(H ))∼−→ H0

S(X,∆δ(H ))→ H0S(X,∆δ(δ∗KX))

∼−→ H0S(X,KX).

We denote the image of 1 ∈ H0(X,ΛX) in H0S(X,KX) by C0

S(F ) and call it the localizedcharacteristic class of F ([4] 5.2). In [23], the author gave another definition of thelocalized characteristic class and proved that the two definitions are equivalent.

5. Ramification of `-adic sheaves

5.1. Let K be a complete discrete valuation field, OK the integer ring, F the residuefield of OK , K a separable closure of K, and GK the Galois group of K over K. Abbesand Saito defined two decreasing filtrations Gr

K and GrK,log (r ∈ Q>0) of GK by closed

normal subgroups called the ramification filtration and the logarithmic ramification

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 65

filtration, respectively ([2], 3.1, 3.2). We denote by G0K,log the inertia subgroup of GK .

For any r ∈ Q>0, we put

Gr+K,log =

⋃

s∈Q>rGsK,log and GrrlogGK = Gr

K,log

/Gr+K,log.

By ([2] 3.15), P = G0+K,log is the wild inertia subgroup of GK , i.e. the p-Sylow subgroup of

G0K,log. For every rational number r > 0, the group GrrlogGK is abelian and is contained

in the center of P/Gr+K,log ([3] Theorem 1).

5.2. Let L be a finite separable extension of K. For a rational number r > 0, we saythat the logarithmic ramification of L/K is bounded by r (resp. by r+) if Gr

K,log (resp.Gr+K,log) acts trivially on HomK(L,K) via its action on K. The logarithmic conductor c

of L/K is defined as the infimum of rational numbers r > 0 such that the logarithmicramification of L/K is bounded by r. Then c is a rational number and the logarithmicramification of L/K is bounded by c+ ([2] 9.5). If c > 0, the logarithmic ramificationof L/K is not bounded by c.

Lemma 5.3 ([16] 1.1). Let M be a Λ-module on which P = G0+K,log acts Λ-linearly

through a finite discrete quotient, say by ρ : P → AutΛ(M). Then,(i) The module M has a unique direct sum decomposition

M =⊕

r∈Q>0

M (r) (5.3.1)

into P -stable submodules M (r), such that M (0) = MP and for every r > 0,

(M (r))GrK,log = 0 and (M (r))G

r+K,log = M (r).

(ii) If r > 0, then M (r) = 0 for all but the finitely many values of r for whichρ(Gr+

K,log) 6= ρ(GrK,log).

(iii) For any r > 0, the functor M 7→M (r) is exact.(iv) For M , N as above, we have HomP−Mod(M (r), N (r′)) = 0 if r 6= r′.

5.4. The decomposition (5.3.1) is called the slope decomposition of M . The valuesr > 0 for which M (r) 6= 0 are called the slopes of M . We say that M is isoclinic if it hasonly one slope. If M is isoclinic of slope r > 0, we have a canonical central characterdecomposition

M = ⊕χMχ,

where the sum runs over finite characters χ : GrrlogGK → Λ×χ such that Λχ is a finiteétale Λ-algebra ([5] 6.7).

5.5. We assume that K has characteristic p and that F is of finite type over k. LetΩ1F (log) be the F -vector space

Ω1F (log) = (Ω1

F/k ⊕ (F ⊗Z K×))/(da− a⊗ a ; a ∈ O×K),

where a denotes the residue class of an element a ∈ F . We denote by OK the integralclosure of OK in K, F the residue field of OK and by v the valuation of K normalizedby v(K×) = Z. For a rational number r, we put mr

K(resp. mr+

K) the set of elements of

K such that v(x) > r (resp. v(x) > r). For any rational number r > 0, GrrlogGK is aFp-vector space, and there exists a canonical injective homomorphism, called the refinedSwan conductor ([18] 1.24),

rsw : HomFp(GrrlogGK ,Fp)→ Ω1F (log)⊗F m−r

K/m−r+

K. (5.5.1)

66 5. RAMIFICATION OF `-ADIC SHEAVES

5.6. Let X be a smooth k-scheme, D a divisor with simple normal crossing on X,Dii∈I the irreducible components of D. A rational divisor on X with support in D isan element R =

∑i∈I riDi of the Q-vector space generated by Dii∈I . We say that R

is effective if ri > 0 for all i. We call generic points of R the generic points of the Di’ssuch that ri 6= 0. We denote by bnRc the divisor

∑i∈IbnricDi on X, where bnric is the

integral part of nri. For two rational divisors R and R′ with support in D, we say thatR′ is bigger than R and use the notation R′ > R if R′ −R is effective.

Let u : P → X a smooth separated morphism of finite type, s : X → P a section ofu and R an effective rational divisor with support on D. Put U = X −D and denoteby j : U → X and jP : PU = u−1(U) → P the canonical injections and by IX theideal sheaf of OP associated to s. We call dilatation of P along s of thickening R anddenote by P (R) the affine scheme over P defined by the quasi-coherent sub-OP -algebraof jP∗(OPU )

∑

n>0

u∗(OX(bnRc)) ·I nX . (5.6.1)

The image of the algebra (5.6.1) by the surjective homomorphism jP∗(OPU )→ s∗j∗(OU)is canonically isomorphic to s∗(OX). Hence we have a canonical section

s(R) : X → P (R)

lifting s ([5] 5.26).

5.7. In the following of this section, let X be a smooth k-scheme, D a divisor withsimple normal crossing on X, Dii∈I the irreducible components of D, and j : U =X −D → X the canonical injection. We denote by (X ×k X)′i the blow-up of X ×k Xalong Di ×i Di, by (X ok X)i the complement of the proper transform of D ×k X in(X ×k X)′i and by (X∗kX)i the complement of the proper transform of Di ×k X andX×kDi in (X×kX)′i. We denote by (X×kX)′ the fiber product of (X×kX)′ii∈I overX×kX, which is also the blow-up of X×kX along Di×kDii∈I ([18] §2.3). We denoteby X ok X the fiber product of (X ok X)ii∈I that we call the left-framed self-productof X along D. We denote by X∗kX the fiber product of (X∗kX)ii∈I over X ×k X,which is the open subscheme of (X ×k X)′ obtained by removing the strict transformsof D×kX and X ×kD in (X ×kX)′, that we call the framed self-product of X along D([5] 5.22).

By the universality of the blow-up, the diagonal map δ : X → X×kX induces closedimmersions that we denote by

δ′ : X → (X ×k X)′ and δ : X → X∗kX.We considerX∗kX as anX-scheme by the second projection. This projection is smooth([18] §2.3).

We denote by D′i the pull-back of δ(Di) by the canonical projection (X ×k X)′i →X×kX and byD′ the pull-back of δ(D) by the canonical projection (X×kX)′ → X×kX.By definition, D′i → Di is a P1-bundle. For a subset J of I, we put DJ =

⋂i∈J Di and

denote by nJ the cardinality of J . Since (X×kX)′ is the fiber product of (X×kX)′ii∈Iover X ×k X, D′ is the union of (P1)nJ -bundles over DJ ([23] 3.12).

We denote by Di the pull-back of δ(Di) by the canonical projection (X∗kX)i →X ×k X and by D the pull-back of δ(D) by the canonical projection X∗kX → X ×kX. By definition, Di → Di is a Gm-bundle. Since X∗kX is the fiber product of(X∗kX)ii∈I over X ×k X, D is the union of (Gm)nJ -bundles over DJ ([22] 2.1).

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 67

5.8. For any effective rational divisor R on X with support on D, we denote by(X∗kX)(R) the dilatation of X∗kX along δ of thickening R (5.6 and 5.7). If weconsider X∗kX as an X-scheme by the first projection, then the dilatation of X∗kXalong δ of thickening R is equal to (X∗kX)(R) ([5] 5.31). There is a canonical morphism

δ(R) : X → (X∗kX)(R)

lifting δ, and a canonical open immersion

j(R) : U ×k U → (X∗kX)(R).

Moreover, the following diagram

U

δU //

j

U ×k Uj(R)

Xδ(R)// (X∗kX)(R)

(5.8.1)

is Cartesian.If R has integral coefficients, then the canonical projection (X ×k X)(R) → X is

smooth ([5] 4.6) and we have a canonical R-isomorphism ([5] 4.6.1)

(X∗kX)(R) ×X R ∼−→ V(Ω1X/k(logD)⊗OX OX(R))×X R. (5.8.2)

5.9. Let F be a locally constant constructible sheaf of Λ-modules on U , R aneffective rational divisor on X with support on D, and x a geometric point of X. PutH = Hom(pr∗2F , pr∗1F ) on U ×k U . Then the base change map

α : δ(R)∗j(R)∗ (H )→ j∗δ

∗U(H ) = j∗(E nd(F )) (5.9.1)

relatively to the Cartesian diagram (5.8.1) is injective ([5] 8.2). We say that the ram-ification of F at x is bounded by R+ ([5] 8.3) if F satisfies the following equivalentconditions ([5] 8.2):

(i) The stalk αx of the morphism α (5.9.1) at x is an isomorphism.(ii) The image of idF in j∗(E nd(F ))x is contained in the image of αx.

We say that the ramification of F along D is bounded by R+ ([5] 8.3) if the ramificationof F at x is bounded by R+ for every geometric point x ∈ X.

5.10. Let F be a locally constant constructible sheaf of Λ-modules on U , R aneffective rational divisor on X with support in D, ξ a generic point of D, ξ a geometricpoint of X above ξ, X(ξ) the corresponding strictly local scheme, η its generic point andr the multiplicity of R at ξ. Then the following conditions are equivalent ([5] 8.8):

(i) The ramification of F at ξ is bounded by R+.(ii) The sheaf F |η is trivialized by a finite étale connected covering η′ of η such

that the logarithmic ramification of η′/η is bounded by r+ (5.2).The conductor of F at ξ is defined to be the minimum of the set of rational numbers

r > 0 such that F |η is trivialized by a finite étale connected covering η′ of η andthat the logarithmic ramification of η′/η is bounded by r+ (5.2). The conductor of Frelatively to X is defined to be the effective rational divisor on X with support in Dwhose multiplicity at any generic point ξ of D is the conductor of F at ξ ([5] 8.10).

Definition 5.11 ([12] 2.6). Let Y be a k-scheme, Z a closed subscheme of Y ,V = Y −Z the complementary open subscheme of Z in Y that is connected and smoothover Spec(k) and F a locally constant and constructible sheaf of Λ-modules on V . For

68 6. CLEAN `-ADIC SHEAVES AND CHARACTERISTIC CYCLES

any geometric point y of Y , Y(y) denotes the strict localization of Y at y. Then F istamely ramified along Z if the following equivalent conditions are satisfied:

(i) For each geometric point y of Y and each geometric point x ∈ Y(y) ×X V , thep-sylow sub-groups of the étale fundamental group π1(Y(y)×Y V, x) act triviallyon Fx.

(ii) For each geometric point y of Y , there exists an étale neighborhood W of yand a Galois étale covering T of W ×Y V of order prime to p, such that thepull-back of F on T is a constant sheaf.

Moreover, if Y is smooth over Spec(k) and Z is a divisor with simple normal crossingon Y , F is tamely ramified along Z if and only if

(iii) For any geometric point ξ of Z localized at a generic point of Z, the pull-backof F on the generic point of the trait X(ξ) is tamely ramified in the usual sense.

Lemma 5.12 ([18] 2.21). Let F be a locally constant and constructible sheaf ofΛ-modules on U . Then the following conditions are equivalent:

(i) F is tamely ramified along D.(ii) The conductor of F vanishes.(iii) The ramification of F along D is bounded by 0+.

5.13. Let F be a locally constant and constructible sheaf of Λ-modules on U . Letξ be a generic point of D, X(ξ) the henselization of X at ξ, ηξ the generic point of X(ξ),ηξ a geometric generic point of X(ξ) and Gξ the Galois group of ηξ over ηξ. We say thatF is isoclinic at ξ if the representation Fηξ of Gξ is isoclinic (5.4). We say that F isisoclinic along D if it is isoclinic at all generic points of D ([5] 8.22).

6. Clean `-adic sheaves and characteristic cycles

Definition 6.1 ([5] 3.1). Let X be a k-scheme, π : E → X a vector bundle, andF a constructible sheaf of Λ-modules on E. We say that F is additive if for everygeometric point x of X and for every e ∈ E(x), denoting by τe the translation by e onEx = E ×X x, τ ∗e (F |Ex) is isomorphic to F |Ex .

6.2. Let Lψ be the Artin-Schreier sheaf of Λ-modules of rank 1 over the additivegroup scheme A1

Fp over Fp, associated to the character ψ (2.2) ([17] 1.1.3). If µ :

A1Fp ×Fp A1

Fp → A1Fp denotes the addition, we have an isomorphism

µ∗Lψ∼−→ pr∗1Lψ ⊗ pr∗2Lψ.

Hence, Lψ is additive (6.1). If f : X → A1Fp is a morphism of schemes, we put Lψ(f) =

f ∗Lψ.

6.3. Let X be a k-scheme, π : E → X a vector bundle of constant rank d andπ : E → X its dual bundle. We denote by 〈 , 〉 : E ×X E → A1

Fp the canonical pairing,by pr1 : E ×X E → E and pr2 : E ×X E → E the canonical projections and by

Fψ : Dbc(E,Λ)→ Db

c(E,Λ)

the Fourier-Deligne transform defined by ([17] 1.2.1.1)

Fψ(K) = Rpr2!(pr∗1K ⊗Lψ(〈 , 〉)).Let π[ : E[ → X be the bidual vector bundle of π : E → X, a : E → E[ the anti-

canonical isomorphism defined by a(x) = −〈x, 〉, and F∨ψ the Fourier-Deligne transform

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 69

for π : E → X. For every object K of Dbc(E,Λ), we have a canonical isomorphism ([17]

1.2.2.1)F∨ψ Fψ(K)

∼−→ a∗(K)(−d)[−2d]. (6.3.1)Let π′ : E ′ → X be a vector bundle of constant rank d′, F′ψ its Fourier-Deligne

transform, f : E → E ′ a morphism of vector bundles, and f : E ′ → E its dual. Forevery object K ′ of Db

c(E′,Λ), we have canonical isomorphisms ([5] 3.4.6, 3.4.7)

Rf! F′ψ(K ′)(d′)[2d′]∼−→ Fψ f ∗(K ′)(d)[2d], (6.3.2)

Rf∗ F′ψ(K ′)∼−→ Fψ Rf !(K ′). (6.3.3)

6.4. Let X be a k-scheme and K an object of Dbc(X,Λ). The support of K is the

subset of points of X where the stalks of the cohomology sheaves of K are not all zero.It is constructible in X.

Proposition 6.5 ([5] 3.6). Let X be a k-scheme, π : E → X a vector bundle ofconstant rank, π : E → X its dual bundle, F a constructible sheaf of Λ-modules on Eand S ⊂ E the support of Fψ(F ). Then, F is additive if and only if for every x ∈ X,the set S ∩ Ex is finite.

Definition 6.6 ([5] 3.8). Let X be a k-scheme, π : E → X a vector bundle ofconstant rank, π : E → X its dual bundle, and F an additive constructible sheaf ofΛ-modules on E. We call the Fourier dual support of F the support of Fψ(F ) in E.We say that F is non-degenerated if the closure of its Fourier dual support does notmeet the zero section of E.

If we replace ψ by aψ for an element a ∈ F×p , then the Fourier dual support of F

will be replaced by its inverse image by the multiplication by a on E. In particular, thenotion of being non-degenerated dose not depend on ψ.

Lemma 6.7 ([18] 2.6). Let X be a k-scheme, π : E → X a vector bundle of constantrank, s : X → E the zero section of π, µ : E ×X E → E the addition and F and Gconstructible sheaves of Λ-modules on E, where F is additive. Let e ∈ Γ(X, s∗F ) be asection and u : F G → µ∗G a map such that the composed map

u|s(X)×E (e× idG ) : G → s∗F G → G

is the identity. Then G is additive and the Fourier dual support of G is a subset of thatof F .

Lemma 6.8 ([5] 3.10). Let X be a k-scheme, π : E → X a vector bundle of constantrank, and F an additive constructible sheaf of Λ-modules on E. If F is non-degenerate,Rπ∗F = Rπ!F = 0.

It follows form (6.3.1), (6.3.2) and (6.3.3) by applying f to the zero section of thedual bundle E of E and K ′ = Fψ(F ).

6.9. Let X be a connected smooth k-scheme of dimension d, D a divisor with simplenormal crossing on X, Dii∈I the irreducible components of D, R an effective Cartierdivisor of X with support in D, U = X −D and V = X − R the complementary opensubschemes of D and R in X respectively, j : U → X, jV : V → X and ν : U → V thecanonical injections. We denote by X∗kX (resp. V ∗kV ) the framed self-product ofX along D (resp. of V along D ∩ V ) (5.7), by δ : X → X∗kX the canonical lifting ofthe diagonal δ : X → X ×k X (5.7) and by (X∗kX)(R) the dilatation of X∗kX alongδ of thickening R, and we take the notation of (5.8). Moreover, we denote by

ν : U ×k U → V ∗kV and j(R)V : V ∗kV → (X∗kX)(R)

70 6. CLEAN `-ADIC SHEAVES AND CHARACTERISTIC CYCLES

the canonical injections, by E(R) the vector bundle (X∗kX)(R) ×X R over R (5.8.2),and by E(R) its dual bundle.

Let F be a locally constant and constructible sheaf of free Λ-modules on U . We put

H = Hom(pr∗2F , pr∗1F )

on U ×k U .Proposition 6.10 ([5] 8.15, 8.17). We keep the assumptions and notation of 6.9,

moreover, we assume that the ramification of F along D is bounded by R+ (5.9). Then,

(i) j(R)∗ H |E(R) is additive. Let S0

R(F ) ⊂ E(R) be its Fourier dual support.(ii) S0

R(F ) is the underlying space of a closed subscheme of E(R) which is finiteover R.

Proposition 6.11. We keep the assumptions and notation of 6.9, moreover, weassume that the ramification of F along D is bounded by R+. Then, for any integerq > 0, Rqj

(R)V ∗ (ν∗(H ))|E(R) is additive. Let SqR(F ) ⊂ E(R) be the Fourier dual support

of Rqj(R)V ∗ (ν∗(H ))|E(R), we have SqR(F ) ⊆ S0

R(F ).

Proof. We focus on the situation q > 1 since the case q = 0 is due to 6.10. For ascheme Y overX×kX, we denote by f1, f2 : Y → X the maps induced by the projectionspr1, pr2 : X ×k X → X, respectively. We denote the fiber product Y ×f2,X,f1 Y simplyby Y ×X Y .

By ([5] 5.34, [18] 2.24), there exists a morphism λ : (X∗kX)×X(X∗kX)→ X∗kXthat lifts the composed map (X ×k X) ×X (X ×k X)

∼−→ X ×k X ×k Xpr13−−→ X ×k X,

and a smooth morphism µ : (X∗kX)(R)×X (X∗kX)(R) → (X∗kX)(R) that makes thediagram

(X∗kX)(R) ×X (X∗kX)(R) µ //

(X∗kX)(R)

(X∗kX)×X (X∗kX)

λ // X∗kXcommutative, where the vertical arrows are the canonical projections. Moreover, thepull-back of µ by the canonical injection E(R) → (X∗kX)(R)

µ(R) : E(R) ×R E(R) → E(R)

is the addition of the bundle E(R) ([5] 5.35). Hence, we have a canonical commutativediagram with Cartesian squares

(U ×k U)×X (U ×k U) = U ×k U ×k Uν

pr13 // U ×k U

j(R)

ν

(V ∗kV )×X (V ∗kV )

µV //

j(R)V

V ∗kVj(R)V

(X∗kX)(R) ×X (X∗kX)(R) µ // (X∗kX)(R)

where ν and j(R)

V are canonical injections. By adjunction, we have canonical maps

ν∗(H )L ν∗(H ) → ν∗(H H ), (6.11.1)

Rj(R)V ∗ (ν∗(H ))L Rj

(R)V ∗ (ν∗(H )) → Rj

(R)

V ∗ (ν∗(H )L ν∗(H )). (6.11.2)

On (U ×k U)×X (U ×k U) = U ×k U ×k U , we have

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 71

H H = Hom(pr∗2F , pr∗1F )⊗Hom(pr∗3F , pr∗2F ),

that gives a mapH H →Hom(pr∗3F , pr∗1F ) = pr∗13H . (6.11.3)

Since µ is smooth, by the smooth base change theorem, we have an isomorphism

µ∗(Rj(R)V ∗ (ν∗(H )))

∼−→ Rj(R)

V ∗ (ν∗(pr∗13(H ))). (6.11.4)

The maps (6.11.1), (6.11.2), (6.11.3) and the inverse of (6.11.4) induce a map

Rj(R)V ∗ (ν∗(H ))L Rj

(R)V ∗ (ν∗(H ))→ µ∗(Rj(R)

V ∗ (ν∗(H ))). (6.11.5)

Consider the following commutative diagram with Cartesian squares

U ×U (U ×k U)δU×id //

ν

(U ×k U)×X (U ×k U)

ν

V ×V (V ∗kV )δV ×id //

j(R)V

(V ∗kV )×X (V ∗kV )

j(R)V

X ×X (X∗kX)(R) δ(R)×id// (X∗kX)(R) ×X (X∗kX)(R)

Notice that µ (δ(R) × id) = id ([5] 5.35). Pulling back (6.11.5) by δ(R) × id, we obtaina commutative diagram

(δ(R) × id)∗(j(R)∗ (H ) Rqj

(R)V ∗ (ν∗(H ))) //

θ

(δ(R) × id)∗µ∗(Rqj(R)V ∗ (ν∗(H )))

j∗δ∗U(H ) Rqj(R)V (ν∗(H ))

ϑ // Rqj(R)V (ν∗(H ))

where θ is an isomorphism induced by the base change isomorphism (5.9.1)

δ(R)∗j(R)∗ (H )

∼−→ j∗δ∗U(H ).

On U ×U (U ×k U) = U ×k U , we have

δ∗U(H )H = Hom(pr∗2F , pr∗2F )⊗Hom(pr∗2F , pr∗1F ),

which induces a map

δ∗U(H )H →Hom(pr∗2F , pr∗1F ) = H . (6.11.6)

The morphism ϑ is the following composed map

j∗δ∗U(H ) Rqj

(R)V (ν∗(H ))→ Rqj

(R)V ∗ ν∗(δ

∗U(H )H )→ Rqj

(R)V (ν∗(H )),

where the second arrow is induced by (6.11.6). The map

ε : Λ→ j∗δ∗U(H ) (6.11.7)

associated to the element idF ∈ Γ(X, j∗δ∗U(H )) = End(F ) induces the identity

H∼−→ ΛH

ε|U×id−−−−→ δ∗U(H )H(6.11.6)−−−−→H .

Hence ε and ϑ induce the identity of Rqj(R)V (ν∗(H )). Restrict (6.11.5) to E(R) ×R E(R),

we obtain a map

(j(R)∗ (H )|E(R)) (Rqj

(R)V ∗ (ν∗(H ))|E(R))→ µ(R)∗(Rqj

(R)V ∗ (ν∗(H ))|E(R)). (6.11.8)

Notice that the zero section s(R) : R → E(R) is just the pull-back of δ(R) : X →(X∗kX)(R) by E(R) → (X∗kX)(R). After restricting (6.11.8) to s(R)(R) ×R E(R), the

72 6. CLEAN `-ADIC SHEAVES AND CHARACTERISTIC CYCLES

map ε|R (6.11.7) induces the identity of Rqj(R)V ∗ (ν∗(H ))|E(R) . Hence, the proposition

follows from (6.7) (applied with F = j(R)∗ (H )|E(R) and G = Rqj

(R)V ∗ (ν∗(H ))|E(R)).

Definition 6.12 ([5] 8.23). We keep the assumptions and notation of 6.9, more-over, we assume that the conductor of F relatively to X is the effective divisor R (5.10)and that F is isoclinic along D (5.13). We say that F is clean along D if the followingconditions are satisfied:

(i) the ramification of F along D is bounded by R+;(ii) the additive sheaf j(R)

∗ H |E(R) on E(R) is non-degenerated (6.6).

Proposition 6.13. We keep the assumptions and notation of 6.9, moreover, weassume that the conductor of F relatively to X is the effective divisor R and that F isisoclinic and clean along D (6.12). Then, we have

RΓE(R)((X∗kX)(R), j(R)∗ (H )(d)) = 0.

Proof. We denote by i : E(R) → (X∗kX)(R) the canonical injection and π :E(R) → R the canonical projection. Notice that ([5] 5.26)

V ∗kV = (X∗kX)(R) ×X V = (X∗kX)(R) − E(R),

then

Rqi!(j(R)∗ H ) =

0 when q 6 1;

i∗Rq−1j(R)V ∗ (ν∗(H )) when q > 2.

Since F is clean along D, for any integer q, the sheaf i∗Rq−1j(R)V ∗ (ν∗(H )) on E(R) is

additive and non-degenerated (6.11). Hence, for any integer q, Rπ∗Rqi!(j(R)∗ (H )) = 0

(6.8). Hence,

RΓE(R)((X∗kX)(R), j(R)∗ (H )(d)) = RΓ(R,Rπ∗Ri

!(j(R)∗ (H ))(d)) = 0.

Remark 6.14. Proposition 6.13 is used in the proof of ([18] 3.4). However, the

proof of loc. cit. relies on ([18] 2.25) which is not enough. We reinforce it in 6.11.

6.15. We keep the assumptions and notation of 6.9 and we denote by

T∗X(logD) = V(Ω1X/k(logD)∨),

the logarithmic cotangent bundle of X, by σ : X → T∗X(logD) the zero section, fori ∈ I, by ξi the generic point of Di, by Fi the residue field of OX,ξi , by Si = Spec(OKi)the henselization of X at ξi, by ηi = Spec(Ki) the generic point of Si, by Ki a separableclosure of Ki and by Gi the Galois group Gal(Ki/Ki).

We assume moreover that the conductor of F is R, and that F is isoclinic and cleanalong D. We denote byMi the Λ[Gi]-module corresponding to F |ηi . Since F is isoclinicalong D, Mi has just one slope ri. We put Iw = i ∈ I; ri > 0 and S =

∑i∈Iw Di. For

i ∈ Iw, letMi = ⊕χMi,χ

be the central character decomposition of Mi (5.4). Note that Mi,χ is a free Λ-moduleof finite type for all χ. By enlarging Λ, we may assume that for all central charactersχ of Mi, we have Λχ = Λ. Since GrrilogGi is abelian and killed by p ([18] 1.24), each χ

factors uniquely as GrrilogGi → Fpψ−→ Λ×, where ψ is the non-trivial additive character

fixed in 2.1. We denote also by χ the induced character and by

rsw(χ) : mriKi/mri+

Ki→ Ω1

Fi(log)⊗Fi F i

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 73

its refined Swan conductor (5.5.1) (where the notation are defined as in 5.5). Let Fχ bethe field of definition of rsw(χ), which is a finite extension of Fi contained in F i. Therefined Swan conductor rsw(χ) defines a line Lχ in T∗X(logD)×X Spec(Fχ). Let Lχ bethe closure of the image of Lχ in T∗X(logD). For i ∈ Iw, we put

CCi(F ) =∑

χ

ri · rkΛ(Mi,χ)

[Fχ : Fi][Lχ], (6.15.1)

which is a d-cycle on T∗X(logD) ×X Di. We define a d-cycle CC∗(F ) on the fiberproduct T∗X(logD)×X S by

CC∗(F ) =∑

i∈IwCCi(F ). (6.15.2)

We define the characteristic cycle of F and denote by CC(F ), the d-cycle onT∗X(logD) defined by ([18] 3.6)

CC(F ) = (−1)d (rkΛ(F )[σ(X)] + CC∗(F )) .

7. Tsushima’s refined characteristic class

7.1. In this section, X denotes a connected smooth k-scheme of dimension d, D adivisor with simple normal crossing on X and Dii∈I the irreducible components of D.We assume that I = It

∐Iw, and we put S =

⋃i∈Iw Di, T =

⋃i∈It Di, U = X −D and

V = X − S. We denote by j : U → X, jV : V → X and ν : U → V the canonicalinjections.

We denote by (X ×kX)′ the blow-up of X ×kX along Di×kDii∈I , by (X ×kX)†

the blow-up of X ×kX along Di×kDii∈It , by X okX the left-framed self-product ofX along D and by X∗kX the framed self-product of X along D (5.7). For any opensubschemes Y and Z of X, we put

(Y ×k Z)′ = (Y ×k Z)×(X×kX) (X ×k X)′,

(Y ×k Z)† = (Y ×k Z)×(X×kX) (X ×k X)†,

Y ∗kZ = (Y ×k Z)×(X×kX) (X∗kX).

Notice that (V ×k V )′ = (V ×k V )† and V ∗kV =∏

i∈It((X∗kX)i ×(X×kX) (V ×k V )).We have the following commutative diagram with Cartesian squares

U ×k U

U ×k U

ν //

ν2

V ∗kVϕ2

V ∗kV //

X∗kX

ϕ

U ×k U

// U ×k V

νo1 // V ok V

ϕ1

U ×k U

// U ×k V

ν†1 // (V ×k V )†

h

j†2 // (V ×k X)†

j†1 //

g

(X ×k X)†

f

U ×k U ν2

// U ×k V ν1

// V ×k V j2// V ×k X j1

// X ×k X

(7.1.1)

where all horizontal arrows are open immersions. We denote by j : U ×k U → X∗kXthe canonical injection.

We denote by δ : X → X×kX the diagonal map. By the universality of the blow-up,δ induces closed immersions

δ† : X → (X ×k X)† and δ : X → X∗kX, (7.1.2)

74 7. TSUSHIMA’S REFINED CHARACTERISTIC CLASS

and hence, by pull-back, the following closed immersions

δ†V : V → (V ×k V )† and δV : V → V ∗kV. (7.1.3)

7.2. In the following of this section, F denotes a locally constant and constructiblesheaf of free Λ-modules on U , tamely ramified along T ∩ V relatively to V . We put

H0 = Hom(pr∗2F , pr∗1F ) on U ×k U,H = RHom(pr∗2F ,Rpr!

1F ) on U ×k U,H = RHom(pr∗2j!F ,Rpr!

1j!F ) on X ×k X,H = j∗H0(d)[2d] on X∗kX.

We have a canonical isomorphism H∼−→H0(d)[2d].

7.3. We denote by H V the restriction of H to V ×k V and by HV the restrictionsof H to V ∗kV . Notice that H V

∼−→ RHom(pr∗2ν!F ,Rpr!2ν!F ). We put

H†V = ϕ1!(Rϕ2∗(HV )) on (V ×k V )†.

Since ν1 is an open immersion, the base change maps give by composition an isomor-phism (7.1.1)

ν†∗1 (H†V ) = ν†∗1 ϕ1!(Rϕ2∗(HV ))

∼−→ νo∗1 (Rϕ2∗(HV ))∼−→ Rν2∗H .

By (3.7.2), we have

h∗(H V )∼−→ h∗ν1!Rν2∗H

∼−→ ν†1!(Rν2∗H )∼−→ ν†1!(ν

†∗1 (H

†V )).

It induces a canonical map

h∗(H V )→H†V , (7.3.1)

that extends the identity of H on U ×kU . Since F is tamely ramified along the divisorT ∩ V relatively to V , the adjoint map

H V → Rh∗(H†V )

is an isomorphism by ([4] 2.2.4).

7.4. We put

H†

= j†1!(Rj†2∗(H

†V ))

on (X ×k X)†, and we consider the following composition of maps

f ∗H = f ∗j1!Rj2∗(H V )∼−→ j†1!g

∗Rj2∗(H V )→ j†1!Rj†2∗h∗(H V )

(7.3.1)−−−→ j†1!(Rj†2∗(H

†V )),(7.4.1)

where the second and the third arrows are induced by the base change maps.

Lemma 7.5 ([23] Lemma 3.13). The adjoint map of (7.4.1)

H → Rf∗H†

is an isomorphism.

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 75

7.6. We put X† = f−1(δ(X)) and S† = f−1(δ(S)). We denote by γ† : (X ×kX)†\X† → (X ×k X)† the canonical injection, which is an open immersion and put

L † = Rγ†∗(Λ). (7.6.1)

The map (7.4.1) induces by pull-back a map

H0X(X ×k X,H )→ H0

X†((X ×k X)†,H†). (7.6.2)

The adjunction Λ→ L † induces a map

H0X†((X ×k X)†,H

†)→ H0

X†((X ×k X)†,H† ⊗L L †). (7.6.3)

By (3.6.1), we have a canonical isomorphism End(j!F )∼−→ H0

X(X×kX,H ). We denotealso idj!F the image of idj!F ∈ End(j!F ) in H0

X(X ×k X,H ). Its image in H0X†((X ×k

X)†,H† ⊗L L †) by the composition of the maps (7.6.2) and (7.6.3) will be denoted by

α(j!F ).

Proposition 7.7 ([23], 3.14 and 3.15). The canonical map

H0S†((X ×k X)†,H

† ⊗L L †)→ H0X†((X ×k X)†,H

† ⊗L L †) (7.7.1)

is injective and there exists a unique element

α0(j!F ) ∈ H0S†((X ×k X)†,H

† ⊗L L †) (7.7.2)

whose image by (7.7.1) is α(j!F ).

7.8. The squares of the following commutative diagram

X

δ′

V

jVoo

V

δ†V

V

δV

Uνoo

δU

(X ×k X)† (V ×k X)†j†1oo (V ×k V )†

j†2oo V ∗kVϕ1ϕ2oo U ×k Uνoo

are Cartesian and all the horizontal arrows are open immersions. By (5.9) and (5.12),since F is tamely ramified along T ∩ V relatively to V , we have an isomorphism

δ∗V (ν∗(H0))∼−→ ν∗(δ

∗U(H0)). (7.8.1)

The base change maps give by composition an isomorphism

δ†∗(H†)∼−→ jV !δ

†∗V (H

†V )

∼−→ jV !δ∗V (HV )

∼−→ jV !ν∗(E nd(F ))(d)[2d], (7.8.2)

where the third arrow is (7.8.1). There exists a unique map

TrV : ν∗(E nd(F ))→ ΛV (7.8.3)

which extends the trace map Tr : E nd(F ) → ΛU ([4] (2.9)). The maps (7.8.2) and(7.8.3) give an evaluation map

ev† : δ†∗(H†)→ jV !(KV ). (7.8.4)

Composing with the canonical map jV !(KV )→ KX , we obtain a morphism

H0S(X, δ†∗(H

†)⊗L δ†∗L †)→ H0

S(X,KX ⊗L δ†∗L †). (7.8.5)

The pull-back by δ† gives a morphism

H0S†((X ×k X)†,H

† ⊗L L †)→ H0S(X, δ†∗(H

†)⊗L δ†∗L †). (7.8.6)

Composing (7.8.5) and (7.8.6), we get a map

H0S†((X ×k X)†,H

† ⊗L L †)→ H0S(X,KX ⊗L δ†∗L †). (7.8.7)

76 7. TSUSHIMA’S REFINED CHARACTERISTIC CLASS

Lemma 7.9 ([22] Lemma 2.3). The canonical map

H0S(X,KX)→ H0

S(X,KX ⊗L δ†∗L †) (7.9.1)induced by the canonical map Λ→ δ†∗L †, is an isomorphism.

7.10. Composing (7.8.7) and the inverse of (7.9.1), we get a map

κ : H0S†((X ×k X)†,H

† ⊗L L †)→ H0S(X,KX). (7.10.1)

We call κ(α0(j!F )) ∈ H0S(X,KX) the refined characteristic cycle of j!F , and we denote

it by CS(j!F ).

Remark 7.11. If T = ∅, we have (X ×k X)† = X ×k X, (V ×k V )† = V ∗kV =

U ×k U , X† = X, S† = S and, by (3.7.2), H†

= H . It is easy to see that (4.10)

CS(j!(F )) = C0S(j!(F )) ∈ H0

S(X,KX).

7.12. In the following of this section, Y denotes a connected smooth k-scheme, Z aclosed subscheme of Y and W = Y −Z the complementary open subscheme of Z in Y .We assume that there exists a proper flat morphism π : X → Y such that V = π−1(W ),that Q = π−1(W ) is a divisor with normal crossing, that S = Qred, that the canonicalprojection πV : V → W is smooth and that T ∩ V is a divisor with simple normalcrossing relatively to W . We have a commutative diagram with Cartesian squares

Uν //

πU

V

jV //

πV

X

π

QiQoo

πQ

WjW // Y Z

iZoo

We make the following remarks:(i) For any locally constant and constructible sheaf of Λ-modules G tamely ramified

along the divisor T∩V relatively to V , πV is universally locally acyclic relativelyto ν!(G ) ([6] Appendice to Th. Finitude, [19] 3.14). Since πV is proper, allcohomology groups of RπU !(G ) are locally constant and constructible on W .

(ii) Since π is proper, we have a push-forward

H0S(X,KX)

∼−→ H0Q(X,KX)→ H0

Z(Y,KY ) (7.12.1)

defined by applying the functor H0(Z,−) to the following composed map

RπQ∗(KQ)∼−→ RπQ∗Rπ

!Q(KZ)

∼−→ RπQ!Rπ!Q(KZ)→ KZ ,

where the third arrow is induced by the adjunction.

Theorem 7.13 (Localized Lefschetz-Verdier trace formula, [23] 5.4). We have(4.10, 7.10)

π∗(CS(j!(F ))) = C0Z(jW !(RπU !(F )))

in H0Z(Y,KY ).

7.14. We assume that Y is of dimension 1 and that Z is a closed point y of Y , andwe denote by y a geometric point of Y localized at y, by Y(y) the strict localization of Yat y, by η a geometric generic point of Y(y). For any object G of Db

ctf(W,Λ) with locallyconstant cohomology groups. We put ([6] Rapport 4.4)

rkΛ(Gη) = Tr(id; Gη),

swy(Gη) =∑

q∈Z(−1)qswy((H

q(G ))η),

dimtoty(Gη) = rkΛ(Gη) + swy(Gη),

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 77

where swy((H q(G ))η) denotes the Swan conductor of (H q(G ))η at y ([21] 19.3). By([23] 4.1), we have

C0y(jW !(G ))− rkΛ(G |η) · C0

y(jW !(ΛW )) = −swy(Gη) (7.14.1)

in H0y(Y,KY )

∼−→ Λ. In fact, the proof of (7.14.1) is simpler than the general casetreated in ([23] 4.1), since Y is of dimension 1, we can use the usual Swan conductorrather than the generalized one ([14] 4.2.2).

Corollary 7.15 ([23] 5.5). Keep the notation and assumptions of 7.14. We have

swy(RΓc(Uη,F |Uη))−rkΛ(F )·swy(RΓc(Uη,Λ)) = −π∗(CS(j!(F ))−rkΛ(F )·CS(j!(ΛU)))(7.15.1)

in H0y(Y,KY )

∼−→ Λ.

Proof. Since F is tamely ramified along T ∩ V relatively to V , we have ([12] 2.7)

rkΛ(RπU !(F )|η) = rkΛ(F ) · rkΛ(RπU !(ΛU)|η).Then, by (7.13) and (7.14.1),

π∗(CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU)))

= C0y(jW !RπU !(F ))− rkΛ(F ) · C0

y(jW !RπU !(ΛU))

= C0y(jW !RπU !(F ))− rkΛ(RπU !(F )|η) · C0

y(jW !(ΛW ))

− rkΛ(F ) ·(C0y(jW !RπU !(ΛU))− rkΛ(RπU !(Λ)|η) · C0

y(jW !(ΛW )))

= −swy(RπU !(F )|η) + rkΛ(F ) · swy(RπU !(Λ)|η).By the proper base change theorem (3.1.4), we have

RπU !(F )|η ∼−→ RΓc(Uη,F |Uη) and RπU !(Λ)|η ∼−→ RΓc(Uη,Λ).

Then (7.15.1) follows.

8. The conductor formula

8.1. In this section, we take again the assumptions of 7.1 and 7.2 and we will takethe notation introduced in §7. Let R be the conductor of F (5.9) that we assume havingintegral coefficients of support in S. We assume also that F is isoclinic and clean alongD (5.13 and 6.12). Notice that if R = 0, a sheaf F is tamely ramified along D and isautomatically isoclinic and clean.

Theorem 8.2. Let F be a sheaf on U as in 8.1. Assume that T ∩ S = ∅ or thatrkΛ(F ) = 1. Then, we have (2.4)

CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU))

= (−1)d rkΛ(F ) · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(logD))XS∩ [X] ∈ H0

S(X,KX),

where the right hand side is considered as an element of H0S(X,KX) by the cycle map

(3.10.2).

The theorem will be proved in 8.20 after some preliminaries. We will deduce fromit the theorem 1.3 in 8.26. The case where rkΛ(F ) = 1 is due to Tsushima ([23] 5.9).

Remark 8.3. Although we follow the same lines as [22] for sheaves of higher ranks,the situation is technically more involved. The assumption S ∩T = ∅ is required for theinjectivity of a map λ defined in (8.6.4), which is a crucial step in my proof (cf. 8.17).We don’t know if it holds without this assumption.

78 8. THE CONDUCTOR FORMULA

8.4. We consider X∗kX as an X-scheme by the second projection, and we denoteby (X∗kX)(R) the dilatation of X∗kX along δ of thickening R (5.6). We have aCartesian diagram (5.8.1)

U

δU //

j

U ×k Uj(R)

Xδ(R)// (X∗kX)(R)

We denote by f (R) : (X∗kX)(R) → X ×k X and ϕ(R) : (X∗kX)(R) → (X ×k X)† thecanonical projections. We put X(R) = f (R)−1(δ(X)) and S(R) = f (R)−1(δ(S)).

We putH (R) = j(R)

∗ (H0)(d)[2d]

on (X∗kX)(R). Notice that H (R)|V ∗kV = HV (7.3).

Proposition 8.5 ([18] Corollary 3.3). There exists a unique homomorphism (7.2)

f (R)∗(H )→H (R) (8.5.1)extending the identity of H on U ×k U .

8.6. We deduce from (8.5.1) by pull-back a map

H0X(X ×k X,H )→ H0

X(R)((X∗kX)(R),H (R)). (8.6.1)We put (7.6.1)

L (R) = ϕ(R)∗(L †). (8.6.2)The canonical map Λ→ L (R) induces a map

H0X(R)((X∗kX)(R),H (R))→ H0

X(R)((X∗kX)(R),H (R) ⊗L L (R)). (8.6.3)

The canonical injection S(R) → X(R) induces a map

λ : H0S(R)((X∗kX)(R),H (R) ⊗L L (R))→ H0

X(R)((X∗kX)(R),H (R) ⊗L L (R)). (8.6.4)

8.7. We denote by V = X(R) − S(R) the complementary open subscheme of S(R) inX(R), by ιV : V → V ∗kV , γV : (V ∗kV )\V → V ∗kV and γU : (U ×k U)\δU(U) →U ×k U the canonical injections. We put D(R) = f (R)−1(δ(D)) and T (R) = f (R)−1(δ(T )).Notice that T (R) ∪ S(R) = D(R) and that δU(U) is the complementary open subschemeof D(R) in X(R). We put

LV = RγV ∗(Λ).

Notice that LV∼−→ L †|V ∗kV = L (R)|V ∗kV (7.6.1) and (8.6.2).

Proposition 8.8 ([22] 2.2). If the sheaf F on U has rank 1, the map λ (8.6.4)is an isomorphism.

Proof. It is sufficient to show that, for any integer q, Hq

V(V ∗kV, HV ⊗L LV ) = 0.

Since F is of rank 1 and is tamely ramified along T ∩V relatively to V , ν∗H0 is a locallyconstant and constructible sheaf on V ∗kV ([4] 4.2.2.1). By ([7] 6.5.5), we have

HV ⊗L LV∼−→ RγV ∗(γ

∗V (HV )).

Since Rι!Rγ∗ = 0 (3.1.5), for any integer q,

Hq

V(V ∗kV, HV ⊗L LV ) = Hq(V ,Rι!RγV ∗(γ

∗V (HV ))) = 0.

Proposition 8.9. If T ∩ S = ∅, the map λ (8.6.4) is injective.

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 79

Proof. Since H0 is locally constant, by ([7] 6.5.5), for any integer q,

HqU(U×kU,H ⊗LRγU∗(Λ))

∼−→ HqU(U×kU,RγU∗γ∗U(H ))

∼−→ Hq(U,Rδ!URγU∗γ

∗U(H )) = 0.

Hence, we have a canonical isomorphism

H0D(R)((X∗kX)(R),H (R) ⊗L L (R))

∼−→ H0X(R)((X∗kX)(R),H (R) ⊗L L (R)).

Since T∩S = ∅, we have T (R)∩S(R) = ∅. Hence, for any object G ofDbctf((X∗kX)(R),Λ),

H0D(R)((X∗kX)(R),G ) = H0

S(R)((X∗kX)(R),G )⊕ H0T (R)((X∗kX)(R),G ).

In particular, the canonical map

H0S(R)((X∗kX)(R),H (R) ⊗L L (R))→ H0

D(R)((X∗kX)(R),H (R) ⊗L L (R))

is injective. Hence λ is injective.

8.10. The pull-back by δ(R) gives a map

H0S(R)((X∗kX)(R),H (R) ⊗L L (R))→ H0

S(X, δ(R)∗(H (R))⊗L δ†∗(L †)).

Since the conductor of F is R and F is isoclinic and clean alongD (8.1), the ramificationof F along D is bounded by R+ (6.12). Hence, we have an isomorphism (5.9)

δ(R)∗(j(R)∗ (H0))

∼−→ j∗(δ∗U(H0)).

We have an evaluation map

δ(R)∗(H (R))∼−→ j∗(δ

∗U(H0))(d)[2d] = j∗(E nd(F ))(d)[2d]→ (j∗(ΛU))(d)[2d] = KX ,

where the third arrow is the push-forward of the trace map Tr : E nd(F ) → ΛU . Itinduces a map

ev(R) : H0S(X, δ(R)∗(H (R))⊗L δ†∗(L †))→ H0

S(X,KX ⊗L δ†∗(L †))∼−→ H0

S(X,KX),

where second arrow is the inverse of the isomorphism (7.9.1).

8.11. By (3.4.2), we have a map

H0(X, δ(R)∗j(R)∗ (H0))× H0

S(X, δ†∗(L †)(d)[2d])∪S−→ H0

S(X, δ(R)∗(H (R))⊗L δ†∗(L †)).

In the following of this section, we denote by e ∈ H0(X, δ(R)∗j(R)∗ (H0)) the unique pre-

image of idF ∈ End(F ) = H0(X, j∗δ∗(H0)). The following diagram is commutative

H0S(X, δ(R)∗(H (R))⊗L δ†∗(L †))

ev(R)// H0

S(X,KX)

(7.9.1)

H0S(X, δ†∗(L †)(d)[2d])

e∪S−OO

· rkΛ(F )// H0

S(X,KX ⊗L δ†∗(L †))

(8.11.1)

since the composition of the following morphisms

H0(X, δ(R)∗j(R)∗ (H0))→ H0(X, j∗δ

∗U(H0))

ev−→ H0(X,Λ)

maps e ∈ H0(X, δ(R)∗j(R)∗ (H0)) to rkΛ(F ) ∈ Λ = H0(X,Λ).

80 8. THE CONDUCTOR FORMULA

8.12. We have a commutative diagram with Cartesian squares (7.1)

V ∗kVj

(R)V

V ∗kV

ϕ2 // V ok Vϕ1 // (V ×k V )†

j†2 // (V ×k X)†

j†1

(X∗kX)(R)

ϕ(R)

11// X∗kX ϕ // (X ×k X)†

(8.12.1)

The base change maps give by composition the following isomorphism

ϕ(R)∗(H†) = ϕ(R)∗j†1!Rj

†2∗(H

†V )

∼−→ j(R)V ! (j†2 ϕ1 ϕ2)∗Rj†2∗(H

†V )

∼−→ j(R)V ! HV ,

which induces a map

ϕ(R)∗(H†)∼−→ j

(R)V ! (HV )→ j

(R)V ∗ (ν∗(H0))(d)[2d] = H (R). (8.12.2)

By (8.5), the composed map

f (R)∗(H ) = ϕ(R)∗(f ∗(H ))(7.4.1)−−−→ ϕ(R)∗(H

†)

(8.12.2)−−−−→H (R)

is equal to (8.5.1). We deduce by pull-back a commutative diagram

H0X(X ×k X,H )

(7.6.2)//

(8.6.1) **

H0X†((X ×k X)†,H

†)

H0X(R)((X∗kX)(R),H (R))

(8.12.3)

8.13. We have the following diagrams with commutative squares

H0X†((X ×k X)†,H

†)

(7.6.3)// H0

X†((X ×k X)†,H† ⊗L L †)

H0X(R)((X∗kX)(R),H (R))

(8.6.3)// H0

X(R)((X∗kX)(R),H (R) ⊗L L (R))

H0X((X∗kX)(R),Λ(d)[2d])

e∪−OO

θ // H0X((X∗kX)(R),L (R)(d)[2d])

e∪−OO

(8.13.1)

H0X†((X ×k X)†,H

† ⊗L L †)

H0S†((X ×k X)†,H

† ⊗L L †)? _(7.7.1)

oo

H0X(R)((X∗kX)(R),H (R) ⊗L L (R))

(1)

H0S(R)((X∗kX)(R),H (R) ⊗L L (R))

λoo

H0X((X∗kX)(R),L (R)(d)[2d])

e∪−OO

H0S((X∗kX)(R),L (R)(d)[2d])

e∪S−OO

λ0oo

(8.13.2)

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 81

H0S†((X ×k X)†,H

† ⊗L L †)

(7.8.6)// H0

S(X, δ†∗(H†)⊗L δ†∗(L †))

(7.8.5)//

H0S(X,KX)

H0S(R)((X∗kX)(R),H (R) ⊗L L (R)) // H0

S(X, δ(R)∗(H (R))⊗L δ†∗(L †))ev(R)

//

(8.11.1)

H0S(X,KX)

H0S((X∗kX)(R),L (R)(d)[2d])

e∪S−OO

// H0S(X, δ†∗(L †)(d)[2d])

· rkΛ(F )//

e∪S−OO

H0S(X,KX)

(8.13.3)where

i. The arrows from the upper row to the middle row are the pull-backs by ϕ(R);ii. The arrows e∪− are the cup products (3.3.2) and the arrows e∪S − are given

in (3.4.2);iii. The arrows θ are induced by the canonical map Λ→ L (R);iv. The arrows λ0 is the canonical map induced by the injection S → X;v. The square (1) is commutative by (3.4.1);vi. The arrows under (7.8.6) are the pull-back by δ(R);vii. We use the canonical isomorphism H0

S(X,KX)∼−→ H0

S(X,KX ⊗L δ†∗(L †)) (cf.7.9).

Lemma 8.14. For any integer q, the canonical maps

HqS((X∗kX)(R),L (R)(d))→ Hq

X((X∗kX)(R),L (R)(d)),

are isomorphisms. In particular, the map λ0 in (8.13.2) is an isomorphism.

Proof. It is sufficient to show that, for any integer q,

HqV (V ∗kV,RγV ∗Λ(d)) = 0,

which follows from the fact that Rδ!V RγV ∗Λ(d) = 0 (3.1.5).

8.15. For any integer q, any object G of Dbctf((X∗kX)(R),Λ) and any closed sub-

scheme Z ∈ (X∗kX)(R), we denote by

HqZ((X∗kX)(R),G )→ Hq

Z((X∗kX)(R),G ⊗L L (R)), x 7→ xa,

the morphism induced by the canonical map Λ → L (R). For any closed immersionZ → Y of closed subschemes of (X∗kX)(R), we denote abusively by

HqZ((X∗kX)(R),G )→ Hq

Y ((X∗kX)(R),G ), x 7→ x,

the canonical map.

Proposition 8.16 ([18] 3.3, 3.4). We denote by [X] ∈ H0X((X∗kX)(R),Λ(d)[2d])

the cycle class of δ(R)(X). Then we have (7.6), (8.6.1)

f (R)∗(idj!(F )) = e ∪ [X] ∈ H0X(R)((X∗kX)(R),H (R)).

The proof in ([18] 3.4) should be modified as in 6.13.

Proposition 8.17. If T ∩ S = ∅ or if rkΛ(F ) = 1, we have (7.10)

CS(j!(F )) = rkΛ(F ) · δ(R)∗(λ−10 ([X]a)) ∈ H0

S(X,KX). (8.17.1)

Proof. By (8.12.3), (8.13.1) and 8.16, we have (7.6)

ϕ(R)∗(α(idj!(F ))) = e ∪ ([X]a) ∈ H0X(R)((X∗kX)(R),H (R) ⊗L L (R)).

82 8. THE CONDUCTOR FORMULA

Then, by 8.8, 8.9, 8.14 and (8.13.2), we have (7.7.2)

ϕ(R)∗(α0(j!(F ))) = e ∪S (λ−10 ([X]a)) ∈ H0

S(R)((X∗kX)(R),H (R) ⊗L L (R)).

Equation (8.17.1) follows form (8.13.3).

Lemma 8.18. We put X(R) = ϕ(R)−1(δ†(X)) and S(R) = ϕ(R)−1(δ†(S)) (8.12.1).Then

(i) There exists a unique element τ ∈ CHd(S(R)) which maps to [X] − ϕ(R)![X] ∈

CHd(X(R)), and we have

δ(R)!(τ) = (−1)d · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(log T ))XS∩ [X] ∈ CH0(S).

(8.18.1)(ii) We consider τ as an element in H0

S(R)((X∗kX)(R),Λ(d)[2d]) by the cycle map

(3.10). We have

τa = λ−10 ([X]a) ∈ H0

S(R)((X∗kX)(R),L (R)(d)[2d]). (8.18.2)

The proof of this lemma is similar to that of ([22] 3.7), in which the author considerthe case where supp(R) = S. It is an immediate application of ([14] 3.4.9).

Corollary 8.19. We have

δ(R)∗(λ−10 ([X]a)) (8.19.1)

= (−1)d · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(log T ))XS∩ [X] ∈ H0

S(X,KX),

where the right hand side is considered as an element of H0S(X,KX) by the cycle map.

Proof. Applying (3.10.3) to the map δ(R) : X → (X∗kX)(R), we have

δ(R)!(τ) = δ(R)∗(τ) ∈ H0S(X,KX), (8.19.2)

where we consider δ(R)!(τ) as an element of H0S(X,KX) by the cycle map. Since the

following diagram

H0S(R)

((X∗kX)(R),Λ(d)[2d])

// H0S(X,KX)

(7.9.1)

H0S(R)

((X∗kX)(R),L (R)(d)[2d]) // H0S(X,KX ⊗L δ†∗L †)

is commutative, where the horizontal arrows are the pull-backs by δ(R), we have

δ(R)∗(τ) = δ(R)∗(τa) ∈ H0S(X,KX). (8.19.3)

Hence, (8.19.1) follows form (8.18.1), (8.18.2), (8.19.2) and (8.19.3).

8.20. Proof of Theorem 8.2. By 8.17 and 8.19, we have

CS(j!(F )) = (−1)d rkΛ(F ) · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(log T ))XS∩ [X],

CS(j!(ΛU)) = (−1)d · cd(Ω1X/k(logD)− Ω1

X/k(log T ))XS∩ [X]

in H0S(X,KX). Hence,

CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU))

= (−1)d rkΛ(F ) · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(logD))XS∩ [X] ∈ H0

S(X,KX).

II. REFINED CHARACTERISTIC CLASS AND CONDUCTOR FORMULA 83

Remark 8.21 ([4] 4.2.1, [22] remark after 3.9). Observe that we have

(−1)d · cd(Ω1X/k(logD)⊗OX OX(R)− Ω1

X/k(logD))XS∩ [X] (8.21.1)

= −c(Ω1X/k(logD)∨) ∩ (1 + c1(OX(R)))−1 ∩ [R]dim 0

= (−1)d · c(Ω1X/k(logD)) ∩ (1− c1(OX(R)))−1 ∩ [R]dim 0 ∈ CH0(S).

8.22. we denote by

T∗X(logD) = V(Ω1X/k(logD)∨),

the logarithmic cotangent bundle of X. Since the R is supported in S, R =∑

i∈Iw riDi,where ri ∈ Z>0. By ([18] 3.16), For i ∈ Iw, we have (6.15.1)

CCi(F ) = ri · rkΛ(F ) · c(Ω1X/k(logD))∩ (1− c1(OX(R)))−1 ∩ [T∗X(logD)×X Di]dim d

in CHd(T∗X(logD)×X Di). Hence, we have (6.15.2)

CC∗(F ) = rkΛ(F ) · c(Ω1X/k(logD)) ∩ (1− c1(OX(R)))−1 ∩ [T∗X(logD)×X R]dim d

(8.22.1)in CHd(T

∗X(logD)×X S).

8.23. In the following, we take the notation and assumptions of 7.12, and we assumethat Y is of dimension 1 and that Z is a closed point y of Y , and we denote by y ageometric point localized at y, by Y(y) the strict localization of Y at y and by η ageometric generic point of Y(y).

Theorem 8.24. We assume that S = D (i.e., T = ∅) or that rkΛ(F ) = 1. Then,for any section s : X → T∗X(logD), we have

swy(RΓc(Uη,F |Uη))− rkΛ(F ) · swy(RΓc(Uη,Λ)) = (−1)d+1 deg(CC∗(F ) ∩ [s(X)])(8.24.1)

in H0y(Y,KY )

∼−→ Λ.

Proof. We denote by $ : T∗X(logD)→ X the canonical projection. Since $s =idX , we have

CC∗(F ) ∩ [s(X)]

= rkΛ(F ) · c(Ω1X/k(logD)) ∩ (1− c1(OX(R)))−1 ∩$∗[R] ∩ [s(X)]dim=0

= rkΛ(F ) ·$∗(c(Ω1X/k(logD)) ∩ (1− c1(OX(R)))−1 ∩ [R]dim=0) ∩ [s(X)]

= rkΛ(F ) · c(Ω1X/k(logD)) ∩ (1− c1(OX(R)))−1 ∩ [R]dim=0 ∈ CH0(S).

By 8.2 and (8.21.1), we get (3.10)

(−1)d(CC∗(F ) ∩ [s(X)]) = CS(j!(F ))− rkΛ(F ) · CS(j!(ΛU)) ∈ H0S(X,KX).

Hence, by (7.15), we have

swy(RΓc(Uη,F |Uη))− rkΛ(F ) · swy(RΓc(Uη,Λ)) = (−1)d+1π∗(CC∗(F ) ∩ [s(X)])

in H0y(Y,KY )

∼−→ Λ. It is easy to see that the composed map

CH0(S)cl−→ H0

S(X,KX)→ H0y(Y,KY )

∼−→ Λ,

where the second arrow is the push-forward (7.12.1), is just the degree map of zerocycles. We obtain (8.24.1).

Remark 8.25. Since π : X → Y is proper and T ∩ V is a divisor with simplenormal crossing relatively to W (7.12), the condition S ∩ T = ∅ in 8.2 implies T = ∅.

84 8. THE CONDUCTOR FORMULA

8.26. Proof of Theorem 1.3. Since F is tamely ramified along T ∩ V relatively toV , F |Uη is tamely ramified along (T ∩ V )η relatively to Vη. By ([12] 2.7, [18] 3.2), wehave

rkΛ(RΓc(Uη,F |Uη)) = (−1)d−1 rkΛ(F ) · cd−1(Ω1Vη/η

(log(T ∩ V )η)) ∩ [Vη]

= rkΛ(F ) · rkΛ(RΓc(Uη,Λ))

in H0(Vη,KVη)∼−→ Λ. Hence, we obtain (1.3.1) by 8.24.

Remark 8.27. We denote by K the function field of Y(y), by K a separable closureof K and by P the wild inertia subgroup of Gal(K/K). We assume that T = ∅ andthat Q is reduced. Notice that X ×Y Y(y) is semi-stable over the strict trait Y(y). Thenthe cohomology group

H∗(Uη,Λ) = H∗c(Uη,Λ)

is tame, i.e., the action of P is trivial ([13] 3.3). Hence, swy(RΓc(Uη,Λ)) = 0.

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