Betti Number Estimates in P-Adic Cohomology

8/10/2019 Betti Number Estimates in P-Adic Cohomology

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arXiv:1411

.2933v1

[math.AG

]11Nov2014

Betti number estimates inp-adic cohomology

Daniel Caro

Abstract

Within the framework of Berthelots theory of arithmetic D-modules, we prove the p-adic analogue of Bettinumber estimates and we give some standard applications.

Contents

1 Lagrangianity in the context of arithmeticD-modules 2

1.1 Convention and preliminaries on filtered modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Purity of the codimension for holonomic modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Preliminaries on tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Lagrangianity for log-extendable overconvergent isocrystals and the case of curves . . . . . . . . . . 7

2 Relative generic O-coherence 9

2.1 Inverse and direct images of complexes of arithmeticD-modules and characteristic varieties . . . . . 92.2 Some kind of Frobenius descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 The result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Betti numbers estimates 15

3.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 The result and some applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Introduction

Letkbe a perfect field of characteristic pand l be a prime number different to p. Whenkis algebraically closed,within the framework of Grothendiecks l -adic etale cohomology ofk-varieties, Bernstein, Beilinson and Delignein their famous paper on perverse sheaves, more precisely in [BBD82, 4.5.1] (or see the p-adic translation here in3.2.2), established some Betti number estimates. The goal of this paper is to get the same estimates in the contextof Berthelots arithmeticD-modules. We recall that this theory of Berthelot gives a p-adic cohomology stable undersix operations (see [CT12]) and admitting a theory of weights (see [AC13b]) analogous to that of Deligne in thel-adic side (see[Del80]). This allows us to consider Berthelots theory as a rightp-adic analogue of Grothendiecksl-adic etale cohomology. By trying to translate thel-adic proof of Betti number estimates in[BBD82,4.5.1] into theframework of arithmetic D-modules, two specific problems appear. The first one is that we do not have a notion oflocal acyclicity within the theory of arithmetic D-modules. We replace the use of this notion by another one that wemight call relative generic O-coherence (more precisely, see Theorem2.3.3). The goal of the chapter two is to provethis property. The proof of this relative generic O-coherence uses the lagrangianity of the characteristic variety of aunipotent overconvergentF-isocrystals. This lagrangian result is the main purpose of the first chapter. In the caseof curves, this yields easily the lagrangianity of the characteristic variety of a holonomic arithmetic D-modules. Werecall that, following Kedlayas semistable reduction theorem of [Ked11], any overconvergentF-isocrystal becomesunipotent after the pullback by some generically etale alteration. This semistable reduction theorem of Kedlaya willbe very useful in the proof of the relative generic O-coherence to reduce by descent to the case of a unipotent F-isocrystal and then to be able to apply the lagrangian result of the first chapter (see the step II.1. of2.3.3). In this first

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chapter, adapting Kashiwaras original proof in [Kas77] to our context, we also check the purity of the dimension ofBerthelots characteristic variety of a holonomic arithmeticD-module endowed with a Frobenius structure (see1.2.9).The second emerging problem when we follow the original l-adic proof of Betti number estimates is that we still do nothave vanishing cycles theory as nice as within thel-adic framework (so far, following [AC13a] we only have a p-adicanalogue of Beilinsons unipotent nearby cycles and vanishing cycles). Here, we replace successfully in the originalproof on Betti number estimates the use of vanishing cycles by that of some Fourier transform and of Abe-Marmora

formula ([AM, 4.1.6.(i)]) relating the irregularity of an isocrystal with the rank of its Fourier transform. We concludethis paper by the remark that these Betti number estimates allow us to state that the results of [BBD82,chapters 4 and5] are still valid (except [BBD82,5.4.78] because the translation is not clear so far).

Acknowledgment

I would like to thank Ambrus Pl for his invitation at the Imperial College of London. During this visit, we noticed thatthep-adic analogue of this Gabbers proof of purity theorem published in [BBD82] was not obvious since contrary tothel-adic case we did not have p-adic vanishing cycles. I would like to thank him for the motivation he inspired toovercome this problem.

Convention, notation of the paper

Let V be a complete discrete valued ring of mixed characteristic(0,p),be a uniformizer, Kits field of fractions,kitsresidue field which is supposed to be perfect. Iflis a field, anl-variety (or simply variety is there is no ambiguity withthe fieldl) is a separated reduced scheme of finite type over l. We recall that ak-varietyXis realizable if there existsan immersion of the formX P, whereP is a proper smooth formal scheme overV. In this paper,k-varieties willalways be supposed to be realizable. We will denote formal schemes by curly or gothic letters and the correspondingstraight roman letter will mean the special fiber (e.g. ifXis a formal scheme over V, thenXis thek-variety equalto the special fiber ofX). The topological space underlying to a k-variety Xis denoted by |X|. WhenM is a V-module, we denote byM its-adic completion and by MQ:=MV K. Concerning the notion of cohomologicaloperations in the theory of arithmetic D-modules, we follow the usual notation (for instance, see the beginning of[AC13b]). Moreover, is f: P P is a morphism of formal schemes overV, we denote by Lf the functor defined byLf(M) =OP,Q

Lf1OP,Q

f1M, for any complexM ofOP,Q-modules. Whenf is flat, we removeL in the notation.

IfT Sis a morphism of schemes and f: X Yis aS-morphism, then we denote by fT: XT YTor simply byf: XT YTthe base change of fbyT S.

If j :U Xis an open immersion of (realizable) varieties, the functor j! : Dbovhol(X/K) Dbovhol(U/K)(or thefunctor j! : Ovhol(X/K) Ovhol(U/K)) will simply be denoted by|Y(in other papers, to avoid confusion, it wassometimes denoted by Ybut, here, there is no risk of ambiguity since we do not work partially).

1 Lagrangianity in the context of arithmetic D-modules

1.1 Convention and preliminaries on filtered modules

We use here the terminology of Laumon in [Lau85, A.1]. For instance, a filtered ring(D,Di)is a ringD, unitary, nonnecessary commutative, with an increasing filtration indexed byZ. If there is no risk of confusion, we will simply saycomplete for ind-pro-complete. In this section,(D,Di)will be a complete filtered ring such that gr(D,Di)is a leftand right noetherian ring.

Definition 1.1.1. 1. A filtered(D,Di)-module(M,Mi)is said to be good if(Mi)iZ is a good filtration (see thedefinition given in [Lau85, A.1.0]). From [Lau85,A.1.2], a filtered(D,Di)-module is complete.

2. Following [Gro61,2.1.2], we say that a filtered(D,Di)-module (M,Mi)is free if(M,Mi)is a direct sum of somefiltered(D,Di)-modules of the form(D,Di)(n)(wherenis some integer and(D,Di)(n):= (D,Di+n)).

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1.1.2. We remark that Lemma [Lau83, 3.3.2] are still valid in our context if we suppose that the filtered (D,Di)-

modules are complete. In particular, let 0 (M,Mi ) f (M,Mi)

g (M,Mi) 0 be a sequence of good filtered

(D,Di)-modules such thatg f=0. Then 0 grM grM grM 0 is exact if and only if, for any i Z, the

sequences 0 Mif

Mig

Mi 0 are exact. When this property is satisfied, we say that 0 (M,Mi )

f

(M,Mi) g (M,Mi) 0 is an exact sequence of good filtered (D,Di)-modules.

Letu : (M,Mi) (N,Ni) be a morphism of good filtered (D,Di)-modules. From [Lau85, A.1.1.2], we noticethat the filtered(D,Di)-modules keruand Cokeruare good. Moreover, we have the exact sequences of good filtered(D,Di)-modules:

0 keru (M,Mi) Coimu 0,

0 Imu (M,Mi) Cokeru 0.

Hence, the category of good filtered(D,Di)-modules is exact and we get a notion of strict morphism of good filtered(D,Di)-modules (see [Lau83, 1.0]). A morphism u : (M,Mi) (N,Ni)of good filtered(D,Di)-modules is strict if andonly if the canonical morphism Coimu Imuis an isomorphism of filtered(D,Di)-modules if and only if, for anyi Z, we haveu(Mi) =u(M) Niif and only if kergr(u) =grker(u)and cokergr(u) =grcoker(u).

Remark 1.1.3. Let(M,Mi)be a filtered(D,Di)-module. We remark that(M,Mi)is a good filtered(D,Di)-module ifand only if there exists a epimorphism of the formu : (L,Li) (M,Mi), where(L,Li)is a free filtered(D,Di)-moduleof finite type, such thatu(Li) =Mi(i.e. such thatuis strict).

Notation 1.1.4(Localisation). Let fbe a homogeneous element of grD. We denote byD[f]the complete filtered ringof(D[f],D[f],i)relatively toS1(f):= {f

n ,n N} grD(see the definition after [Lau85,Corollaire A.2.3.4]).Let(M,Mi)be a good filtered(D,Di)-module. We put

(M[f],M[f],i):= (D[f],D[f],i) (D,Di)(M,Mi), (1.1.4.1)

the localized filtered module of(M,Mi) with respect to S1(f). We recall that(M[f],M[f],i) is also a good filtered

(D[f],D[f],i)-module (see [Lau85,A.2.3.6]) and grM[f]

grD[f] grD grM(see [Lau85, A.1.1.3]).

The results and proofs of Malgrange in [Mal76,IV.4.2.3] (we can also find the proof in the book[HTT08,D.2.2])can be extended without further problem in the context of complete filtered ring:

Lemma 1.1.5. Let(M,Mi)be a good filtered(D,Di)-module. Then there exist some free filtered(D,Di)-modules offinite type (Ln,Ln,i)with n Nand strict morphisms of good filtered(D,Di)-modules(Ln+1,Ln+1,i) (Ln,Ln,i)and(L0,L0,i) (M,Mi)such that LM is a resolution of M (in the category of D-modules).

In that case, we say that(L,L,i)is a good resolution of(M,Mi).

Proof. This is almost the same than [Mal76,IV.4.2.3.2]. For the reader, we recall the construction: with the remark1.1.3, there exists a strict epimorphism of good filtered (D,Di)-modules of the form0 : (L0,L0,i) (M,Mi), with(L0,L0,i)a free filtered(D,Di)-module of finite type. Let(M1,M1,i)be the kernel of0(in the category of good filtered(D,Di)-modules: see1.1.2). Since(M1,M1,i)is good, there exists a strict epimorphism of the form1 : (L1,L1,i) (M1,M1,i), with(L1,L1,i)a free filtered (D,Di)-module of finite type. Hence, the morphism (L1,L1,i) (L0,L0,i)isstrict. We go on similarly.

Remark 1.1.6. Let(L,L,i)be a good resolution of(M,Mi). Then gr(L,L,i)is a resolution of gr(M,Mi)by freegr(D,Di)-modules of finite type.

Lemma 1.1.7. Let K be a complex. Let(FiK)iZbe an increasing filtration of K. We put

FiHr(K):=Im(Hr(FiK

) Hr(K)).

Thengri(Hr(K))is a subquotient of Hr(griK

).

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Proof. For instance, we just have to copy the end of the proof of[HTT08, D.2.4] (or also Malgranges description ofthe corresponding spectral sequence in [Mal76,IV.4.2.3.2]).

Proposition 1.1.8. Let(M,Mi)be a good filtered(D,Di)-module. Let(N,Ni)be a filtered(D,Di)-module. For anyinteger r, there exists a filtration F ofExtrD(M,N)such that

1. ExtrD(M,N) = iZFi Extr

D(M,N),

2. 0= iZFi ExtrD(M,N)

3. grFExtrD(M,N)is a subquotient ofExtrgrD(grM,grN).

In particular, we have the implication

ExtrgrD(grM,grN) =0 Extr

D(M,N) =0.

Proof. We can follow the usual arguments: from 1.1.5,there exists a good resolution (L,L,i)of(M,Mi). We putK :=HomD(L,N). SinceLis a resolution ofMby projectiveD-modules, we getHr(K) =ExtrD(M,N).

LetFiKn be the subset of the elementsofHomD(Ln,N)such that, for any integer j Z,(Ln,j) Ni+j . SinceLn is a D-module of finite type, we get iZFiKn =Kn, iZFiKn =0. With the canonical induced filtration onHr(K) =ExtrD(M,N) (see1.1.7), this yields the first two properties. SinceLn is a free filtered (D,Di)-modules

of finite type, we check that the canonical morphism grKn HomgrD(grLn,grN)is an isomorphism. Since grL isa resolution of grMby projective grD-modules, this impliesHr(grK) =ExtrgrD(grM, grN). We conclude by usingLemma1.1.7.

1.2 Purity of the codimension for holonomic modules

Lemma 1.2.1. Let X be an affine smooth variety over k, D:=(X,D(0)

X/k),(Di)iNbe its filtration by the order of the

operators, f be an homogeneous element ofgrD. Let(M,Mi)and(N,Ni)be two good filtered(D[f],D[f],i)-modulesand r be an integer.

1. We havecodimExtrgrD[f](grM,grN) r.

2. If r


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1) Since ExtrD(0)X,Q

(N,D(0)X,Q) is coherent (right)

D(0)X,Q-module, from Theorem A and B of Berthelot, we get the

equality(X,ExtrD(0)X,Q

(N,D(0)X,Q)) =Ext

rDQ(N,

DQ). From1.2.5,this implies ExtrDQ(N,DQ) DQD[h],Q=0. Since theextensionDQD[h],Qis flat (see1.2.3), we get ExtrD[h],Q(N[h],D[h],Q)

ExtrDQ(N,

DQ) DQD[h],Q=0.2) a) Since Car(0)(N) =Supp(grM), then D(h0) Car(0)(N) =Supp((grM)h0 ). Since we have alsoD(h0)

Car(0)(N) =D(h0) V, then in particular we get Codim(grM)h0=r. Since(grM)h0 =gr(M[h0]), then from1.2.1forany i < rwe obtain ExtigrD[h0 ]

(gr(M[h0]), grD[h0]) = 0. Then it follows from 1.1.8 that for i < r, Exti

D[h0 ](M[h0],D[h0]) = 0.

On the other hand, from1.2.1we get for anyi>rthe inequality Codim(ExtigrD[h0]

(gr(M[h0]),grD[h0]))>r. Hence, by

reducingD(h0)if necessary (use again the remark1.2.6), for anyi >rwe get ExtigrD[h0 ](gr(M[h0]),grD[h0]) =0 and

then ExtiD[h0 ]

(M[h0],D[h0]) =0. To sum up, we have found an homogeneous elementh grDsuch that D(h0)and

fori =r, ExtiD[h0 ]

(M[h0],D[h0]) =0.

2) b) Now, sinceM[h]is withoutp-torsion,RHomD[h] (M[h],D[h]) LD[h]D[h0]

RHomD[h0 ]

(M[h0],D[h0]). From the

exact sequence of universal coefficients (e.g. see the beginning of the proof of[Vir00,I.5.8]), we get the inclusionExtiD[h] (M[h],

D[h]) D[h]D[h0] ExtiD[h0] (M[h0],D[h0]). Hence, for anyi =r, from the step 2) a) of the proof, we obtain

the vanishing ExtiD[h] (M[h],D[h]) D[h]D[h0] = 0. By using [Ber96, 3.2.2.(ii)], since Ext

iD[h] (M[h],D[h]) is a coherentD[h]-module, fori =rwe get ExtiD[h] (M[h],D[h]) =0 and then Ext

iD[h],Q(N[h],

D[h],Q) =0 (becauseD[h]D[h],Qis flat).3) From steps 1) and 2), we have checked that RHomD[h],Q(N[h],

D[h],Q) =0. By using the biduality isomorphism(see[Vir00,I.3.6]), we getN[h]=0, which is absurd following Lemma1.2.5because D(h0).

Theorem 1.2.8. LetX be a smooth V-formal scheme. Let r be an integer, Nbe a coherentD(0)X,Q-module such that

ExtsD(0)X,Q

(N,D(0)X,Q) =0for any s =r. Then, the characteristic variety Car

(0)(N)ofN is purely of codimension r.

Proof. IfVis an irreducible component of Car(0)(N)of codimension s, then from1.2.7 we get ExtsD(0)X,Q

(N,D(0)X,Q) = 0

since it contains V. Hences=r.

Corollary 1.2.9. LetX be a smooth integral V-formal scheme of dimension d. LetN=0be a holonomic F-DX,Q-

module. Then, the characteristic varietyCar(N)ofN is purely of codimension d.

Proof. The is a consequence of Virrions holonomicity characterization (see Theorem [Vir00,III.4.2] and of Theorem1.2.8.

1.3 Preliminaries on tangent spaces

Notation 1.3.1. Let f: X Ybe a morphism of smoothk-varieties. We denote byTX(resp. TY) the cotangentspace ofX(resp.Y), and byX: TX X,f: XYTY TYthe projections and byf: XYTY TX thecanonical morphism. We setTXY:=

1f (T

XX). We denote by Ufthe function from the set of subvarieties ofT

Xto

the set of subvarieties ofTYdefined by posing, for any subvarietyVofTX,Uf(V) := f(1f (V)). The application

U : f Uf is transitive (with respect to the composition), i.e. we have the equalityUg Uf=Ugffor anyg :YZ(see[HTT08,I.2.4.1]).

Lemma 1.3.2. Let u : Z X be an immersion of smooth k-varieties and f: XP be a morphism of smooth k-varieties.

1. Then we the equality1f (T

ZX) =T

ZP. In particular, if f is a immersion we get the identification Uf(T

ZX) =TZP (as subspaces of T

P).

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2. Moreover,1u (T

XX) =ZXT

XX T

ZX. When u is an open immersion, we have ZXT

XX=T

ZX.

Proof. The equality1f (T

ZX) =T

ZPcan be checked by considering the following diagram with cartesian squares:

TZZ

TZX

TZP

TZ ZXTX

u

u

ZP TP

TX XP TP,f

(1.3.2.1)

where, from[HTT08, I.2.4.1], the composition of the second row isfu(which justifies the cartesianity of the upperright square).

Concerning the second assertion, bothZXTXXand T

ZXare subvarieties ofZXTX. So, the inclusion is local

inXand Zand this is checked by computations in local coordinates.

Lemma 1.3.3. Let f: X P be a morphism of smooth k-varieties and(Xi)1irbe a family of smooth subvarietiesof X (resp. open subvarieties of X) such that X ri=1Xi. Then, T

XP

ri=1T

X

i

P (resp. TXP= ri=1T

X

i

P).

Proof. From the first equality of Lemma1.3.2, we reduce to the case X=P. Using the second part of Lemma1.3.2,we get the inclusionsXi XTXX T

Xi

X(resp. the equalitiesXi XTXX=T

XiX), which yields the desired result when

X=P.

Proposition 1.3.4. Let f: X Y be a morphism of smooth varieties.

1. If f is tale, thenfis an isomorphism, TXY= 1f (T

YY), Uf(T

XX) =f(T

XY) T

YY .

2. If f is smooth thenUf(TXX) T

YY .

3. When f is a smooth surjective morphism, we have the equality Uf(TXX) =T

YY .

Proof. First, suppose that fis etale. Using[Gro61,1.7.11.(iv)], we check thatfis an isomorphism. By computation

in local coordinates, we getT

XY=1

f (T

YY), which implies the end of the first assertion. Let us prove the secondassertion. Since this is local inX(indeed, for instance use Lemma1.3.3and the transitivity of the applicationU), wecan suppose that there exists an etale morphism of the formX AnYwhose composition with the projection A

nY Y

gives f. By transitivity ofUwith respect to the composition (see1.3.1) and by using the tale case, we reduce to thecase where fis the projection AnY Y. Since this is local relatively to Y, we can suppose that Yhas local coordinates.Then, this is an easy computation. When fis a smooth surjective morphism, we get the last assertion because in thatcaseY(Uf(TXX)) = Y.

1.4 Lagrangianity for log-extendable overconvergent isocrystals and the case of curves

1.4.1. Let P be a smooth formal scheme over V. We recall that for any exact sequence 0 E E E 0 ofcoherentF-D

P,Q-modules, we have the equality Car(E) =Car(E) Car(E)([Ber02,5.2.4.(i) and 5.2.7]). Let Ebe

a coherentF-DX,Q-module. We say that E is Lagrangian (resp. isotropic) if Car(E)is Lagrangian (resp. isotropic).

From1.2.9, E is Lagrangian if and only ifE is holonomic and isotropic.Let Ebe a complex ofF-Dbcoh(D

P,Q). We say that E is Lagrangian if for any integern, the characteristic variety

ofHn(E)is Lagrangian. IfE E E E[1]is a exact triangle ofF-Dbcoh(DP,Q)then E

and E are Lagrangianif and only ifE is Lagrangian.

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1.4.2. LetXbe an affine smooth variety overkadmitting local coordinatest1, . . . , td. We denote byX(m) the basechange ofXunder themth power of Frobenius ofS:=Spec k, byFm :X X(m) the relative Frobenius morphism.From the equalities [Ber96, 1.1.3.1, 2.2.4.(iii)], we compute that for any j


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Corollary 2.1.3. Let f: X Y be an tale morphism of integral smooth k-varieties. LetE Dbcoh(D(0)

X ), F

Dbcoh(D(0)Y ).

1. We have the equality |Car(0)(f!(0)

(F))| =f(1f (|Car

(0)(F)|)).

2. If f is moreover finite and surjective of degree d, then

|Car(0)(f+(0) (E))| =Uf(|Car(0)(E)|); (2.1.3.1)

ZCar(0)(f+(0)f!(0) (F)) =dZCar(0)(F). (2.1.3.2)

Proof. First, let us check the first equality. Since this is local inXand Y, we can suppose thatXand Yare affine with

local coordinates. We have fgrD(0)Y

grD(0)X (i.e. fis an isomorphism) and the etale homomorphism of rings

(Y, grD(0)Y ) (X,grD(0)

X ). We get (X,grf!(0) (F)) = (X, grD(0)X ) (Y,grD(0)Y )

(Y,grF). Since the extension

(Y, grD(0)Y ) (X,grD(0)

X )is flat, we get Supp((X,grf!(0) (F))) =1f (Supp((Y, grF))), which gives the desired

equality.Since the second part of the lemma is local in Y, we can suppose thatXand Yare affine with local coordinates and

f(OX)is a free OY-module of rankd. We have(Y,f+(0) (E)) =(X,E)and since(Y,D(0)Y ) (X,D

(0)X )is finite,

we get(Y,grf+(0) (E)) =(X,grE), which gives2.1.3.1.Morevoer,(X, grf+(0)f!(0) (F)) = ((Y, grF))d.

2.1.4. Let f: P P be a proper morphism of smooth formal V-schemes. Let(E,)be a coherentF-DP,Q-module.

From the equivalence of categories of [Ber00,4.5.4], there exist (unique up to isomorphism) a coherentD(0)P,Q-module

F(0) and an isomorphism(0) :D(1)P,Q D(0)

P ,Q

F(0) FF(0) which induced(E,)by extension. Fix an integer

i Z. From the isomorphisms

D(1)P,Q D(0)

P,Q

Hif+(0) (F(0))

[Ber02,3.5.3.1]Hif+(1) (

D(1)P,Q D(0)

P ,Q

F(0)) (0)

Hif+(1) (FF(0))

[Ber02,3.5.4.1]FHif+(0) (F

(0)),

we get Car(Hif+(E)) = Car(0)(Hif+(0) (F(0))). Choose a coherentD(0)

P-module without p-torsionE(0) such that

E(0)Q

F(0). SinceHif

+

(0) (F(0)) (Hif

+

(0) (E(0)))Q, then by puttingG(0) equal to the quotient ofHif+

(0) (E(0))

by itsp-adic torsion part, Car(0)(Hif+(0) (F(0))) =Car(0)(G(0)) Car(0)(Hif+(0) (E

(0))). Hence,

|Car(Hif+(E))| |Car(0)(Hif+(0) (E

(0)))| := |Car(0)(kVHif+(0) (E

(0)))|.

By using a spectral sequence (the result is given in the beginning of the proof of [ Vir00,I.5.8]), we obtain themonomorphism kVHif+(0) (E

(0))Hi(kLV

f+(0) (E(0))). Hence, |Car(0)(kVHif+(0) (E

(0)))| |Car(0)(Hi(kLV

f+(0) (E(0))))|. We havekL

Vf+(0) (E

(0)) f+(0) (E

(0)), where E

(0):=kV E(0)

kL

VE(0). Finally we get

|Car(Hif+(E))| |Car(0)(Hif+(0) (E

(0)))|. (2.1.4.1)

From2.1.2,since f is proper then |Car(0)(Hif+(0) (E(0)

))| Uf(|Car(0)(E(0)

)|). By Berthelots definition of the

caracterisc variety ofE

, we have Car(E

) =Car(0)

(E(0)

). Hence, we have checked the following proposition.Proposition 2.1.5. Let f: P Pbe a proper morphism of smooth formal V-schemes. Let(E,) F-Dbcoh(D

P,Q).

We have the inclusion|Car(f+(E

))| Uf(|Car(E)|).

2.1.6. Let f: P P be a finite tale surjective morphism of smooth formalV-schemes.

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1. In that case, since f+(0) = f(and then preserves the property ofp-torsion freeness) and f+= f, the inclusion2.1.4.1is an equality. In fact, with the usual notation of Abe (see[Abe10]), we get the equality

ZCar(Hif+(E)) =ZCar(0)(Hif+(0) (E

(0))).

2. Moreover, let (E,) be a coherent F-DP,Q-module. From the equivalence of categories of [Ber00, 4.5.4],

there exist (unique up to isomorphism) a coherentD(0)P,Q-moduleF(0) and an isomorphism(0) :D(1)P,QD(0)P,Q

F(0) FF(0) which induced(E,) by extension. Choose a coherentD(0)

P-module without p-torsionE(0)

such that F(0) E

(0)Q . Since D

P,Q = f

DP,Q = f

!DP,Q and

D(0)P

= fD(0)P

= f!(0)D(0)

P, we check that

f!(0)E(0) = fE(0) has nop-torsion and that E

D

P,Q D(0)P

f!(0)E(0). Moreover, putting E

(0):=kV E(0),

we get

ZCar(f!(E)) =ZCar(f!(0)E

(0)). (2.1.6.1)

Proposition 2.1.7. Let f: P P be a finite tale surjective morphism of degree d of integral smooth formal V-schemes. LetE F-Dbcoh(D

P,Q). Then we get

ZCar(f+f!E) =dZCar(E); (2.1.7.1)

(P,f+f!(E)) =d(P,E). (2.1.7.2)

Proof. From2.1.3.2and 2.1.6, we getZCar(f+f!E) = dZCar(E). Then, the equality2.1.7.2is a consequence of2.1.7.1and of Berthelots index theorem[Ber02,5.4.4].

2.2 Some kind of Frobenius descent

We will need some kind of Frobenius descent so we introduce the following notation.

2.2.1. LetFk:k kbe the Frobenius map and0 : V Vbe a lifting of Frobenius. Let sbe a positive integer and= s0 the corresponding lifting of the sth power of the Frobenius map. LetXbe ak-variety and P be a formalV-scheme. We denote byX and P the corresponding k-variety (resp. formal V-scheme) induced by the base changeby. We will denote byFsX

/k:XX

andFsP

/V

: P P the corresponding relative Frobenius.

Lemma 2.2.2. Let f: X P be a dominant morphism of smooth integral k-varieties. Let ZX be a proper closedsubset. Then there exists a dense open subvariety U of P, a universal homeomorphism g :U U with U normal, a

projective generically finite and etale U-morphism of the form f:V (XP U)redsuch thatV is smooth over U,f1(ZP U)redis a strict normal crossing divisor inV relatively to U.Proof. Let l be the field of fraction ofP, X(l):=XPSpec (l), Z(l):=ZXX(l). Let l be an algebraic closure

ofl , L:=lGal(l/l) the fixed field by Gal(l/l). We putY(L):=X(l) Spec (l)Spec(L), X(L):= (Y(L))red and Z(L):=(Z(l) Spec(l)Spec (L))red. From desingularisation de Jongs theorem (see[dJ96] or [Ber97, 4.1]), since L is per-fect, there exists a projective and generically finite and etale morphism L : X(L) X(L) such that X

(L) is smooth

and Z(L) := 1

L (Z(L)) is a strict normal crossing divisor in X(L). By using [Gro66, 8.8.2.(ii)], there exists a fi-

nite (radicial) extension l of l included in L such that there exists l-varieties X(l) and X(l) satisfying X(L)

X(l) Spec(l)Spec(L)and X(L)

X(l) Spec (l)Spec (L). By increasingl is necessary, we can suppose thatX(l)andX(l)are reduced. (Indeed, sinceX(L)is reduced then so isX(l) Spec (l) Spec(L). From [Gro60, 5.1.8], this yields

X(l) Spec(l)Spec (L) = (X(l) Spec(l)Spec(L))red= ((X(l))red Spec(l)Spec(L))red. Hence, the closed immersion(X(l))red Spec(l) Spec (L)X(l) Spec(l) Spec (L)is in fact an isomorphism. By increasingl

is necessary, it followsfrom [Gro66,8.10.5.(i)] that(X(l))redX(l)is an isomorphism.)

We putY(l):=X(l) Spec (l) Spec(l). By increasing l is necessary, it follows from [Gro66,8.8.2.(i)] that thereexists

a morphisml:=X(l) X(l)(resp. X(l) Y(l)) inducingL(resp. the surjective closed immersionX(L) Y(L)).

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By using [Gro67,17.7.8] and[Gro66,8.10.5] (by increasingl is necessary), we can suppose that X(l) Y(l) is asurjective closed immersion (i.e.X(l)= (Y(l))redsinceX(l)is reduced), thatlis projective and generically finite andetale morphism, thatX(l)is smooth andZ

(l):=

1l (Z(l))is a strict normal crossing divisor inX

(l).

LetP be the normalization ofP in l . Then the canonical morphismg : P Pis a universal homeomorphism,i.e. is finite (e.g. use [Liu02,4.1.27]), surjective and radicial (e.g. use the exercise [Liu02,5.3.9.(a)]). By using[Gro66, 8.8.2.(ii)] (this times, we consider the projective system of open affine dense subvarieties ofP), there exists

a dense open affine subvarietyU ofP, two morphismsV U andV U such thatX(l) V USpec(l)andX(l)

V USpec (l). With the same arguments as above, we can suppose thatV andV are reduced and

thatV = (XP U)red. By shrinking U is necessary, from [Gro66,8.8.2.(i)], there exists aU-morphism f:V Vwhich inducesl . By shrinkingU

is necessary, by using [Gro67,17.7.8] and[Gro66,8.10.5] we get the desiredproperties.

Remark 2.2.3. Letg :U Ube a universal homeomorphism of normal integral k-varieties. Then, for slarge enough,there exists a universalhomeomorphism h :U U such that gh = FsU/k:U U

(we can have alsohg = FsU/k).

Proposition 2.2.4. Let f: X P be a dominant morphism of smooth integral k-varieties. Let Z X be a properclosed subset. Then, for s large enough, there exists a dense open subvariety U of P, such that, by putting W :=(X PU)red (where X PU means the base change of X by the composition of FsU/k:U U

with the open

immersion U P), there exists a projective generically finite and etale U-morphism of the form :W W suchthatW is smooth over U, Z :=1(Z P U)redis a strict normal crossing divisor in W relatively to U.

Proof. Using the lemma2.2.2 and the remark 2.2.3,we get with their notation the diagram of morphisms ofk-varieties

W

Vf

T

(X P U)red

X P U

X P U

X P U

X

U

h U g U P

whereT := (X PU) (XPU)(X PU)red andW :=V UU. SinceW is smooth overU andU is

smooth, thenW is smooth (and in particular reduced). SinceW Tis projective, generically finite and etale (i.e.there exists a dense open subvarietyT ofTsuch that its base change by the open immersion T Tbecomes finiteand etale), this implies that the factorization :W Tredis also projective, generically finite and etale. We remarkthatTred= (X P U)red. FinallyZ :=1((Z P U)red)is a strict normal crossing divisor relatively to U.

Lemma 2.2.5. Let f: P P be a finite, surjective morphism of smooth formal V-schemes. LetE be a coherentD

P,Q-module. Then E is OP,Q-coherent if and only if f

!(E)is OP,Q-coherent.

Proof. Recall that we denote by f the functor defined by f(M) = OP,Q f1OP,Q f1M, for any complex M of

OP,Q-modules (remark that f is flat). IfE is OP,Q-coherent then f!(E) f(E)is OP,Q-coherent. Conversely,suppose f!(E) is OP,Q-coherent. First, suppose that f is finite and tale. Since f+= f, we get that f+f

!(E) isOP,Q-coherent. SinceEis a direct factor of f+f!(E), then Eis also OP,Q-coherent. By splitting fby a universalhomeomorphism followed by a finite etale morphism, then we reduce to the case where fis a universal homeomor-phism. In that case, fors large enough,Fs

P/V: P P factors throught f. Hence, since from [Ber00] the functor

(FsP/V)

induces an equivalence of categories (for coherentDQ-modules)and using the easy implication of the lemma,

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we can suppose that f=FsP/V (and P = P

). In that case, from the isomorphism given just after Theorem [Ber00,

4.2.4] and from the definition off! given in [Ber02,4.3.2.2], we get thatf!(E) f(E)(this is harder to check when

Eis only DP,Q-coherent). We can supposeP affine. SinceE is a coherent D

P,Q-module, this is sufficient to check

that(P,E)is of finite type over(P,OP,Q)(see [Car06, 2.2.13]). Since the extension(P,OP,Q) (P,OP,Q)isfaithfully flat (because f: P Pis faithfully flat), since(P,f!E)

(P,OP,Q) (P,OP,Q) (P,E)is of finite

type over(P

,OP

,Q), we conclude.

2.3 The result

Lemma 2.3.1. Let :X X be a finite tale surjective morphism. LetE Dbovhol(X/K)(resp.E Dbovhol(X/K)).The propertyEDbisoc(X/K)is equivalent to the property +(E)Dbisoc(X/K). The propertyEDbisoc(X/K) isequivalent to the property+(E) Dbisoc(X/K).Notation 2.3.2. LetXbe an integral variety and E F-Ovhol(X/K). Then, there exists a smooth dense open sub-varietyY ofXsuch that E|YF-Isoc(Y/K)(see the notation [AC13b, 1.2.14] and use[Car07, 2.3.2]). Then, bydefinition, rk(E)means the rank ofE|Y(which does not depend on the choice of such open dense subvariety Y).

Theorem 2.3.3. LetP1 be a smooth separated formal V-scheme, P2 be a proper smooth formal V-scheme, P:=P1 P2andpr : P P1be the projection. LetE be a complex of F-Dbovhol(D

P,Q).

Then there exists an open dense formal subscheme U1ofP1such that, for any finite tale surjective morphisms ofthe form 1 :P1 P1and2 :P2 P2, puttingP=P1 P2, :PP andE := +(E), we have(pr )+(E)|U1

Dbcoh(OU1,Q).

Proof. I) We can suppose that1=Id. Indeed, consider the following diagram

P1 P2 2

1

P1 P21

pr

P11

P1 P2 2 P1 P2 pr P1.

Suppose there exists an open dense formal subscheme U1ofP1 such that pr+ 2+(+2(E))|U1 D

bcoh(OU1,Q). By

base change theorem we get

1++1pr+ 2+(

+2(E))

1+pr+2+

+1(

+2(E))

1+pr+2+(E) pr++(E).

From Lemma2.3.1, this implies that(pr )+(E)|U1Dbcoh(OU1,Q).II) We proceed by induction on the dimension of the supportXofE. There exists a smooth dense open subvariety

Y ofXsuch that E|YF-Dbisoc(Y,P/K)(see the notation [AC13b,1.2.14] and use [Car07, 2.3.2]). Let j :YXbe the inclusion. By devissage, we can supposeYintegral, that E is a module and that E

j+j!(E)with j!(E)

F-Isoc(Y,P/K). We putZ:=X\Y. By abuse of notation (to simplify them), P1will mean a dense open set U1ofP1(be careful that the open set has to be independent of the choice of2),Xand P will mean the base change ofXand P by the inclusion U1 P1.

1) From Kedlayas semistable reduction theorem (more precisely the global one, i.e. [Ked11,2.4.4]), there exists asurjective, projective, generically finite tale morphism a :X X, withX integral and smooth such thatZ := a1(Z)is the support of a strict normal crossing divisor ofX and, putting Y := a1(Y),a(E)|Y extends to a log-F-isocrystalwith nilpotent residues on the smooth log scheme (X,Z) (i.e. the underlying scheme isX and the log structurecomes from the strict normal crossing divisor Z). Sincea is projective, there exists a closed immersion of the formu :X PNP (this is the product in the category of formal scheme over V) such that the composition ofu with theprojectionf:PNPP inducesa. Sincea+ a!(E)is a direct factor ofE, we reduce to the case whereXis smooth,

Zis a strict normal crossing divisor ofXand E|Yextends to a log-F-isocrystal with nilpotent residues on(X,Z).2) From Proposition 2.2.4, by replacingP1by an open dense formal subscheme if necessary and for slarge enough,

putting W:= (X P1 P1)red, there exists a projective generically finite and etaleP1-morphism of the form :W W

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such thatW is smooth overP1,Z :=1(Z P1 P1)redis a strict normal crossing divisor in W relatively toP1. We

setY :=1((Y P1 P1)red). Hence, we get the commutative diagram:

Z

W

P = P3 P1pr

f

P1

(Z P1 P1)red

red

W= (X P1 P1)red

a

red

P2P1pr

P1

FsP1/V

Z

X P2P

1

pr P1 ,

whereP3is a proper smooth integral formal scheme over V, pr : P3 SpfVis the structural morphism, fis in factinduced by a morphism of the form f: P3 P2 (since is projective, this f is more precisely chosen to be equalto the projection f:PN P2 P2 for some integer N). We put G:=a!(E) = (FsP1/V)(E) and G :=!(G) =RWf

!(G). Sincea induces the morphism of smooth log-schemes(W,Z) (X,Z), sinceE|Y extends to alog-F-isocrystal with nilpotent residues on (X,Z)then G |Y extends to a log-F-isocrystal with nilpotent residueson(W,Z).

3) LetZ1, . . . ,Zrbe the irreducible components ofZ. For any subsetI of{1, . . . ,r}, we setZI:= iIZi . Then|Car(G)| I{1,...,r}T

Z

IP. Indeed, since the check is local onP, we can supposeP affine with local coordinates

t1, . . . ,td ofP

inducing local coordinatest1, . . . ,tn ofW

and such thatZi = V(ti )for i

=1, . . . , r. From [Elk73],there exists a smooth affine formal V-scheme W whose special fiber isW. Letu : W P be a lifting ofW P

and letF :=u!(G). From1.4.3.1,|Car(F)| I{1,...,r}T

ZI

W. SinceG u+(F), from [Ber02,5.3.3], we get

|Car(G)| =Uu(|Car(F)|). Using1.3.2, we get Uu(TZI

W) =TZI

P, which gives the desired result.

4) Let 3 :P3 P3 be the finite etale morphism of integral proper smooth formal V-schemes induced from2 by base change by f: P3 P

2 (hence, 3 =id

2:PN P2 PN P2 ). PuttingG :=+3(G), we get

|Car(pr+3+(G))| TP1 P1.Proof: Since pr is proper, from2.1.5,we get the inclusion

|Car(pr+3+(G))| Upr (|Car(3+(G))|).Using2.1.7.1,we get |Car(3+

+3(G

)| = |Car(G)|. Hence,

Upr (|Car(3+(G))|) = Upr (|Car(G)|) Upr (I{1,...,r}TZI

P) = I{1,...,r}(Upr (T

ZI

P)),

where the inclusion comes from the step 3). Moreover, from1.3.2, UuI (T

ZI

ZI ) =T

ZI

P, whereuI:Z

I P is the

closed immersion. By transitivityof the applicationU (see 1.3.1),Upr (T

ZI

P) =Upr (UuI (T

ZI

ZI )) =UpruI (T

ZI

ZI ).

Sincepr uI:Z

I P1is smooth, using1.3.4.2, we get the inclusion UpruI (T

ZI

ZI ) T

P1P1, which yields the de-

sired result.5) G is a direct factor of+(G)(which is by definition f+(G)).a) Put:= Y (Y P1 P1)redand b : (Y

P1 P1)red Y the morphisms induced respectively byanda. We

have the isomorphismb! b+. Indeed, from [AC13b,1.3.12], sincebis a universal homeomorphism, the functorsb! andb+ induce quasi-inverse equivalence of categories (for categories of overholonomic complexes). Sinceb isproper, thenb+= b!(i.e., via the biduality isomorphism,b+ commutes with dual functors). Hence, we get thatb!

commutes also with dual functors.b) Since:=b is a morphism of smooth varieties and E|Y is an isocrystal, then!(E|Y)

+(E|Y).

Hence, since is proper, we get the morphisms by adjunction E|Y ++(E|Y) +!(E|Y) E|Y.

The composition is an isomorphism. Indeed, since E|Y is an isocrystal, we reduce to check it on a dense open

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subset ofY. Hence, we can suppose thatandbare morphisms of smooth varieties. We reduce to check this kindof property separately forband. The case ofbcan be checked using [AC13b, 1.3.12]. Sinceis generically finiteand etale, the second case is already known.

c) We have just checked that E|Y is a direct factor of+!(E|Y). This implies thatb!(E|Y)is a directfactor ofb!+!(E|Y)

+!(E

|Y) +(G

|Y). Since (Y P1 P1)red = a1(Y), with the convention

of the paper, we get the morphism j : (Y P1 P1)red(X P1 P1)red = W. We have also j

:Y W. We

get j+b!(E|Y)is a direct factor of j++(G|Y). By base change isomorphism, sinceE j+j!E we obtainG

j+j

!G. This yields+(G) +j+(G

|Y) j++(G

|Y)and G a!j+(E

|Y) j+b

!(E|Y),which gives the desired result.

6) PuttingG:=2+(G) (and with the abuse of notation f:P3 P1P2 P1 of the paper), the step 5) im-plies (using some base change isomorphism), thatGis a direct factor of+(G) = f+(G). Since pr+3+(G) pr+

2++(

G), we obtain |Car(pr+2+(G))| |Car(pr+3+(G))|. From part 4), this yields |Car(pr+2+(G))| TP1 P1. Letpr:P2SpfVbe the structural morphism. Since pr+2+(G) =pr+(G), this inclusion is equivalent tosay thatpr+(G) Dbcoh(OP1,Q)(recall that in the proof of the theorem, by abuse of notation, P1means a dense opensubscheme). SinceG a!(E), we obtainpr+(G) pr+(a!(E)) (FsP1/V)! pr

+(

E). From Lemma2.2.5,thisimplies thatpr+(E) Dbcoh(OP1 ,Q), which yieldspr+(E) Dbcoh(OP1,Q).

3 Betti numbers estimates

3.1 Preliminary

Lemma 3.1.1. Let U be an affine smooth variety. LetF F-Isoc(U/K). Then we have the inequalities:

dimHdpU+(F) rk(F); (3.1.1.1)

dimHdpU!(F) rk(F). (3.1.1.2)

Proof. SincepU!(F) (pU+(F)), where means the O-linear dual functor. Hence, we reduce to check the first

inequality. SinceU is affine,HdpU+(F)corresponds toF=0, thehorizontal sections ofF(i.e. the 0th cohomologicalspace of the Monsky-WashnitzercohomologyofF). SinceF=0 KOUis included inF(whereOUmeans the constantobject ofF-Isoc(U/K)), we get the desired inequality.

Lemma 3.1.2. Let X be a smooth irreducible curve, j :U X an open immersion such that Z:=X\ U is a closedpoint. LetF F-Isoc(U/K). Let i :ZX be the closed immersion. Then, for any integer n {0,1},

dimHni!j!(F) rk(F). (3.1.2.1)

Proof. Let j : j! j+ be the canonical morphism. Sincei!j+= 0, since the cone ofj has its support in Z, thenby using Berthelot-Kashiwara theorem (see [Ber02,5.3.3]) we get ker(j)

i+H0i!j!and coker(j)

i+H1i!j!.

From [AC13b,1.4.3], we obtainDXi+H0i!j!(F) i+H1i!j!(DU(F)). Hence,H0i!j!(F)

DZH1i!j!(DU(F))and

we reduce to check3.1.2.1forn=1.We have the exact sequence 0 j!+(F) j+(F) i+H1i!j!(F)0. From Kedlayas semistable theorem

[Ked03], there exists a finite surjective morphism f: P X, withP smooth integral, such that f!(F)comes from

a convergent isocrystal onP

with logarithmic poles along f1

(Z). SinceP

andXare smooth and fis finite andsurjective then fis flat (e.g. see [Gro67,IV.15.4.2]). LetX be an open dense subset ofP such thatZ :=f1(Z) X

is a closed point. We get the (quasi-finite) flat morphisma :X Xand the open immersion j :U :=a1(U) U.Sincea is flat, thena! is exact. Hence, we get the exact sequence 0 a!j!+(F) a!j+(F) a!i+H1i!j!(F) 0.From [AC13b,1.4.8], the inclusionj!+(a!(F))j+(a!(F))factors through the composition a!j!+(F) a!j+(F)

j+a!(F). Then, we get the epimorphism

i+H1i!j!(a

!F) j+(a

!(F))/j!+(a!(F)) a!j+(F)/a

!j!+(F) a!i+H

1i!j!(F) i+H

1i!j!(F)

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(we have the last isomorphism because a induces the isomorphism a1(Z) Z). By applyingi! (and by using

Berthelot-Kashiwara theorem), we get dimH1i!j!(F) dimH1i!j!(a!F). Since a!Fis log-extendable, then from[AC13b,3.4.12] we obtain the inequality dimH1i!j!(a!F) rk(a!F) =rk(F).

3.1.3. LetSbe ak-variety. Let us briefly review the geometric Fourier transform[NH04], but only in case ofA1S/S.Fix Qp such that

p1 =p. We assume that K. We denote by L the Artin-Schreier isocrystal in

F-Isoc

(A1

k/K)(see [Ber84]).We denote by : A1k (A1k)

A1kthe canonical duality bracket given by txy. We denote the composition byS: A1SS(A

1S)

A1k (A1k)

A1k.

Now, consider the following diagram: (A1S) p2A1SS(A

1S)

p1A1S, where(A1S)

is the dual affine space overS,which is nothing but A1S. Similarly than Katz and Laumon in [KL85,7.1.4, 7.1.5] (in fact, this paragraph describe theparticular case wherer=1), for any E F-Dbovhol(A

1S), the geometric Fourier transformF(E)is defined to be

F(E):=p2+p!1EA2S

!SL

(cf. [NH04, 3.2.1]1). Here is compatible with Laumons notation (see [KL85,7.0.1, page 192]) and was defined inthe context of arithmetic D-modules in [AC13b, 1.1.6].

3.1.4. An important property for us of Fourier transform is the following. The functorF is acyclic, i.e. ifE

F-Ovhol(A1S/K)then F(E) F-Ovhol((A

1S)

/K)(cf. [NH04, Theorem.3.1]).Lemma 3.1.5. Let f: T S be a morphism of k-varieties. LetE F-Dbovhol(A

1S/K)andF F-D

bovhol(A

1T/K). We

have the canonical isomorphisms

f!F(E) F(f

!E); (3.1.5.1)

f+F(F) F(f+F). (3.1.5.2)

Proof. From some base change isomorphism, we get the first isomorphism:

f!F(E) = f!p2+

p!1EA2S

!SL

p2+f

!p!1EA2S!SL

p2+

f!p!1EA2Tf

!!SL

.

SinceT=Sf: , we obtainp2+f!p!1EA2Tf!!SL

p2+

p!1(f

!E)A2T!TL

=F(f!E), which gives3.1.5.1.

Moreover, similarly and by using the projection formula (see [AC13b,A.6])

f+F(F) = f+p2+p!1(F)A2T

!TL

p2+f+

p!1(F)A2Tf

!!SL

p2+f+p

!1(F)A2S

!SL

p2+

p!1(f+F)A2S

!SL

= F(f+F).

Since the proof of the main result on Betti estimate (see3.2.2) in the case of curves is easier (e.g. remark that wedo not need in this case the Lemma2.3.3) and since its proof is made by induction, we first check separately this curvecase via the following lemma.

Lemma 3.1.6(Curve case). Suppose k is algebraically closed. Let X1be a projective, smooth and connected curve,EF -D0(X1/K) (see the notation of [AC13b, 1.2]). There exists a constant c(E) such that, for any finite tale

morphism of degree d1of the form :X1X1withX1connected, by puttingE :=+(E), we have1. dimH1pX1+(

E) c(E);

2. For any r 0,dimHrpX1+(E) c(E)d1.

1Notice that our twisted tensor product and hers are the same.

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Proof. There exists an open dense affine subvarietyU1ofX1such that E|U1 F-Dbisoc(U1/K), LetZ1 be the closedsubvarietyX1 \U1, j :U1X1andi :Z1X1be the immersions. We putU1:=11 (U1),Z1:=11 (Z1)i.e we getthe cartesian squares:

Z1 i

1

X11

U11

j

Z1i X1 U1.

j

By considering the exact trianglej!j!(E) E i+i+(E) +1 we reduce to check the proposition forE=j!j!(E)or E =i+i+(E)(and because the functors j !j! andi+i+ preserveD0).

1) In the case where E =i+i+(E), we can suppose thatZ1is a point. We put G :=i+(E). SinceZ1is d1copies ofZ1, then we get1+

+1 G

Gd1 and thenp X1+(

E) pX1+(i+Gd1 ) Gd1 , which gives the desired result.2) Suppose now E = j!j!(E). We put F= j!(E),F= j !(E). By devissage, we can suppose that Fis a module, i.e.

F F-Isoc(U1/K)(and thenF F-Isoc(U1/K)).SinceU1is affine, since j!is exact (andE j!(F)), for anyr {0,1}, we getHrpX1+(E)

Hrp U1!(

F) =0.From3.1.1.2,we get dimH1pX1+(

E) = dimH1p U1!(F)rk(F) = rk(F). It remains to estimate|(X1,E)|. From

Lemma2.1.7we get the second equality (X1,E) = (X1,1++

1(E)) =d1 (X1,E).Hence, we can choose in thatcasec(E) =max{|(X1,E)|; rk(F)}.

3.2 The result and some applications

We will use the following remark during the proof of the main theorem.

Remark 3.2.1. LetMbe a solvable differential module on the Robba ring over K. We get the differential slopedecompositionM= M, whereMis purely of slope. By definition Irr(M):= 0 rk(M). Hence, we get that

Irr(M) rk(M) + Irr(M]1,[), (3.2.1.1)

whereM]1,[:= ]1,[M.

Theorem 3.2.2. Suppose k is algebraically closed. Let(Xa)1anbe projective, smooth and connected curves, X=na=1Xa, E F-D

0(X/K)(see the notation of [AC13b,1.2]). There exists a constant c(E)such that, for any finitetale morphism of degree da of the form a :Xa Xa withXa connected, by puttingX= na=1Xa, :X X andE :=+(E), we have

1. For any r,dimHrpX+(E) c(E)na=1 da.2. For any r 1,dimHrpX+(E) c(E) max{aA da | |A| =n r}.

Proof. We proceed by induction the integer n 1. The casen = 1 has already been checked in 3.1.6. Supposen 2. Leta :XaXabe some finite tale morphism of degree dawithXaconnected,X= na=1Xa, :XXandE :=+(E). We put Y:=a=1Xa,Y:=a=1Xa, :Y Y. Let pr:Y Spec kandpr:Y Spec kbe the projections

(recall that from the convention of this paper, for instance,pr means also the projectionpr:X=X1 YX1etc.).From Lemma2.3.3, thereexists an affine open dense subvarietyU1(independent of the choiceofi) ofX1suchthatpr+(+E)|U1 Dbisoc(U1/K). LetZ1be the closed subvarietyX1 \U1,U1:=11 (U1),Z1:=11 (Z1). Letj :U1X1i :Z1X1be the inclusions.

We havej ! pr+(E) =pr+(E)|U1 F-Dbisoc(U1/K). Indeed, from2.3.1,this is equivalent to prove1+(pr+(E))|U1 Dbisoc(U1/K). Then, we get the desired property from the isomorphism1+(pr+(E)) pr+(+E).Step I) With the notation2.3.2,we check that there exists a constantc(only depending on E) such that

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For anys, rkHs pr+(E) cnb=2 db. For anys 1, rkHs pr+(E) cmax{bB db |B {2, . . . , n} and |B| =n 1 s}.Proof. Lettbe a closed point ofU1,tbe a closed point ofU1 such that1(t) =t. Letit:t X1,it:t X1,

t:t U1be the closed immersions. Since the functor !t[1]is acyclic onF-Dbisoc(U1/K), sincei!t[1] (!t[1]) j

!,

we obtaini!t[1](H

s

pr+(E))

Hs

(i!tpr+(E[1])). Moreover, for sucht, we have rk(H

s

pr+(E)) = dim i!t[1]H

s

pr+(E).We put E1:=i!t(E)[1]andE1:=+(E1). Sincet Y is a smooth divisor ofX, theni!t[1]is right exact. Hence

E1 F-D0(Y/K). Since+ = ! and it=it :t YX, we get the isomorphismi!tE[1] =i!t+(E)[1] +i!t(E)[1] =+(E1) =E1.

By base change isomorphism, we get:i!tpr+(E) pr+i!t(E). Hence,

i!tpr+(E)[1] pr+(E1) =pY+(E1),

where we have identifiedY witht Y. By composition, we obtain i!t[1](Hs pr+(E)) HspY+(E1) and then

rk(Hs pr+(E)) =dimHspY+(E1). We conclude by applying the induction hypothesis to E1(notice the fact thatU1isindependent on the choice of

iis fundamental, i.e. we do need Theorem2.3.3).

Step II)1) By considering the exact triangle j!j!(E) E i+i+(E) +1 we reduce to check the proposition for E=j!j!(E)or E =i+i+(E)(and because the functors j !j! andi+i+ preserveD0).2) Suppose E =i+i+(E). We can suppose thatZ1is a irreducible (i.e. sincekis algebraically closed,Z1

Speck).

Consider the diagram with cartesian squares:

Z1 Y 1

i

Z1 Y i

Z1 Y pr

i

Z1

i

X 1 X1 Y X pr X1.

We put G :=i+(E),G :=+(G). SinceZ1isd1copies ofZ1, then we get1++1G

(G)d1 . This impliespX+(E) =p X+(+(i+G)) pX+(i++1+G) pY+1++1(G) (pY+(G))d1 , (3.2.2.1)

where in the last isomorphism we have identifiedYwithZ1 Y. We conclude by applying the induction hypothesis toG.

3) Suppose now E = j!j!(E), i.e.i+(E) =0.a) We check that there exists a constantc(only depending on E) such that

For anys, dimH1pX1+(Hs pr+(E)) cnb=2 db.

For anys 1, dimH1pX1+(Hs pr+(E)) cmax{bB db |B {2, . . . , n} and |B| =n 1 s}.

Proof: We put F= j!(E),F:= +(F). By using base change isomorphism, we get j!F E. Moreover,pr+(F) j! pr+(j!F) j! pr+(E) Dbisoc(U1/K). Moreover, j!pr+(F) pr+j!(F) pr+(E). Since j!isexact, we get j!Hs pr+(F) Hs pr+(E). From3.1.1.2,this implies

dimH1pX1+(Hs pr+(E)) =dimH1p U1!(Hs pr+(F)) rkHs pr+(F) =rkHs pr+(E).

From the step I), we get the desired estimate.

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b) We have the spectral sequence

Er,s2 =HrpX1+H

s pr+(E) Hr+spX+(E).

SinceU1is affine of dimension 1 and since p X1+(Hs pr+(E)) p U1!(H

s pr+(F)), we getEr,s2 =0 whenr {0,1}.Hence, by using the step a) (we can also vary the order ofX1, . . . ,Xn), it remains to check that there exists a constantc

(E

)such that For anys, |(X1,Hs pr+(E))| c(E)na=1 da. For anys 1, |(X1,Hs pr+(E))| c(E)max{aA da | |A| =n s}.i) In this step, we reduce to the case d1=1. For this purpose, consider the following diagram

X1 Y 1 pr

X1 Y pr

X

pr

X1 1 X1.We have (

X1,Hs

pr+(

E)) = (X1,1+Hs

pr+(

E)). By base change, we get the second isomorphism:1+

pr+(

E)

1+pr++1(+(E)) 1++1pr+(+(E)). Since1+and +1 are exacts, we get1+Hs pr+(E) 1++1 Hs pr+(+(E)).Hence, (X1,1+Hs pr+(E)) =(X1,1++1Hs pr+(+(E))). From Lemma 2.1.7, we have (X1,1++1Hs pr+(+(E))) =d1(X1,Hs pr+(+(E))). Hence, we have checked that(X1,Hs pr+(E)) = d1(X1,Hs pr+(+(E))), which yields thedesired result.

ii) We suppose from now thatd1= 1. We prove that we can reduce to the case whereX1= P1kand U1=A1k. Indeed,

from Kedlayas main theorem of[Ked05], by shrinkingU1is necessary, there exists a finite morphism f: X1 P1ksuch that U1= f1(A1k)and the induced morphism f:U1 A

1kis etale. We get the cartesian squares:

X

pr

f

X

f

pr X1

f

U1

g

j

P1kY

prP1kY

pr P1k A1k

j

(3.2.2.2)

Set E := f+(E). Then we get thatE :=+(E) f+(E). We have already seen in Step II.3.a) thatpr+(E) j!pr+(F). Since f+j! j!g+,g+is exact (becausegis finite and etale), and j!is exact, we get

Hsf+pr+(E) Hsf+j!pr+(F) j!g+Hs pr+(F) f+j!Hs pr+(F) f+Hs pr+(E).Morever, sinceHs pr+(E) Hsf+pr+(E), we obtain(X1,Hs pr+(E)) =(P1k,f+Hs pr+(E)) =(P1k,Hs pr+(E)).

iii) We suppose from nowX1= P1kand U1= A1k. From3.1.2,for anym {0, 1}, we have

H

m

(i

!

j!Hs

pr+(F)) rk(Hs

pr+(F)) =rk(Hs

pr+(E)).From the step I, this latter is well estimated. Since(X1,Hs pr+(E)) =(X1,j!Hs pr+(F)) =(X1, i+i!j!Hs pr+(F))+(X1,j+Hs pr+(F)), we reduce to estimate(X1,j+Hs pr+(F)).

From Christol-Mebkhouts Theorem [CM01,5.0-10] (as described in the introduction), we have the followingp-adic Euler-Poincare formula:

(X1,j+Hs pr+(F)) =(U1,Hs pr+(F)) =rk (Hs pr+(F))(U1) Irr(Hs pr+(F)),

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whereis the complement ofU1inX1, i.e. ofA1Kin P1k.

From Abe-Marmoras formula [AM,2.3.2 and 4.1.6.(i)], with the inequality3.2.1.1,we get the following one:

Irr(Hs pr+(F)) rk(F(Hs pr+(F)))+ 2rk(Hs pr+(F)),

where Fmeans the Fourier transform (see3.1.3). Hence, we reduce to check the (last) step iv).

iv) In this step, we estimate rk(F

(Hs

pr+(F))).From the step I) applied to M := j+F(j!E), there exists a constantc(only depending on E) such that For anys, rk(Hs pr+(+M)) cnb=2 db. For anys 1, rk(Hs pr+(+M)) cmax{bB db |B {2, . . . , n} and |B| =n 1 s}.

It remains to check that rk(F(Hs pr+(F))) =rk(Hs pr+(+M)). Since=(i.e.d1=1), we get the diagram

X

pr

X

pr X1

U

j

pr

U

j

pr

U1

j

(3.2.2.3)

By base change (recall that! =+) and next by using3.1.5, we have

pr+(+M) =pr+(+j+F(F)) pr+j++(F(F)) j+pr++(F(F)) j+pr+(F(F)) j+(F(pr+F)).

Since the Fourier transform F is acyclic (see3.1.4) and so is j+, we get Hs pr+(+M) j+(F(Hs pr+F)).Since rk(F(Hs pr+F) =rk(j+(F(Hs pr+F))(recall the notation of2.3.2), we can conclude.

From Theorem3.2.2,the reader can check the p-adic analogues of corollaries [BBD82, 4.5.25] by copyingthe proofs. Moreover, from [AC13b], we have a theory of weight within the framework of arithmetic D-modules.

For instance, we have checked the stability properties under Grothendieck six operations, i.e. thep-adic analogueof Deligne famous work in[Del80]), which is also explained in [BBD82, 5.1.14]. In [BBD82,5.2.1], a reverseimplication was proved. The reader can check that we can copy the proof without further problems (i.e., we onlyhave to check that we have nothing new to check, e.g. we already have3.1.1or the purity of the middle extension ofsome pure unipotentF-isocrystal as given in [AC13b,3.6.3]). For the reader, let us write this p-adic version and itsimportant corollary[BBD82,5.3.1] (this corollary is proved in[AC13b]in another way, but Theorem3.2.3below is anew result).

Theorem 3.2.3([BBD82,5.2.1]). We suppose k= Fps is finite and that F means the sth power of Frobenius. Choosean isomorphism of the form : Qp

C. Let X be a k-variety andE F-Ovhol(X/K). We suppose that, for any etale

morphism U X with U affine, the K-vector space H0(pU+(+(E))is -mixed of weight w. Then Eis -mixed ofweight w.

Corollary 3.2.4([BBD82, 5.3.1]). With the notation3.2.3, ifE is -mixed of weightw (resp. w), then anysubquotient ofE is-mixed of weight w (resp. w).

Finally, except[BBD82,5.4.78], the reader can check easily the other results of the chapter 5 of [BBD82] bytranslating the proofs in ourp-adic context.

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Daniel CaroLaboratoire de Mathmatiques Nicolas OresmeUniversit de Caen Campus 214032 Caen CedexFrance.email: [email protected]

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Betti Number Estimates in P-Adic Cohomology

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