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arXiv:1608.03796v2 [math.NT] 30 Nov 2017

BRUNO CHIARELLOTTO AND CHRISTOPHER LAZDA

ABSTRACT. In this article we study various forms ofℓ-independence (including the caseℓ=p) for the coho-mology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form ofℓ-independence for the unipotent fundamental group of smooth and projective varieties over finite fields, by then proving a certain ‘spreading out’ result we are able to deduce a much weaker form of

ℓ-independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduceℓ-independence results for the cohomology of smooth and proper varieties over equicharactersitic local fields from the well-known results onℓ-independence for smooth and proper varieties over finite fields. As another consequence of this ‘spreading out’ result we are able to de-duce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristicpwe show a similar weak version ofℓ-independence for the unipotent fundamental group of a semistable curve in mixed characteristic.

CONTENTS

1. Introduction 1

2. Preliminaries on fundamental groups andp-adic cohomology 4

3. The formalism ofℓ-independence, statement of the conjectures 5

4. Unipotent fundamental groups over finite fields 8

5. Cohomology over equicharacteristic local fields I 10

6. Cohomology over equicharacteristic local fields II 13

7. L-functions and the proof of Proposition5.8 15

8. Unipotent fundamental groups over equicharacteristic local fields 19

9. Curves in mixed characteristic 22

References 23

1. INTRODUCTION

One intriguing aspect of the proof of the Weil conjectures by Grothendieck and his school, via the theory of ´etale cohomology, is in the need to choose an auxiliary primeℓ; the fieldQℓofℓ-adic numbers then providing the coefficient field for a Weil cohomology theory. (To fix terms, letF be a field,X/Fan algebraic variety,Fsepa separable closure andGFthe corresponding absolute Galois group.) Whilea priori

To account for this plethora of different but subtly related cohomology groups, Grothendieck proposed the theory of pure motives, which for smooth and proper varieties at least should provide some form of algebro-geometric objecth∗(X) =Lihi(X)such thatH´eti(XFsep,Q_ℓ)∼=hi(X)⊗Q_ℓ(again, suitably inter-preted). Thus the groupsH_´eti(XFsep,Q_ℓ)should be ‘independent ofℓ’, since all information they contain can be deduced from the ‘absolute’ cohomology groupshi(X). In particular for curves, we essentially have h1(C) =Jac(CFsep)∨.

While the theory of motives, despite significant recent advances, is still incomplete, its prediction that various cohomology groups should be ‘independent ofℓ’ can, over certain ground fields, be precisely for-mulated in ways which do not depend on such a theory. For example, one such statement is that for smooth and proper varietiesX over finite fields of characteristic p, the trace of Frobenius onH_´eti(XFsep,Q_ℓ)(for ℓ6=p) has values inZand is independent ofℓ. Since the absolute Galois group of finite fields is generated by Frobenius, this tells us that, up to semisimplification, the various Galois representationsH_´eti(XFsep,Q_ℓ) are, once restricted to FrobZ⊂GF, pairwise isomorphic (at least after base changing to any suitably large

fieldΩ).

Over more general fields, formulating similarly precise statements is somewhat tricky, since the Galois representationH_´eti(XFsep,Q_ℓ)depends on the interaction between the topology ofQ_ℓand the profinite topol-ogy onGF. For local fields (i.e. complete, discretely valued fields with finite residue fields) one can, thanks

to the theory of Weil–Deligne representations and Grothendieck’sℓ-adic monodromy theorem (and at least forℓdifferent to the residue characteristic) package up the information contained inHi

´et(XFsep,Qℓ)in a way

that only depends onQ_ℓas an abstract field, it therefore makes sense to conjecture that these groups become ‘pairwise isomorphic’ over suitably large fields.

Forℓ=pone needs a slightly different approach. Over finite fields, one needs to replace ´etale cohomol-ogy by crystalline cohomolcohomol-ogyHi

cris(X/K), and over mixed characteristic local fields one needs to replace

theℓ-adic monodromy theorem by it’sp-adic analogue, stating that the ´etale cohomology of varieties over mixed characteristic local fields is potentially semistable. Over finite fields, the formalism of crystalline cohomology (including weights) shows that the ‘independence ofℓ’ results can be extended to include the caseℓ=p, and over mixed characteristic local fields it was shown in [Fon94] how to use p-adic Hodge theory to attach Weil–Deligne representations to de Rham Galois representations, thus again making sense of theℓ-independence conjectures ‘at p’. WhenF is an equicharacteristic local field, then a more subtle use of crystalline cohomology allows one to view the cohomology of smooth and proper varieties overF as(ϕ,∇)-modules over the Robba ring, and again the appropriate local monodromy theorem allows one to construct associated Weil–Deligne representations (see for example§3 of [Mar08]). Thus following Fontaine [Fon94] one can again formulateℓ-independence conjectures in this case, including atℓ=p.

One can also ask about other expected ‘motivic’ invariants, the most basic being the unipotent funda-mental group. Whenℓ6=pthen this can be defined as theQℓ-pro-unipotent completion of the geometric ´etale fundamental group. WhenF is finite andℓ6=pit is therefore acted on by Frobenius, and again we can ask whether or not this ‘non-abelian’ representation of FrobZis independent ofℓ. WhenFis local (and again whenℓ6=p) it is acted on ‘continuously’ byGF, so we can apply Grothendieck’s monodromy theorem

over the bounded Robba ringE†_{. Again, appealing to the p-adic local monodromy theorem allows us to}

produce a non-abelianp-adic Weil–Deligne representations in this case. Hence in both these cases (i.e. F finite or equicharacteristic local) one can formulate preciseℓ-independence conjectures for the unipotent fundamental group, including the caseℓ=p. WhenF is mixed characteristic local andℓ=pthings are more tricky, since in order to apply Fontaine’s methods one first needs to know that thep-adic unipotentπ1

is de Rham as aGF-representation. For semistable curves this follows from the results of [AIK15] and in

general from results of D´eglise and Nizioł [DN15].

The purpose of this article, then, is to prove various cases of these ℓ-independence conjectures, all including the caseℓ=p. After recalling the various facts we need concerning cohomology and fundamental groups in§2, the precise formulation of the conjectures is given in§3. In the rest of the article we will then prove certain versions of them in the following four cases.

(1) Unipotent fundamental groups of smooth projective varieties over finite fields. (2) Cohomology groups of smooth, proper varieties over equicharacteristic local fields.

(3) Unipotent fundamental groups of smooth, proper varieties with semistable reduction over equichar-acteristic local fields.

(4) Unipotent fundamental groups of smooth, proper curves with semistable reduction over mixed characteristic local fields.

In the first case, the subject of§4, we prove the strongest possible form ofℓ-independence, stating that the various unipotentπuni

1 ’s become isomorphic as ‘non-abelian’ Weil–Deligne representations over suitably

large fields. Here the basic idea behind the proof is formality, in that the rational homotopy type is a formal consequence of the cohomology ring.

In the second case, treated in §5and§6, we show a weaker form of ℓ-independence, stating that the Frobenius seimsimplifications of the various cohomology groups become isomorphic over suitably large fields. The proof proceeds by first considering the case when the variety in question has semistable reduction (Theorem5.1). In this situation we use a ‘spreading out’ result to reduce to the case of a global semistable family, and then apply a result of Deligne in [Del80] stating that ‘ℓ-independence’ of a suitable local system on an open curve impliesℓ-independence at the missing points. In fact Deligne’s result only applies for

ℓ6=p, but it is not difficult to follow through his proof ‘atp’, as we do so in§7. The general case is then deduced by using alterations and an argument involving weights (Theorem6.1).

In the third case, the topic of§8, we prove a fairly weak form ofℓ-independence, stating that the Frobe-nius semisimplifications of the graded pieces of the universal enveloping algebra of the Lie algebra ofπuni 1

(for the filtration coming from the augmentation ideal) become isomorphic over suitably large fields. The proof here follows that of the second case, again reducing to the case of a global semistable family and applying the same independence result of Deligne. Finally, in§9we consider the last case, proving the same result for curves in mixed characteristic by reducing to the equicharacteristic situation.

Notations and conventions. We will letF denote either a finite fieldkor a local field with residue field k(usually equicharacteristic). As always, p will be the characteristic ofk, and we will letq denote its

cardinality. The term ‘Frobenius’ without further qualification will always be understood to meangeometric Frobenius, and Frobenius structures on isocrystals will, unless specified otherwise, meanq-power Frobenius structures. We will letW =W(k)denote the ring of Witt vectors of k,K its fraction field andKun the maximal unramified extension ofK. By ‘variety’ we will mean ‘separated scheme of finite type’. We will denote a separable closure ofF byFsep, and the corresponding absolute Galois group by GF. We will

denote byR_{the Robba ring overK, and byMΦR}∇ _{the category of(ϕ,∇)-modules over}R_{. We will denote} byE†

K⊂RKthe bounded Robba ring.

2. PRELIMINARIES ON FUNDAMENTAL GROUPS AND p-ADIC COHOMOLOGY

The purpose of this section is to recall some general facts about unipotent fundamental groups, as well as some of the material from [LP16] concerningp-adic cohomology over equicharacteristic local fields. So let X/Fbe smooth, geometrically connected, quasi-projective variety andx∈X(F). Then for anyℓ6=char(F)

theℓ-adic version of the unipotent fundamental group has two different but equivalent interpretations. First of all we may consider the ´etale fundamental groupπ´et

1(X,x¯)based at a geometric point abovex, together

with the homotopy exact sequence

1→π´et

1(XFsep,x¯)→π´et

1(X,x¯)→GF→1.

The pointxinduces a splitting of this exact sequence, and this gives rise to an action ofGFonπ₁´et(XFsep,x¯). Applying the Malcev completion functor overQℓthen gives a pro-unipotent group schemeπ1´et(XFsep,x¯)_Q

ℓ

overQ_ℓwhich by functoriality is acted on byGF.

Alternatively, we can use a Tannakian approach. That is, we consider the category UniQℓ(XFsep) of

unipotent lisseQ_ℓ-sheaves onX. The pointxgives rise to a fibre functorx∗: UniQ_ℓ(XFsep)→Vec_Q

ℓ and

hence we may defineπ´et

1(XFsep,x¯)_Q

ℓ to be the (pro-unipotent) affine group scheme representing

automor-phisms ofx∗. By functorialityGF acts on UniQ_ℓ(XFsep), preservingx∗, and hence by Tannaka duality on

π´et

1(XFsep,x¯)_Q

ℓ. As the notation suggests, this coincides with the previous construction by [SGA5, Expos´e

VI,§1.4.2].

While the first definition does not behave well whenℓ=char(F), the second has an clear analogue, at least whenF=kis finite, replacing lisseQℓ-sheaves onXFsep by overconvergent isocrystals onX/K, the category of which we will denote by Isoc†(X/K). Let UniK(X)denote its full subcategory of unipotent

objects. Again, the pointxinduces a fibre functorx∗: UniK(X)→VecK and by definitionπ1rig(X/K,x)

is the (pro-unipotent) affine group scheme representating automorphisms ofx∗. The Frobenius pullback functorF∗: UniK(X)→UniK(X)is an autoequivalence, and induces an isomorphismF∗:π1rig(X/K,x)→

π₁rig(X/K,x).

WhenF =k((t))is equicharacteristic local, we need to use the machinery of [LP16] to define the p -adic unipotent fundamental groups. Since we will also need the results fromloc. cit. concerningp-adic cohomology for varieties over suchF, we will take the opportunity now to quickly recap some of the main points from [LP16]. There we constructed, for any varietyX/F, cohomology groupsH_rigi (X/E†

K)as

(ϕ,∇)-modules over the bounded Robba ringE†

K, satisfying all the expected properties of an ‘extended’

Weil cohomology theory. In other words, we have finite dimensionality, vanishing in the expected degrees, versions with compact support or support in a closed subscheme, Poincar´e duality, K¨unneth formula &c. We also constructed a category Isoc†(X/E†

K)of overconvergent isocrystals onXrelative toE †

Corollary 2.2] it was proved that this category is Tannakian. The pointxprovides a fibre functor

x∗: Uni_E†

K(

X)→Vec_E†

K

from the subcategory of unipotent isocrystals, and again we may defineπ₁rig(X/E†

K,x)to be the

correspond-ing pro-unipotent group scheme overE†

K. It was also shown in [Laz16,§5] how to put a canonical ‘(ϕ,∇)

-module structure’ onπ₁rig(X/E†

K,x), that is a (ϕ,∇)-module structure on its Hopf algebra, which simply

as a(ϕ,∇)-module (i.e. forgetting the Hopf algebra structure) is a direct limit of its finite dimensional sub-(ϕ,∇)-modules.

Base changing the whole situation toR_{Kwe therefore obtain a non-abelian(ϕ,∇)-module}

π₁rig(X/R_K,x):=πrig

1 (X/E † K,x)⊗E†

K

R_K

overR_{K, as well as abelian ones}

H_rigi (X/R_K):=Hi

rig(X/EK†)⊗E†

K

R_K.

These should be considered the p-adic analogues of theℓ-adic Galois representationsπ´et

1(XFsep,x¯)_Q

ℓ and

Hi_´et(XFsep,Q_ℓ)respectively.

3. THE FORMALISM OFℓ-INDEPENDENCE,STATEMENT OF THE CONJECTURES

In this section we will review some of the formalism behind the notion ofℓ-independence over finite and equicharacteristic local fields, following [Del80,Fon94]. For this section,F will stand for either a finite fieldkor a local field with residue fieldk.

Definition 3.1. We define the Weil groupWF⊂GF as follows.

• IfF=kthen we will letWFdenote the subgroup ofGF∼=FrobZkb consisting ofintegerpowers of

Frobenius.

• IfFis local, then we will letWFdenote the inverse image ofWkunder the surjectionGF։Gk.

It is topologised as follows: ifF is finite thenWF is given the discrete topology, ifF is local thenWF is

given the unique topology such that the inertia groupIFis open.

SinceWFis pro-discrete, we may form the associated group schemeW_FalgoverQby writingWF=lim_←−iGi

in the category of groups, viewing each discrete groupGi as a group scheme over Q and then setting

W_Falg=lim_←−

iGiin the category of group schemes.

Lemma 3.2. For any field E of characteristic0there is an equivalence of categories between continuous representationsρ :WF →GL(V)in finite dimensional E-vector spaces and finite dimensional algebraic

representationsρalg_:Walg

F →GL(V)of the group scheme W alg F over E.

IfFis local then there is a natural action ofW_Falgon the additive groupGaby setting

gxg−1=q−v(g)x

wherev(g)is such thatgmaps to Frob_kv(g)inWk.

Definition 3.3. We define the Weil–Deligne groupW_F′ to beW_Falg ifF =kis finite orW_Falg⋉ GaifF is

Note that the usage of the term ‘Weil–Deligne’ group to refer toW_Falg over finite fields is not at all standard, and we employ it here only to be able to make uniform statements for finite and local fields.

We will now letEbe a field of characteristic 0, and letC _{→AffEbe a category fibred over the category} of affineE-schemes. In other words for everyE-algebraRwe have a categoryC_{Rand for every morphism} R→R′we have a pullback (or ‘base extension’) functorC_R→C_R′. The examples the reader should keep

in mind are the following.

• the category of (quasi-)coherentO_-modules;

• the category of pro-nilpotent LieO_-algebras;

• the category of (quasi-coherent) HopfO_-algebras;

• the category of pro-unipotent group schemes.

In particular, given an objectX∈C_{E it makes sense to speak of the functor onE-algebras}

Aut_E(X)(R) =AutC_R(X⊗R).

Definition 3.4. AC_{E-representation ofW}′

F is an object X ∈CE together with a morphism of functors

W_F′ →Aut_E(X). The category of such objects is denoted RepC_E(W_F′).

Note that whenC _{is the category of coherent}O_{-modules, then one recovers the usual notion of anE} -valued Weil(–Deligne) representation, in general the point of this slightly fiddly approach is in order to have a well-defined notion for non-abelian objects such as Hopf algebras or group schemes. We can now finally make precise the concept ofℓ-independence for (not necessarily abelian) Weil–Deligne representations.

Definition 3.5( [Del73],§8). (1) LetE′/E be a field extension, and X ∈RepC

E′(W

′

F)aCE′-valued

Weil–Deligne representation. Then we say thatX is defined overEif for any algebraically closed fieldΩcontainingE′,X⊗Ω∈RepC_Ω(W_F′)is isomorphic to all of its Aut(Ω/E)-conjugates. (2) Let{Ei}i∈Ibe a family of field extensions ofE, and{Xi}i∈Ia family ofCEi-valued Weil–Deligne

representations. Then we say that{Xi}i∈IisE-compatibleif eachXiis defined overE, and for any

i,j and any algebraically closed fieldΩcontainingEi andEj, the objectsXi⊗ΩandXj⊗Ω in

RepC_Ω(W_F′)are isomorphic.

WhenC _{is the category of coherent}O_{-modules, then we also have a weaker notion of compatibility} described as follows. IfF is finite then we define the ‘monodromy’ filtrationM•on a representationV of W_F′ to be the trivial one, i.e. 0 in negative degrees andV otherwise. IfFis local then we will letM•be the usual monodromy filtration coming from viewing theGaaction as a nilpotent endomorphismN:V →V.

Definition 3.6. Let{Vi}i∈I be a family of finite dimensionalEi-valued Weil–Deligne representations as

above. We say that{Vi}i∈Iisweakly E-compatibleif for allkthe character

Tr(−|GrM_kVi):WF→Ei

of thek-th graded piece of the monodromy filtration has values inEand is independent ofi.

Note that the difference between weak compatibility and compatibility is essentially that of Frobenius semisimplicity. For a representationV ofW_F′, we will letVF-ss denote its Frobenius semisimplification. Thus whenF =k is finite this is just the semisimplification, and when F is local we semisimplify the underlyingWF-representation.

Lemma 3.7. Let{Vi}be a family of W_F′-representations. Then{Vi}is weakly E-compatible if and only if

Proof. WhenF is finite this is clear, whenFis local this is [Del73, Proposition 8.9].

Now, ifX/F is a variety andℓ6=p, then one can view theith ´etale cohomology groupH_´eti(XFsep,Q_ℓ) as a continuous representation ofGF, and hence as anQℓ-valued algebraic representation ofWF′, when we

consider it as the latter we will generally writeH_ℓi(X). Forℓ=pandF=kis finite, the natural Frobenius action on the rigid cohomologyHi

rig(X/K)allows us to consider it as aK-valued algebraic representation

H_pi(X)ofW_F′. Forℓ=pandF∼=k((t))equicharacteristic local, then we may consider the rigid cohomology groupsH_rigi (X/R_{K)ofX as(ϕ,∇)-modules over the Robba ring}R_{K. Using Marmora’s functor [Mar08]} from(ϕ,∇)-modules overR_KtoKun_{-valued Weil–Deligne representations, we may therefore consider the}

Weil–Deligne representationH_pi(X)associated toH_rigi (X/R_{K). Whenℓ=pandFis mixed characteristic} local, then the ´etale cohomologyH_´eti(XFsep,Q_p) is de Rham as a representation ofG_F, hence we may follow Fontaine’s construction [Fon94] to produce aKun-valued Weil–Deligne representation, which we will again denoteH_pi(X). Of course, we may also consider versionsH_ci_,ℓ(X)with compact support. For cohomology, then, theℓ-independence conjectures (first formulated forFlocal in mixed characteristic by Fontaine in [Fon94]) are as follows.

Conjecture 3.8. • C_ℓ(X,Hi_{): the system{H}i

ℓ(X)}ℓisQ-compatible.

• Cℓ,w(X,Hi): the system{H_ℓi(X)}ℓis weaklyQ-compatible.

• C_ℓ(X,Hi

c): the system{Hci,ℓ(X)}ℓisQ-compatible.

• Cℓ,w(X,Hci): the system{Hci,ℓ(X)}ℓis weaklyQ-compatible.

WhenF is finite, thenC_ℓ,w(X,Hi)is known wheneverX is smooth and proper (forℓ6=p this follows

from Deligne’s proof of the Riemann hypothesis [Del74], the fact that the same is true atℓ=pthen follows from [KM74, Corollary 1]). Even in this case, however,C_ℓ(X,Hi_{)is still wide open in general; by Lemma} 3.7it would follow from the Frobenius semisimplicity conjecture. Hence for X proper and smooth of dimensiond we knowC_ℓ(X,Hi_{)fori=0,1,2d−1,2d, as well as for abelian varieties in all degrees. Let}

us also remark that in [MO] the authors proveCℓ,w(X,Hi)andCℓ,w(X,Hci)whenever dimX≤2.

WhenFis local, then a much weaker version ofC_ℓ,w(X,Hi_)andC

ℓ,w(X,Hci)forms part of the main result

of [Zhe09], however, even for smooth and projective varieties the fact that ‘purity’ forHiis more compli-cated for local fields means that there is no straightforward deduction ofC_ℓ,w(X,Hi_{)from Zheng’s results.}

In [LP16, Theorem 5.85] it was proved that whenF∼=k((t))is equicharacteristic local, thenCℓ,w(X,Hi)

holds for smooth (possibly open) curves and abelian varieties, or for smooth and proper varieties with good reduction. In fact, it was claimed there that in factCℓ(X,Hi)holds for smooth curves and abelian varieties, however there is a gap in the proof which we shall correct in Lemma3.12below.

Next let us turn to the unipotent fundamental groupπ1, whose definition we recalled in the previous

section. WhenX/Fis a geometrically connected variety with base pointx∈X(F), andℓ6=p, then again we may consider theℓ-adic unipotent fundamental groupπ´et

1(XFsep,x¯)_Q

ℓas aGF-representation, we will write

πℓ

1(X,x)for the associatedWF′-representation (with values in the fibred category of pro-unipotent group

schemes). When char(F) =ℓ=pwe use the rigid fundamental group (π₁rig(X/K,x)whenF=kis finite,

π₁rig(X/R_K,x)whenF∼_{=k((t))is local) to produce ap-adic representation ofW}′

F, with values in the fibred

category of pro-unipotent group schemes, which we will denote byπ₁p(X,x)(again using Marmora’s functor from(ϕ,∇)-modules to Weil–Deligne representations whenF∼=k((t))). When char(F) =0,ℓ=pandX is a semistable curve we note that by [AIK15, Theorem 1.8] thep-adic ´etale pro-unipotent fundamental groupπ´et

1(XFsep,x¯)_Q

aKun-valued (non-abelian) Weil–Deligne representationπ₁p(X,x). For more generalX, there are results of D´eglise and Nizioł in [DN15] which again state thatπ´et

1(XFsep,x¯)Qp is de Rham.

We will writeAℓ,X,xfor the Hopf algebra ofπ1ℓ(X,x),Lℓ,X,xfor its Lie algebra, ˆUℓ,X,xfor the completed

universal enveloping algebra ofLℓ,X,xandaℓ,X,xfor the augmentation ideal of ˆUℓ,X,x(these are

representa-tions in Hopf algebras, pro-nilpotent Lie algebras and associative algebras respectively, and for eachk≥1 we may consider ˆU_ℓ,X,x/ak_ℓ,

X,xas a Weil–Deligne representation in vector spaces). The strongest form of ℓ-independence one can state for fundamental groups is the following.

Conjecture 3.9(Cℓ(X,π1uni)). The collection{π1ℓ(X,x)}ℓof pro-unipotent groups with an action of WF′ is

Q-compatible. Equivalently the collection{A_ℓ,X,x}ℓ(resp.{Lℓ,X,x}ℓ,{Uˆℓ,X,x}ℓ) of Hopf algebras (resp. Lie

algebra, resp. associative algebras) isQ-compatible.

We will also be interested in the following weakening of the above conjecture.

Conjecture 3.10 (Cℓ(X,Uˆ/a•),Cℓ,w(X,Uˆ/a•)). Fix k ≥1. Then the system {Uˆℓ,X,x/akℓ,X,x}ℓ of WF′

-representations is (weakly)Q-compatible.

Remark 3.11. Note that in this conjecture we are only requiring compatibility as W_F′-representations, i.e. we are saying nothing about the algebra structure. We clearly haveC_ℓ(X,πuni

1 )⇒Cℓ(X,Uˆ/a•)⇒ Cℓ,w(X,Uˆ/a•).

We would like to finish this section by fixing a hole in the proof of [LP16, Theorem 5.85], stating that whenF∼=k((t))is equicharacteristic local andX/Fis a smooth curve or an abelian variety, thenCℓ(X,Hi) holds, in fact we only showed there thatC_ℓ,w(X,Hi)holds, that is weakℓ-independence. Given Lemma3.7,

the next result completes the proof of ‘strong’ℓ-independence in these cases.

Lemma 3.12. Let F∼=k((t)) and X/F be either a smooth curve or an abelian variety. Then H_ℓ1(X)is

Frobenius semisimple for allℓ.

Proof. First supposeX=Ais an abelian variety, the proof of [LP16, Theorem 5.88] shows that the graded

pieces of the monodromy filtration onH_ℓ1(A)(for anyℓ, includingℓ=p) are given by the following:

• the cohomologyH1

ℓ(T)of a torus overF;

• the cohomologyH_ℓ1(B)of an abelian variety with potentially good reduction;

• the Weil–Deligne representation coming from a continuous representationGF→Zsfor somes≥0.

Frobenius semisimplicity of the first and third are clear (since they both factor through a finite quotient of GF), the second follows from Frobenius semisimplicity for abelian varieties over finite fields. Since the

three graded pieces are of different weights, Frobenius semisimplicity ofH_ℓ1(A)follows. The deduction for smooth (possibly open) curves now follows exactly as in [LP16,§5.4].

4. UNIPOTENT FUNDAMENTAL GROUPS OVER FINITE FIELDS

In this section, we will prove the following result.

Theorem 4.1. Let F=k be a finite field, and X/F a smooth, projective variety. Then Cℓ(X,π1uni)holds.

Let us first note the following consequence of the weak Lefschetz theorem.

Proof. If dimX>2 then choose an iterated hyperplane sectionY ֒→X passing throughxwith dimY =

2. The weak Lefschetz theorem [SGA2, Expos´e XII, Corollaire 3.5] says that the map π´et

1(YFsep,x¯)→

π´et

1(XFsep,x¯)is an isomorphism, hence the same is true of theirQℓ-unipotent completions, in other words

πℓ

1(Y,x)→π1ℓ(X,x)is an isomorphism forℓ6=p.

Whenℓ=pwe use the weak Lefschetz theorem slightly differently. We claim that for any unipotent isocrystalEonX/K, the induced map

H_rigi (X/K,E)→H_rigi (Y/K,E|Y)

is an isomorphism fori=0,1 and injective for i=2, this suffices to prove π₁p(Y,x)→π₁p(X,x)is an isomorphism by combining Propositions 1.2.2 and 1.3.1 of [CLS99] with Tannakian duality. IfEis constant then this is simply the usual weak Lefschetz, in general one inducts on the unipotence degree and uses the

five lemma.

The key ingredient in the proof of Theorem4.1is the following concrete description of the Lie algebra Lieπℓ

1(X,x), due to Pridham [Pri09].

Proposition 4.3. Let L(H1

ℓ(X)∨)denote the free Lie algebra on the dual of Hℓ1(X), and Iℓthe ideal gener-ated by the image of the dual of the cup product

H_ℓ2(X)∨ ∪

∨ ℓ

→H_ℓ1(X)∨⊗H_ℓ1(X)∨.

Then there is a WF-equivariant isomorphism

Lℓ,X,x=Lieπ1ℓ(X,x)∼=

L(H1

ℓ(X)∨) I_ℓ .

Proof. Forℓ6=pthis is [Pri09, Corollary 2.15], whenℓ=pexactly the same proof works, using thep-adic

formalism of weights (see for example [Ked06]).

Since Frobenius semisimplicity is known forH_ℓ1(X)forXsmooth and projective of any dimension, and anyℓ, we deduce the following corollary.

Corollary 4.4. To prove Theorem4.1, it suffices to show that the image of the cup product

H_ℓ1(X)⊗H_ℓ1(X)→∪ℓH_ℓ2(X)

satisfies weakℓ-independence, i.e. the family of representations{im(∪ℓ)}ℓis weaklyQ-compatible.

Clearly this holds whenX is a smooth, projective curve, since then the cup product map is surjective. WhenXis a surface, we use the existence of K¨unneth projectors onX.

Proposition 4.5 (Corollary 2A10 and Lemma 2.4, [Kle68]). Let X/F be a smooth projective surface.

Then there exist correspondencesϖi∈CH2(X×X)such that for all ℓ, the action ofϖi on cohomology

H_ℓ∗(X) =LiHℓi(X)is to project to the summand Hℓi(X).

We can now complete the proof of Theorem4.1.

Proof of Theorem4.1. We may assume that dimX=2. Chooseϖ1∈CH2(X×X)as above, and letϖ1⊗1:=

ϖ1×ϖ1∈CH4(X4). Then the effect ofϖ1⊗1onHℓ∗(X×X)is to project to the summandHℓ1(X)⊗Hℓ1(X). Now letϖ1∪1be the pullback ofϖ1⊗1to CH2(X×X)via∆×∆:X2→X4. The effect ofϖ1∪1onH_ℓ∗(X)

is therefore to project onto the image of∪ℓ insideHℓ2(X). Using Corollary4.4the result now follows by composingϖ1∪1with the graph of (some power of) Frobenius and applying the Lefschetz fixed point

5. COHOMOLOGY OVER EQUICHARACTERISTIC LOCAL FIELDSI

The main result of this section is the following.

Theorem 5.1. Suppose that F∼=k((t))is equicharacteristic local, and let X/F be a smooth, proper variety

with semistable reduction. Then Cℓ,w(X,Hi)holds.

This is the main step in showing suchℓ-independence unconditionally for smooth and proper varieties overF. The first key result we need in the proof of Theorem5.1essentially allows us to reduce to the ‘globally defined’ case, and in fact works in slightly larger generality than we will need. So for now let Sdenote a Dedekind scheme,s∈S a closed point andOb_{S,s the completed local ring ats. By definition a}

semistable scheme overOb_{S,sis one that is ´etale locally smooth over Spec}Ob_S,s[x1, . . . ,xr]/(x1· · ·xr)

.

Proposition 5.2. LetX _→SpecOb_{S,sbe a semistable scheme of finite type and n≥2an integer. Then}

there exists an ´etale neighbourhood(U,u)→(S,s)of s and a flat schemeY _→U , smooth away from u and semistable at u, such that

Y _×U OU,u

mn u

∼

=X _×b

O_S,s

b

O_S,s

mn s

as schemes overO_U,u/mn

u∼=ObS,s/mns.

Proof. Choose a local parametert at s. After shrinking S we may assume that S=Spec(R) is affine andt ∈R. Then there exists an ´etale cover {Ui→X} of X such that each Ui is ´etale over some

SpecOb_{S,s[x1, . . . ,xd+1]/(x1· · ·xr−t)(withranddallowed to vary withi.)}

Choose some finitely generated, normalR-algebraAcontained withinOb_{S,ssuch that everything in sight} is defined overA, i.e.X_{, the ´etale cover{}Ui→X}and the ´etale maps

Ui→Spec

_{A[x1, . . . ,x}

d+1]

(x1· · ·xr−t)

.

Now by Artin’s approximation theorem, the pointα: SpecOb_S,s→Spec(A)of the finite typeR-scheme

Spec(A), coming from the inclusionA⊂Ob_{S,s, can be approximated modulo}tnby a Henselian point, in other words there exists a morphismαh,n_{: SpecO}h

S,s

→Spec(A)which agrees withαmodulotn_{. Then pulling}

back viaαh,n_{and using the fact thatt is a local parameter atswe can see that there exists a semistable}

familyY _→SpecOh S,s

which agrees withX _modulotn_.

Now finally we note that there must be some finitely generated sub-algebraB⊂Oh

S,s, ´etale overSsuch

thatY _{is defined over}Band smooth away froms, settingU=Spec(B)then completes the proof.

To apply this result, we will specialise, and letR∼=kJtKbe the ring of integers insideF∼=k((t)). Suppose that we are given a semistable schemeX_/R. We will letX×_denoteX _{endowed with the log structure} given by the special fibre, andX₀×the special fibre endowed with the inverse image log structure fromX_. LetX denote the generic fibre ofX_{. Then we may consider, as in [Nak98], the log-´etale cohomology} Hi_´et(X₀×,tame,Qℓ)of the log schemeX0×, as anℓ-adic representation ofGF(on which the wild inertia group

PF⊂IF acts trivially). We have the following logarithmic analogue of the smooth and proper base change

theorem.

Proposition 5.3( [Nak98], Proposition (4.2)). There is an isomorphism

of GF-representations.

Of course, there is a similar result for p-adic cohomology, which goes as follows. Let MΦN K denote

the category of(ϕ,N)-modules overK, that is finite dimensional vector spaces together with a semilinear, bijective Frobeniusϕand a nilpotent endomorphismN, such thatNϕ=qϕN. Given such a(ϕ,N)-module V, we can put a connection onV⊗KR_{Kby setting∇(v⊗1) =N(v)⊗t}−1_{, and this induces a fully faithful}

functor

− ⊗KR_K:MΦN

K→MΦ∇R_K

from(ϕ,N)-modules overKto(ϕ,∇)-modules overR_{K, whose essential image is exactly the objects for} which the connection acts unipotently. ForX _{as above, we may consider the log-crystalline cohomology} H_{log -cris}i (X₀×/K×):=H_{log -cris}i (X₀×/W×)⊗K of its special fibre as a(ϕ,N)-module overK, as in [HK94]

(the notationW×,K×refers to the fact that we are endowingW with the Teichm¨uller lift of the log structure of the punctured point).

Proposition 5.4( [LP16], Theorem 5.46). There is an isomorphism

H_log-crisi (X₀×/K×)⊗KR∼_=Hi

rig(X/RK)

insideMΦ∇R_K.

We therefore obtain the following.

Corollary 5.5. LetX _→Spec(R)be a proper, semistable scheme with generic fibre X . Then there exists

a smooth, geometrically connected curve C/k, a k-rational point c∈C(k), a proper, semistable scheme Y _→C, smooth away from c, and an isomorphism R∼=Ob_C,c(inducing F∼=kd(C)_c) such that

H_´eti(XFsep,Q_ℓ)∼=Hi

´et(YFsep,Q_ℓ)

as GF representations for all i, ℓ6=p, and

H_rigi (X/R_K)∼_=Hi

rig(YF/RK)

as(ϕ,∇)-modules, for all i. In particular, Cℓ,w(X,Hi)is equivalent to Cℓ,w(YF,Hi)

Proof. By Proposition5.2we know that there exists a globally defined semistable scheme which agrees withX _{up to order 2, hence the special fibres coincide as log schemes. Hence by Propositions5.3and5.4}

the cohomologies also coincide.

Our primary interest here is inℓ-independence results, however, let us note in passing that Corollary5.5

implies that the Clemens–Schmid exact sequence, obtained by the first author in collaboration with Tsuzuki in [CT14], exists forX_.

Corollary 5.6. Let d=dim(X_/R) =dimX ₋₁. There is a Clemens–Schmid long exact sequence

. . .→H_rigm(X0/K)→H_logm _-cris(X₀×/K×)→N H_logm _-cris(X₀×/K×)(−1)→H_rig2d−m(X0/K)∨(−d−1)→. . .

Proof. WhenX _{extends to a global semistable family, this is the main result of [CT14]. Given Proposition}

To return to our main focus, then, to prove Theorem5.1it suffices to do so in the ‘globally defined’ case, i.e. we may assume that we have a smooth geometrically connected curveC/k,c∈C(k)such that F=kd(C)_c, and a semistable schemeX _{→C, smooth away fromc, such thatX=}X_F.

LetU=C\c, and consider a collection{F_{ℓ}ℓof local systems onU, where}F_{ℓis a lisse}Q_ℓsheaf onU forℓ6=pandF_{pis an overconvergentF-isocrystal onU/K(hereF is theq=#k-power Frobenius). Then} for any closed pointx∈Uthe action of geometric Frobenius onF_{ℓ,x¯forℓ6=p, plus the action ofF}deg(x)_on F_{p,x, defines a collection{}F_{ℓ,x}ℓof representations of the Weil groupW}

k(x).

Definition 5.7. (1) We say that the system{F_{ℓ}is weaklyQ-compatible if{}F_{ℓ,x}ℓis so for all closed}

pointsx∈U.

(2) We say that the system{F_{ℓ}is pure of weightnif for each closed pointx∈Uthe eigenvalues of} Frobk(x)(resp.Fdeg(x)) onFℓ,x¯(resp.Fp,x) are Weil numbers of weightn.

Given such a system, we may also restrict to the Weil–Deligne group atc. Whenℓ6=p, this is fairly standard, since we may consider the Galois groupGFas a decomposition group atc, and hence restrict the ℓ-adic representationπ1(U,η)¯ →Fℓ,η¯ of the fundamental group ofU (based at some geometric generic

point) to get anℓ-adic representation of GF, and hence a Weil–Deligne representation, which we shall

callF_{ℓ,c. Whenℓ=p we use the construction of [Tsu98,§6.1]. Pulling back to an appropriate tubular} neighbourhood ofcgives a functor

F-Isoc†(U/K)→MΦ∇R_c

whereR_{cis a copy of the Robba ring at}c. Then applying Marmora’s functor from(ϕ,∇)-modules over the Robba ring to Weil–Deligne representations [Mar08] gives us aKun-valued representation ofW_F′ which we shall callF_{p,c. The key result is then the following, whose proof we will defer to§7below.}

Proposition 5.8. If{F_ℓ}ℓis a weaklyQ-compatible system on U , which is moreover pure of some weight

n, then{F_{ℓ,c}ℓis a weaklyQ-compatible system of Weil–Deligne representations.}

We can now complete the proof of Theorem5.1

Proof of Theorem5.1. We may assume thatX is globally defined, i.e. we haveC,c,U,X _{as above such}

thatF=kd(C)_c andX_F∼_{=X. We will apply Proposition5.8, taking}F_{ℓto be theith higher direct image} of the constant sheaf under the map f :X_|U→U. Hence forℓ6=pwe takeF_ℓ=Ri_f

∗Qℓ, and forℓ=p we takeF_{pto be the overconvergentF-isocrystalR}i_f

∗O_X†_/K onU/Kconstructed by Matsuda and Trihan in [MT04]. By smooth and proper base change in ´etale cohomology we have, for any closed pointx∈U

F_ℓ,x=Hi

´et(Xx,¯ Qℓ)

and by the same result for Ogus’ convergent cohomology [Ogu84, Proposition 3.5] we have

F_p,x=Hi

rig(Xx/K(x))

whereK(x)is the fraction field of the Witt vectors ofk(x). Hence byCℓ,w(Xx,Hi)we know that{Fℓ}ℓis a

Q-compatible system onU, and by Proposition5.8it follows that{F_{ℓ,c}ℓis a weaklyQ-compatible system}

of Weil–Deligne representations. But again applying smooth and proper base change in ´etale cohomology tells us thatF_ℓ,c∼_=Hi

ℓ(X)forℓ6=p, and the proof of [LP16, Proposition 5.52] tells us thatFp,c=Hpi(X).

6. COHOMOLOGY OVER EQUICHARACTERISTIC LOCAL FIELDSII

In this section, we generalise Theorem5.1to apply to all smooth and proper varieties over an equichar-acteristic local field, not just those with semistable reduction.

Theorem 6.1. Suppose that F∼=k((t))is equicharacteristic local, and let X/F be a smooth, proper variety.

Then Cℓ,w(X,Hi)holds.

The proof will be via a rather delicate weight argument for Weil–Deligne representations. We will therefore letQ′_ℓdenote the fieldQ_ℓifℓ6=p, andKunifℓ=p, and throughout considerQ′_ℓ-valued Weil– Deligne representations (i.e.Q′_ℓ-valued representations ofW_F′). If

(V,N,ρ:WF→GL(V))

is such a representation then we have a canonical filtration

Mk:=

∑

kerNi+1∩imNj

onV, called the monodromy filtration. This is functorial inV in the sense that if f :V →W is a morphism of Weil–Deligne representations, then f(MkV)⊂MkW for all k. Moreover eachMk is preserved by the

WF-action, and hence there is an induced action ofWFon GrMkV.

Definition 6.2. We say thatV is quasi-pure of weightiif the eigenvalues of any liftϕ∈WF of geometric

Frobenius areq-Weil numbers of weighti. By convention, the zero object is pure ofallweights.

Since the inertiaIFacts through a finite quotient, this does not depend on the choice of Frobenius liftϕ.

We will need the following results concerning purity and Weil–Deligne representations.

Proposition 6.3. (1) The subcategory RepQ′_ℓ(WF′)(i) ⊂RepQ′_ℓ(WF′) of Weil–Deligne representations

which are pure of some fixed weight i is abelian, i.e. stable under taking kernels and cokernels. (2) Any morphism f :V →W between Weil–Deligne representations which are pure of weights i> j

respectively is zero.

(3) Let X/F be smooth and proper. Then theQ′_ℓ-valued Weil–Deligne representation H_ℓi(X)attached to theℓ-adic cohomology of X is quasi-pure of weight i.

Proof. Ifℓ=pthen (1) is [LP16, Lemma 5.61], and (2) is [LP16, Lemma 5.59]; exactly the same proof

works forℓ6=p. Ifℓ=6 pthen (3) follows from [Ito05, Theorem 1.1], and ifℓ=pfrom [LP16, Theorem

5.33].

We will also need the following result.

Lemma 6.4. Let F′/F be a finite extension (not necessarily Galois or even separable), and let X′/F′be smooth and proper. Let X denote X′considered as a variety over F, and fix a primeℓ.

(1) If theℓ-adic cohomology H_ℓi(X′)is quasi-pure of weight i as a W_F′′-representation, then H_ℓi(X)is

quasi-pure of weight i as a W_F′-representation.

(2) If C_ℓ,w(Hi)holds for X′/F′, then Cℓ,w(Hi)holds for X/F.

Proof. It suffices to treat the cases whereF′/F is separable, and whereF′=F1/p. IfF′/F is separable,

then we may viewW_F′′⊂WF′ as a finite index subgroup. Then we have thatHℓi(X) =Ind

W_F′ W′

F′

H_ℓi(X′)is simply

Next suppose thatF′=F1/p, or in other words that we are looking at the finite extension Frob :F→F. LetX1 be a given smooth and proper variety overF, andX0 the F-variety by considering X1 with the

composite structure morphismX1→Spec(F)

Spec(Frob)

→ Spec(F). Since the mapX1֒→X0⊗F,FrobF is a

nilpotent thickening, we haveHi

ℓ(X1)∼=H_ℓi(X0⊗F,FrobF)asWF′-representations, and since the Frobenius

mapX0→X0⊗F,FrobF is a universal homeomorphism, it follows thatHℓi(X0)∼=Hℓi(X0⊗F,FrobF)asWF′

-representations. Hence we can deduce thatH_ℓi(X1)∼=H_ℓi(X0)asWF′-representations, which allows us to

conclude.

With these preliminaries in place, we can now prove Theorem6.1.

Proof of Theorem6.1. let X/F be smooth and proper. Then by de Jong’s theorem on alterations [dJ96,

Theorem 8.2], we may choose a proper hypercoverX•→X and finite extensionsFn/F such that:

• the structure morphismXn→Spec(F)factors through Spec(Fn); • Xn/Fnhas strictly semistable reduction.

In particular, by applying Theorem5.1, Proposition6.3(3) and Lemma6.4we can see that the cohomology groupsH_ℓi(Xn), whereXn is viewed as anF-variety, are quasi-pure of weighti, and moreover that they

satisfy weakℓ-independence. We now consider the spectral sequence

E₁n,i=H_ℓi(Xn)⇒H_ℓi+n(X).

SinceE₁n,iis quasi-pure of weighti, it follows from Proposition6.3(1) thatE₂i,nis also quasi-pure of weight n, and hence from Proposition6.3(2) that the spectral sequence degenerates atE2.

We will now consider the induced filtrationP•onH_ℓi+n(X), whose associated graded GrPiHℓi+n(X) =E

n,i 2

is therefore quasi-pure of weighti. If we look at the quotient map

H_ℓi+n(X)→Gr_iP_+n=E₂0,i+n

then it follows from Proposition6.3(1) that the kernelPi+n−1H_ℓi+n(X)is quasi-pure of weighti+n, and

hence that the quotient map

Pi+n−1Hℓi+n(X)։GrPi+n−1=E 1,i+n−1 2

must be zero by Proposition6.3(2). In particular, we havePi+n−1Hℓi+n(X) =Pi+n−2Hℓi+n(X), and again applying Proposition6.3(2) we can see that the quotient

Pi+n−2H_ℓi+n(X)։GrPi+n−2=E 2,i+n−2 2

must be zero. Continuing thus, we find thatPi+n−jHℓi+n(X) =P0Hℓi+n(X) =0 for all j>0, from which we deduce that theE2-page of the spectral sequence only has non-zero terms in the first column. Put differently,

we have a long exact sequence

0→H_ℓi+n(X)→H_ℓi+n(X0)→H_ℓi+n(X1)→H_ℓi+n(X2)→. . .

for alli,n, ℓ. Since we know weakℓ-independence for all the termsH_ℓi+n(Xn), we can therefore deduce

7. L-FUNCTIONS AND THE PROOF OFPROPOSITION5.8

The purpose of this section is to prove Proposition5.8, as well as give an application of Theorem6.1

showing that the Hasse–WeilL-functions of smooth and proper varieties over global function fields (includ-ing those defined us(includ-ingp-adic cohomology) are independent ofℓ. The basic idea is that away fromℓ=p Proposition5.8follows from [Del73, Th´eor`eme 9.8], and in fact Deligne’s proof also works atℓ=ponce certain facts aboutp-adic cohomology of smooth curves are in place. The deduction ofℓ-independence for the Hasse–WeilL-function is then entirely straightforward. In fact, it is not entirely accurate to describe this as an application of Theorem6.1, since it is in fact a version ofℓ-independence for localL-functions that forms a key component of theproofof Proposition5.8, and therefore of Theorem6.1. However, we thought it worth isolating this special case. All the results presented in this section are more or less special cases of more general results of Abe–Caro proved in [AC14], however, in the simple case that we are interested in one can give proofs using simpler machinery from the theory of arithmeticD_{-modules on curves and} formal curves, in particular that from [Car06] and [Cre12].

Since the situation we are interested in in this section is somewhat different to that in the rest of the article, we will change notation slightly (only for this section, we will return to the usual notations in the next section). So we will letFbe a global function field with field of constantsk, and letX be a smooth, projective model forF. We will letU be an open subset ofX, and letS=X\U. For any placev∈ |X|

(with some fixed local parametertv) we will letkv denote the residue field atv,Kv the unique unramified

extension ofKwith residue fieldkv,Wvits ring of integers andRva copy of the Robba ring atv, considered

as a subring ofKvJtv,tv−1K. Letϕv denote a #kv-power Frobenius onRv. We will also denote #kv-power

Frobenius operators byFv.

If we are given an overconvergentF-isocrystalF _∈F-Isoc†_{(U/K)and a pointv∈X we can consider}

Tsuzuki’s functor

i∗_v:F-Isoc†(U/K)→MΦ∇R_v

to the category of(ϕv,∇)-modules overR_{v, and therefore define the localL-factor of}F _atvby Lv(F,t):=det(1−tdegvϕv|(i∗vF)∇=0).

The globalL-function ofF _{is then defined to be}

b

L(F_,t):=

_∏

v∈|X|

Lv(F,t),

the notation bL used here is in order to distinguish it from the Etesse–Le Stum L-function LEL defined

in [ELS93], which is defined by only considering local´ L-factors at points ofU. Note that at placesv∈U the localL-factor can also be described (by Dwork’s trick) as the inverse characteristic polynomial det(1−

tdegvFdegv|F_{v)of the linearised Frobenius of}F _{acting on the stalk}F_vofF _atv. Our first task will be

to prove a function equation forbL(F_{,t), and to do so we will need to interpret the local factorsLv(}F_,t)in terms of arithmeticD_{-modules on}X.

So we will denote byX _{a lift ofXto a smooth projective formal curve overW, by}D†

X_,QBerthelot’s ring

of arithmetic differential operators onX_{, and byF-Modhol(}D†

X_,Q)(resp.F-Dbhol(D †

X_,Q)) the category of

holonomicF-D†

X_,Q-modules (resp. the bounded derived category ofD †

X_,Q-modules equipped with

Frobe-nius structure, whose cohomology sheaves are holonomic). For any complexK _∈F-Db hol(D

†

X,Q)we will

can consider the pullbacki+_vK _∈F-Db hol(D

†

Spf(Wv),Q). Its cohomology sheaves are thereforeF-isocrystals

overKv, we may therefore considerFdegvas aK-linear operator on these cohomology sheaves. Caro then

defines

Lv(K,t):=

∏

detK(1−tdegvFdegv|Hi(i+vK))

(−1)r

degv

L(K_,t):=

_∏

v∈|X|

Lv(K,t).

The functional equation that we require will follow from the existence of the ‘intermediate extension’ functor

j!+:F-Isoc(U/K)→F-Modhol(DX† _,Q)

which will be such that:

(1) bL(F_{,t) =L(j!+}F_,t);

(2) D(j!+F)∼=j!+(F∨), whereDis the dual functor forD-modules, and(−)∨that for isocrystals.

As noted at the beginning of this section, this is a special case of much more general results of [AC14], however, we thought it worthwhile to go through the construction in detail in the simple case that we are interested in. To construct j!+, first let us consider Berthelot’s ring DX† _,Q(†S) of arithmetic

dif-ferential operators on X _{with overconvergent singularities along} S, and the corresponding categories F-Modhol(DX† _,Q(†S))andF-Dbhol(D

†

X_,Q(†S))of (complexes of) holonomicF-D †

X_,Q(†S)-modules. Then

thanks to [Car06, Th´eor`eme 2.3.3] we may consider anyF_∈F-Isoc†_{(U/K)as a holonomicF-D}† X,Q(†S)

-module, which we shall do. We have canonical functors

j+:F-Dbhol(DX† _,Q(

†_S))→F

-Db_hol(D† X_,Q)

j+:F-Db_hol(D†

X,Q)→F-D

b hol(D

† X,Q(

†_S))

given by restriction and extension of scalars alongD†

X,Q→D

†

X,Q(†S)respectively, these are both exact

for the naturalt-structures on both sides. Define j!to beDj+DwhereDis the duality functor, note that this is exact for holonomicF-D_{-modules by (the proof of) [Vir00, Proposition III.4.4]. By [Vir00, Proposition} I.4.4] we have j+D=Dj+and hence we get adjoint pairs(j+,j+)and(j!,j+). Also by [Car06, 1.1.8] we havej+j!∼=j+j+∼=id.

The closed immersioni:S→Xlifts to a closed immersioni:S_→X _{and hence we obtain Berthelot’s} pullback and pushforward functors

i!:F-Db_hol(D†

X_,Q)→F-D

b

hol(DS†_,Q)

i+:F-Dbhol(D †

S_,Q)→F-Dbhol(D † X_,Q),

we definei+=Di!D. It follows from [Ber02, Th´eor`eme 4.3.13] thati+D=Di+ and hence by [Car06, Th´eor`eme 1.2.21] we have two adjoint pairs(i+,i+)and(i+,i!). Also by [Ber02, Th´eor`eme 5.3.3] we get i!i+∼=i+i+=∼id. By [Car06, (1.1.6.5)] we have, for anyE ∈F-Dbhol(D

†

X_,Q), exact triangles

j!j+E →E →i+i+E +→1

i+i!E →E → j+j+E +→1.

In particular takingE_=j+F _{we get an exact trianglej!}F_→j+F_→i+i+_j