Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic D -modules
aa r X i v : . [ m a t h . AG ] M a y Explicit calculation of Frobenius isomorphisms and Poincar´eduality in the theory of arithmetic D -modules Tomoyuki Abe
Abstract
The aim of this paper is to compute the Frobenius structures of some cohomologicaloperators of arithmetic D -modules. To do this, we calculate explicitly an isomorphismbetween canonical sheaves defined abstractly. Using this calculation, we establish the relativePoincar´e duality in the style of SGA4. As another application, we compare the push-forwardas arithmetic D -modules and the rigid cohomologies taking Frobenius into account. Thesetheorems will lead us to an analog of “Weil II” and a product formula for p -adic epsilonfactors. Introduction
In this paper, we prove several results concerning Frobenius structures in the theory of arithmetic D -modules. There are mainly three goals in this paper.(G1) Compute and describe Frobenius structures of some cohomological operators, appearingin [Be2], concretely in terms of differential operators.(G2) Establish a relative Poincar´e duality in the style of SGA4 in the theory of arithmetic D -modules.(G3) Compare the push-forwards in the theory of D -modules and the rigid cohomologies withFrobenius structure .First, (G1) is the starting point of other two goals. We describe some isomorphisms appearingin [Be2] explicitly by taking local coordinates. Apart from (G2) and (G3), this calculation isused in [AM] to compute the geometric Fourier transform defined by C. Noot-Huyghe explicitly.With this description, we are able to re-prove Gross-Koblitz formula [GK] using arithmetic D -modules. This calculation will be discussed in other places. We expect that these ideas can begeneralized to a calculation of p -adic ε -factors.For (G2), a duality theory was established by A. Virrion in [Vi2] to some extent. However,we need two more ingredients to call it the Poincar´e duality: 1) comparison of the extraordinarypull-back and the normal pull-back for a smooth morphism, and 2) taking Frobenius structuresinto account. Using our result 1) on the comparison of two types of pull-backs, we are also able tocompare duality functors of the theory of rigid cohomology and that of arithmetic D -modules,which completes a work in [Ca5]. For 2), even without Frobenius structures, her duality isvery powerful tool, but in practical uses of arithmetic D -module theory, Frobenius structureis another important ingredient that contain arithmetic information. For example, L -functionsfor holonomic D † -modules cannot be defined without Frobenius structures, and thus to showthe functional equation for L -functions, it is necessary to consider Frobenius structures in theduality.(G3) is another application of (G1). If we do not consider Frobenius structures, this is awell-known result of Berthelot [BeI, 4.3.6.3]. This type of comparison theorem is necessary when1e want to exploit results of the theory of D -modules in the theory of rigid cohomologies and vice versa . For example, in [Ca2, 3.3], the author discussed the relations of L -functions definedusing the theory of rigid cohomologies and that of arithmetic D -modules. This result can bereinforced and re-stated much clearer using our result (cf. Remark 3.12).Now, let us go into more details of the results. Let R be a complete discrete valuation ring ofmixed characteristic (0 , p ), and we denote by k its residue field which is assumed to be perfect, K its field of fractions. Let X be a smooth formal scheme over Spf( R ), and X be the reductionof X over k . Let s be a positive integer, and we put X ′ := X ⊗ k,F sk k where F sk : Spec( k ) → Spec( k ) denotes the s -th absolute Frobenius isomorphism. In this introduction, we also assumethat there exist liftings σ : Spf( R ) ∼ −→ Spf( R ) of F sk and F : X → X ′ := X ⊗ R,σ R of therelative Frobenius morphism F sX /k : X → X ′ for simplicity. A coherent F - D † X , Q -module is acouple of a coherent D † X , Q -module M and an isomorphism F ∗ M σ ∼ −→ M where M σ denotesthe D † X ′ , Q -module induced by M by the base change σ . Frobenius structures are known tobe stable under reasonable cohomological operations of arithmetic D -modules such as push-forwards, extraordinary pull-backs, tensor products, etc (cf. [Be2]).When we try to calculate Frobenius structures of some cohomological operations ( e.g. push-forward functor), an obstacle lies in the isomorphism ω X ∼ −→ F ♭ ω X ′ of [Be2, 2.4.2]. The con-struction of this isomorphism is formal using general facts of [Ha]. However, we need to tracemany isomorphisms of [Ha] to compute it explicitly, which is monotonous but messy. The ad-vantage of this computation is that it makes us possible to calculate Frobenius structures in“brutal” but very direct ways, at least locally. As an example of the explicit computation, wecalculate the Frobenius structure of push-forwards (cf. paragraph 2.5). We can also prove aproper base change type lemma (cf. Lemma 2.6). With an aid of a result of Caro, we get theproper base change theorem in paragraph 5.7. Another application will be to prove the followingtheorem. Theorem 3.10. —
Let f : X → Y be a smooth morphism of relative dimension d betweensmooth formal schemes. For a coherent F - D † Y , Q -module M , (b) f ! ( D Y ( M )) ∼ = D X ( f ! M )( d )[2 d ] where ( d ) denotes the d -th Tate twist (cf. paragraph ). The construction of the isomorphism without Frobenius structures requires only standardmethods of the theory of arithmetic D -modules, but to see the compatibility with Frobeniusstructures, we need the explicit calculation of the isomorphism of canonical sheaves. Using aresult we get on the way we prove this theorem, we compare the rigid cohomologies and thepush-forwards in the arithmetic D -module theory, which is (G3). This theorem can be seen asa part of Poincar´e duality. See the last section for an account of this interpretation. Moreoverthis theorem leads us to complete a work of Caro in [Ca3] (cf. Corollary 3.12) comparing theduality functors in the theory of rigid cohomology and that of arithmetic D -modules.In this paper, we also include some small but useful results concerning Frobenius pull-backs. Namely, we prove: 1. commutation of the dual functor and the tensor product in somecases, 2. the K¨unneth formula, 3. compatibility of the relative duality homomorphism withFrobenius. The result 1 uses (b) in the proof, but results 2 and 3 are independent of the explicitcomputations. The results 1 and 2 are included in this paper with the intention of use in [AM].The result 3 is aimed to establish the Poincar´e duality as we have already mentioned.Finally, let us point out some notable applications of our result. Currently, we have thefollowing two important applications: 2 establishing the “yoga of weight” in p -adic cohomologies, especially an analog of “WeilII” in the theory of arithmetic D -modules. This will be treated in a paper of the authorjointly with D. Caro (in preparation). • a product formula for p -adic epsilon factors. See [AM] for more details.In the proofs of those two results, (G2) and (G3) are used extensively.Let us see the structure of this paper. In §
1, we describe isomorphisms which are keyisomorphisms to construct the commutativity. In §
2, we calculate the Frobenius structure ofpush-forwards explicitly. We should mention that this calculation is a key to calculate theFrobenius structure of Fourier transforms explicitly, which is carried out in [AM]. With thesetwo sections, (G1) is attained. As an application, we show a proper base change type lemmaalso in this section. In §
3, we show that the dual functor and the extraordinary pull-back functorcommute up to some degree shift and Tate twist in the smooth case. Using a lemma we proveto show this commutativity, we will compare the rigid cohomology and the push-forward ofarithmetic D -modules, and we get (G3). In §
4, we will show some complementary results, whichare used in [AM]. The idea of the proof of the K¨unneth formula is due to P. Berthelot. Inthis section, we also prove that the relative duality isomorphism of Virrion is compatible withFrobenius. Together with §
3, (G2) is completed. In §
5, we interpret the results in terms of thephilosophy of “six functors” by Grothendieck, which clarifies the meaning of the results in thispaper.
Acknowledgments
The author would like to thank Professor P. Berthelot for letting him know the proof of theK¨unneth formula. Most of the work of this paper was done when the author was visiting toIRMA of
Universit´e de Strasbourg in 2010. He would like to thank A. Marmora and the institutefor the hospitality. He also like to express his gratitude to Professor A. Shiho for stimulatingdiscussions. This work was supported by Grant-in-Aid for JSPS Fellows 20-1070, and partiallyby l’agence nationale de la recherche
ANR-09-JCJC-0048-01.
Notation0.1.
In this paper we fix a complete discrete valuation ring R with mixed characteristic (0 , p ).We denote the residue field by k , the field of fractions by K . For a non-negative integer i , weput R i to be R/π i +1 R where π is a uniformizer. We denote by e the absolute ramification indexof K .In general, we use Roman fonts ( e.g. X ) for schemes and script fonts ( e.g. X ) for formalschemes. For a formal scheme X over Spf( R ), we usually denote by X i the reduction X ⊗ R R i over Spec( R i ). For a scheme X over Spec( k ), we denote by F X : X → X the absolute Frobenius homo-morphism: it sends a section f of O X to f p . We fix a positive integer s , and put q := p s . Weput X ( s ) := X ⊗ k,F s ∗ k k , and call it the relative s -th Frobenius of X . Let D be a sheaf of rings on a topological space X . When we simply say D -module, itmeans left D -module. We denote by D ∗ coh ( D ) ( ∗ ∈ { + , − , b } ) the full subcategory of D ∗ ( D ) suchthat the objects consist of complexes whose cohomology sheaves are coherent. We denote by D perf ( D ) the full subcategory whose objects consist of perfect complexes ( i.e. complexes locallyquasi-isomorphic to bounded complexes of locally projective D -modules). We denote by D ftd ( D )the full subcategory consisting of finite Tor-dimensional complexes ( i.e. complexes possessing3ounded flat resolutions). We put D b perf ( D ) := D perf ( D ) ∩ D ftd ( D ). When X is quasi-compactand D is coherent, D b perf ( D ) coincides with D perf ( D ) ∩ D b coh ( D ). See SGA6 Exp. I for details.When we denote by D ( D ) g (resp. D ( D ) d ) we consider complexes of left (resp. right) D -modules(g and d stand for French words “gauche” and “droit”). When we put Q as an index, this meanstensor with Q . In this paper, we freely use the language of arithmetic D -modules. For details see [Be1],[Be2], [BeI]. In particular, we use the rings D ( m ) X , b D ( m ) X , D † X for a smooth scheme X and asmooth formal scheme X . We use the category LD −→ b Q , qc ( b D ( • ) X ) whose definition is written in[BeI, 4.2]. Let Z be a divisor of the special fiber of X . Then by the same construction, we canconsider the category LD −→ b Q , qc ( b D ( • ) X ( Z )). For this category, see also [Ca2, 1.1.3]. Let X be a smooth formal scheme, and Z be a divisor of its special fiber. Let U := X \ Z , X and U be the special fibers of X and U respectively. Let M be a coherent ( F -) D † X , Q ( † Z )-module such that it is coherent as an O X , Q ( † Z )-module. Let C be the full subcategory of thecategory of coherent ( F -) D † X , Q ( † Z )-modules consisting of such M . Then we know that thespecialization functor induces an equivalence between C and the category ( F -)Isoc † ( U, X/K )by [Be1, 4.4.12] and [Be2, 4.6.3, 4.6.7]. We say that M is a convergent ( F -)isocrystal on U overconvergent along Z by abuse of language.
1. Explicit calculation of isomorphisms of canonical sheaves
In this section, we will explicitly calculate the isomorphisms of [Be2, 2.4.3, 2.4.4] (cf. Theorem1.7), from which some commutation results of Berthelot [Be2] are derived. The existence of theseisomorphisms are direct consequences of fundamental properties of the functors of Hartshorne[Ha], and for the explicit calculations, we need to go back to the proofs of these fundamentalproperties, and trace these isomorphisms step by step. We follow the notation of [Ha, III].
First we review the notations and functors of Hartshorne [Ha, III] in short. Let f : X → Y be a morphism of schemes. When f is smooth, we denote by ω X/Y the canonical sheaf V d Ω X/Y where d denotes the relative dimension of f . When f is regular closed immersion, let J be thesheaf of ideals of O Y defining X . Then we put ω X/Y := ( V d J / J ) ∨ where d is the codimensionof X in Y , and ∨ is the dual as an O X -module. In both cases, ω X/Y is a locally free O X -moduleof rank 1.Suppose f is smooth. We define a functor f ♯ : D ( O Y ) → D ( O X ) as follows. See [Ha, III, § C ∈ D ( O Y ), we put f ♯ ( C ) := f ∗ ( C ) ⊗ O X ω X/Y [ d ] where d is the relativedimension of f . We see that this functor takes D b qc ( O Y ) into D b qc ( O X ) (here D ∗ qc denotes the fullsubcategory of the derived category consisting of objects whose cohomologies are quasi-coherentsheaves).In turn, suppose f is a finite morphism. We denote by f the morphism of ringed spaces( X, O X ) → ( Y, f ∗ O X ). Then we define a functor f ♭ : D + ( O Y ) → D + ( O X ) as follows. See [Ha,III, §
6] for more details. For C ∈ D + ( O Y ), we put f ♭ ( C ) := f ∗ R H om O Y ( f ∗ O X , C ). We knowthat this functor takes D +qc ( O Y ) into D +qc ( O X ), and if f has finite Tor-dimension (cf. [Ha, II § e.g. flat morphism), then it takes bounded complexes into bounded complexes.4 .2. Now, consider the following diagram of schemes Y g / / ZX f ` ` AAAAAAA h > > ~~~~~~~ where f and h are regular closed immersions of codimension d >
0, and g is a finite flat morphism.Since g is finite flat, we get that g ♭ ( O Z ) ∼ = g ∗ H om O Z ( g ∗ O Y , O Z ) . There exists the natural equivalence f ♭ g ♭ ∼ = h ♭ by [Ha, III Proposition 6.2]. By taking the d -thcohomology, we get an isomorphism ι : f ∗ E xt d O Y ( f ∗ O X , g ∗ H om O Z ( g ∗ O Y , O Z )) ∼ −→ h ∗ E xt d O Z ( h ∗ O X , O Z ) . Now, suppose Z is an affine scheme. Then the other two schemes are also affine schemes. Wedenote the global sections of X (resp. Y , Z ) by R X (resp. R Y , R Z ). By [Ha, III, Proposition6.1], the source and target of ι are quasi-coherent O X -modules. Thus, ι is associated to thefollowing isomorphism of R X -modulesExt dR Y ( R X , Hom R Z ( R Y , R Z )) ∼ −→ Ext dR Z ( R X , R Z ) , and we also denote this isomorphism by ι . We calculate this isomorphism in terms of thefundamental local isomorphism [Ha, III, Proposition 7.2].Suppose moreover that there exists a system of local parameters defining X in Y (resp. in Z )denoted by { y i } i ≤ i ≤ d (resp. { z i } ≤ i ≤ d ). Let I := Ker( R Y → R X ). The sheaf ω X/Y is the quasi-coherent sheaf associated to Hom R X ( V d I/I , R X ). Since y ∧ · · · ∧ y d defines a basis of V d I/I ,we denote by ( y ∧ · · · ∧ y d ) ∨ its dual basis. In the same way, we define a basis ( z ∧ · · · ∧ z d ) ∨ of ω X/Z . We preserve the notation. We define a homomorphism α in the followingdiagram so that it is commutative. Ext dR Y ( R X , Hom R Z ( R Y , R Z )) ∼ / / ι ∼ (cid:15) (cid:15) ω X/Y ⊗ R Y Hom R Z ( R Y , R Z ) α (cid:15) (cid:15) Ext dR Z ( R X , R Z ) ∼ β / / ω X/Z
Here the horizontal isomorphisms are the isomorphisms of [Ha, III, 7.2] . Let g ∗ ( z i ) = P ≤ j ≤ d f ij y j where f ij ∈ R Y . Here the expression may not be unique, but take one. We put G := ( f ij ) i ≤ i,j ≤ d ∈ Mat d × d ( R Y ) . Then α (( y ∧ · · · ∧ y d ) ∨ ⊗ ϕ ) = ϕ (det( G )) · ( z ∧ · · · ∧ z d ) ∨ where the over-line denotes to take the image of the homomorphism R Z → R X inducing h .Proof. On the way we prove the lemma, we will review the definition of the homomorphism β .Let R Z ζ i be a free R Z -module of rank 1 whose generator is ζ i . Let K • := V • ( L di =1 R Z ζ i ) be theKoszul complex. By definition, the differential homomorphism K r → K r − is defined by sending5 i ∧ · · · ∧ ζ i r to P ( − j z i j ζ i ∧ · · · ∧ c ζ i j ∧ · · · ∧ ζ i r where c ζ i j means omit ζ i j . The canonicalhomomorphism R Z → R X defines a complex ^ • (cid:16) d M i =1 R Z ζ i (cid:17) → R X → , which is known to be a free resolution of R X . Now, let us define a homomorphism ^ r (cid:16) d M i =1 R Z ζ i (cid:17) → ^ r (cid:16) d M i =1 R Y ν i (cid:17) by mapping ζ i ∧ · · · ∧ ζ i r to (cid:16) X ≤ j ≤ d f i ,j ν j (cid:17) ∧ · · · ∧ (cid:16) X ≤ j ≤ d f i r ,j ν j (cid:17) = X ( j ,...,j r ) ∈ [1 ,d ] r f i ,j . . . f i r ,j r · ν j ∧ · · · ∧ ν j r where [1 , d ] is the set { i ∈ Z | ≤ i ≤ d } . Then it is a standard calculation to check that thesehomomorphisms define a homomorphism of Koszul complexes: γ : ^ • (cid:16) d M i =1 R Z ζ i (cid:17) → ^ • (cid:16) d M i =1 R Y ν i (cid:17) . This induces the following commutative diagram.Hom R Y ( V d ( L di =1 R Y ν i ) , R Y ) ⊗ R Y Hom R Z ( R Y , R Z ) (cid:15) (cid:15) / / ω X/Y ⊗ R Y Hom R Z ( R Y , R Z ) α (cid:15) (cid:15) Hom R Z ( V d ( L di =1 R Z ζ i ) , R Z ) / / ω X/Z
Here, the left vertical arrow is induced by γ . The bottom horizontal arrow is the surjectivehomomorphism defined by sending φ to φ ( ζ ∧ · · · ∧ ζ d ) · ( z ∧ · · · ∧ z d ) ∨ . This factors throughExt dR Z ( R X , R Z ), and this is β by definition. The top horizontal arrow is defined in the samemanner. The homomorphism γ sends ζ ∧ · · · ∧ ζ d to det( G ) · ( ν ∧ · · · ∧ ν d ). Thus, we get thelemma. (cid:4) Now we will calculate the isomorphism µ X in [Be2, Lemme 2.4.2], which is one of thetwo ingredients to calculate the Frobenius isomorphisms explicitly. The other ingredient is theexplicit calculation of Frobenius by Garnier, which we will review in paragraph 2.2.Let us fix the situation and notations. We fix a positive integer s >
0, and put q := p s as inNotation. Let S be a scheme endowed with quasi-coherent m -PD ideal ( a , b , α ) such that p ∈ a and p is nilpotent, and X be a smooth scheme over S of relative dimension d . Let S be thesubscheme of S defined by a , and X := X × S S . Suppose S and X are affine schemes, and X possesses a system of local coordinates { x i } ≤ i ≤ d over S ( i.e. the homomorphism X → A dS induced by { x i } is ´etale). Recall X ( s )0 := X ⊗ S ,F sS S is the s -th relative Frobenius of X over S . Let X ′ be a smooth lifting of X ( s )0 over S . There exists a system of local coordinates { y i } ≤ i ≤ d of X ′ . Since S and X are affine, we may lift the relative Frobenius homomorphismdenoted by F : X → X ′ over S uniquely such that F ∗ ( y i ) = x qi .We will use multi-index notation. For an integer i , we put i := ( i, . . . , i ) in Z d . For k =( k , . . . , k d ) and k ′ = ( k ′ , . . . , k ′ d ) in Z d , we denote by k < k ′ (resp. k ≤ k ′ ) if k i < k ′ i (resp. k i ≤ k ′ i ) for any 1 ≤ i ≤ d . We define k − k ′ := ( k − k ′ , . . . , k d − k ′ d ).6e know that F ∗ O X ∼ = M ≤ k Before starting the proof, we remind that Conrad pointed out in [Co] that with signconvention of [Ha], many compatibilities stated in [Ha] do not hold, and we need to use themodified convention as in [Co, 2.2]. In this proof, since the outcome does not change, we followthe conventions of [Ha]. For skeptical readers, we put signs (cid:13) through (cid:13) below arrows of thehomomorphisms whose sign change if we use the conventions of [Co], and see how they differ atthe very end of this proof.To avoid confusions, we put Y := X ′ in this proof. Consider the following diagram. X × S X × F / / p $ $ IIIIIIIII X × S Y q Y $ $ HHHHHHHHH q X z z vvvvvvvvv X ∆ d d IIIIIIIII F / / f $ $ IIIIIIIIII s : : vvvvvvvvv Y g z z uuuuuuuuuu S Here p is the second projection, ∆ is the diagonal morphism, q X is the first projection, s isthe graph morphism of F , q Y is the second projection, and f and g are structural morphisms.For an affine scheme Z , we denote the global sections of Z by R Z . Note that all the schemesappearing in the diagram are affine. Let us consider X × S X (resp. X × S Y ) as a scheme over X by the projection p (resp. q X ) unless otherwise stated. We put x ,i := x i ⊗ R X × S X and R X × S Y x ∆ ,i := x i ⊗ − ⊗ x i in R X × S X x s,i := F ∗ ( y i ) ⊗ − ⊗ y i in R X × S Y . The set { x ∆ , , . . . , x ∆ ,d , x , , . . . , x ,d } (resp. { x s, , . . . , x s,d , x , , . . . , x ,d } ) forms a system oflocal coordinates of X × S X (resp. X × S Y ), and { x ∆ , , . . . , x ∆ ,d } (resp. { x s, , . . . , x s,d } ) de-fines a local system of parameters defining ∆( X ) (resp. s ( X )). We also note that dx ∆ , • :=7 x ∆ , ∧ · · · ∧ dx ∆ ,d defines a basis of ω X × S X/X . For ϕ ∈ Hom R Y ( R X , R Y ), we denote by ϕ ′ ∈ Hom R X × SY ( R X × S X , R X × S Y ) the homomorphism defined by 1 ⊗ ϕ .Now, let d ′ : R X → R X × S X ∼ = R X ⊗ R S R X be a homomorphism of R S -algebras defined by d ′ ( b ) := b ⊗ − ⊗ b . We consider R X × S X as an R X -algebra by the first component for a while.Then for a, b ∈ R X , we get d ′ ( ab ) = a d ′ ( b ) + b d ′ ( a ) − d ′ ( a ) · d ′ ( b ) . Thus, (1 × F ) ∗ ( x s,i ) = d ′ ( F ∗ ( y i )) = d ′ ( x qi ) = (cid:16) − d ′ x q − i + X q − >j ≥ f i,j d ′ x ji (cid:17) · d ′ x i =: F i · x ∆ ,i with f i,j ∈ R X . By definition, we have ( ( Hx − ( q − i ) ′ ( d ′ x kj ) = 0 for any i = j or i = j and k = q − Hx − ( q − i ) ′ ( d ′ x q − i ) = − ⊗ . Let(1.5.1) G := diag( F , . . . , F d ) ∈ Mat d × d ( R X × S X )where diag denotes the diagonal matrix. Then, we obtain(1.5.2) ( Hx − ( q − ) ′ (det( G )) = ( Hx − ( q − ) ′ (( − d d ′ x q − . . . d ′ x q − d ) = ( − ⊗ d = 1 ⊗ . We set back the convention, and consider X × S X as a scheme over X by p . The homo-morphism µ X is defined in the following way: ω X ∼ = f ♯ O S [ − d ] ∼ = F ♭ g ♯ O S [ − d ] ∼ = F ♭ ω Y/S where the first and third isomorphisms are by definition [Ha, III, § 2] and the second isomorphismis induced by [Ha, III, Proposition 8.4]. In the rest of this proof, we will drop the section numberIII when we cite [Ha]. Since the sheaves we are considering are quasi-coherent and schemes areaffine, we do not make any difference between sheaves and its global sections.We will start to calculate from F ♭ ω Y/S . In the rest of this proof, we will use the identification F ♭ M ∼ = O ∨ X ⊗ O X ′ M to describe the elements contrary to the standard convention (1.4.1) of thispaper. Thus the sheaf F ♭ ω Y/S is identified with Hom R Y ( R X , R Y ) ⊗ R Y ω Y/S . Take an element ϕ ⊗ dy • ∈ Hom R Y ( R X , R Y ) ⊗ R Y ω Y/S . First, we need to calculate the isomorphism a : F ♭ ω Y/S ∼ −→ s ♭ q ♯Y ω Y/S , which is the third isomor-phism in the proof of [Ha, 8.4]. This isomorphism is the isomorphism of [Ha, 8.2]. To calculatethis, first, we get an isomorphismHom R Y ( R X , R Y ) ⊗ R Y ω Y/S ∼ = F ♭ ω Y/S ∼ −−→ (cid:13) ∆ ♭ p ♯ F ♭ ω Y/S ∼ = ω X/X × S X ⊗ R X × X ω X × S X/X ⊗ R X Hom R Y ( R X , R Y ) ⊗ R Y ω Y/S . This isomorphism is defined in [Ha, 8.1]. Let x ∨ ∆ , • denotes the dual basis of x ∆ , ∧ · · · ∧ x ∆ ,d in ω X/X × S X , dy • denotes dy ∧ · · · ∧ dy d , and dx , • denotes dx , ∧ · · · ∧ dx ,d . We define x ∨ s, • and8 x s, • in the same way. Then the isomorphism sends ϕ ⊗ dy • to x ∨ ∆ , • ⊗ dx , • ⊗ ϕ ⊗ dy • . Secondly,we get an isomorphism ω X/X × S X ⊗ R X × X ω X × S X/X ⊗ R X Hom R Y ( R X , R Y ) ⊗ R Y ω Y/S ∼ = ∆ ♭ p ♯ F ♭ ω Y/S ∼ −→ ∆ ♭ (1 × F ) ♭ q ♯Y ω Y/S ∼ = ω X/X × S X ⊗ R X × X Hom R X × SY ( R X × S X , R X × S Y ) ⊗ R X × Y ω X × S Y/Y ⊗ R Y ω Y/S . This isomorphism is defined in [Ha, 6.3], and sends x ∨ ∆ , • ⊗ dx , • ⊗ ϕ ⊗ dy • to x ∨ ∆ , • ⊗ ϕ ′ ⊗ dx , • ⊗ dy • .Thirdly, we get an isomorphism ω X/X × S X ⊗ Hom R X × SY ( R X × S X , R X × S Y ) ⊗ ω X × S Y/Y ⊗ ω Y/S ∼ = ∆ ♭ (1 × F ) ♭ q ♯Y ω Y/S ∼ −→ s ♭ q ♯Y ω Y/S ∼ = ω X/X × S Y ⊗ R X × SY ω X × S Y/Y ⊗ R Y ω Y/S . This isomorphism is defined in [Ha, 6.2], and this is the homomorphism we calculated in Lemma1.3. Thus, using this lemma, we get that it sends x ∨ ∆ , • ⊗ ϕ ′ ⊗ dx , • ⊗ dy • to ϕ ′ (det( G )) · x ∨ s, • ⊗ dx , • ⊗ dy • where G is the matrix defined in (1.5.1). Combining these three isomorphisms wegot, we obtain a : Hom R Y ( R X , R Y ) ⊗ R Y ω Y/S ∼ −→ ω X/X × S Y ⊗ R X × SY ω X × S Y/Y ⊗ R Y ω Y/S ϕ ⊗ dy • ϕ ′ (det( G )) · x ∨ s, • ⊗ dx , • ⊗ dy • where the over-line denotes taking the image of the canonical homomorphism R X × S Y → R X inducing the morphism s .Now, we come back to the definition of the isomorphism of [Ha, 8.4]. We need to calculatethe isomorphism b : s ♭ q ♯Y ω Y/S ∼ −→ s ♭ q ♯X f ♯ O S [ − d ], which is the second isomorphism in the proofof [Ha, 8.4]. We have an isomorphism s ♭ q ♯Y ω Y/S ∼ = ω X/X × S Y ⊗ R X × SY ω X × S Y/Y ⊗ R Y ω Y/S ∼ −−→ (cid:13) s ♭ ( f ◦ q X ) ♯ O S [ − d ] ∼ = ω X/X × S Y ⊗ R X × SY ω X × S Y/S . This isomorphism is defined in [Ha, 2.2], and sends x ∨ s, • ⊗ dx , • ⊗ dy • to x ∨ s, • ⊗ (( − d dx s, • ∧ dx , • ).Then we get an isomorphism ω X/X × S Y ⊗ R X × SY ω X × S Y/S ∼ = s ♭ ( f ◦ q X ) ♯ O S [ − d ] ∼ −−→ (cid:13) s ♭ q ♯X f ♯ O S [ − d ] ∼ = ω X/X × S Y ⊗ ω X × S Y/X ⊗ ω X/S . This is also an isomorphism of [Ha, 2.2], and sends x ∨ s, • ⊗ ( dx s, • ∧ dx , • ) to ( − d x ∨ s, • ⊗ dx s, • ⊗ dx • .Since d + d is even, we get b : ω X/X × S Y ⊗ R X × SY ω X × S Y/Y ⊗ R Y ω Y/S ∼ −→ ω X/X × S Y ⊗ ω X × S Y/X ⊗ ω X/S x ∨ s, • ⊗ dx , • ⊗ dy • x ∨ s, • ⊗ dx s, • ⊗ dx • . At last, we get an isomorphism c : ω X/X × S Y ⊗ ω X × S Y/X ⊗ ω X/S ∼ = s ♭ q ♯X ω X/S ∼ −−→ (cid:13) ω X/S . This isomorphism is defined in [Ha, 8.1], and sends x ∨ s, • ⊗ dx s, • ⊗ dx • to dx • .Now, by definition, µ − X = c ◦ b ◦ a . The above calculation shows that µ − X ( ϕ ⊗ dy • ) = ϕ ′ (det( G )) · dx • . ϕ = Hx − ( q − and considering (1.5.2), we get the lemma.As we noted at the beginning of this proof, we need some modification for the calculationof the homomorphisms (cid:13) through (cid:13) if we use the convention of [Co, 2.2]. Precisely for thehomomorphisms (cid:13) and (cid:13) , we need to multiply by ( − d ( d − / , and for the homomorphisms (cid:13) and (cid:13) , we need to multiply by ( − d . Thus, µ − X is multiplied by ( − d ( d − d = 1, andthe result remains to be the same as we stated. (cid:4) When F ∗ ( y i ) is not equal to x qi , we can also calculate in the same way. Wecan write d ′ ( F ∗ ( y j )) = P i f i,j · d ′ x i where f i,j ∈ R X × S X using the notation of the proof of theproposition. We put g i,j := ((1 ⊗ Hx q − )( f i,j )) − where the over-line denotes to take the imageof the canonical homomorphism R X × S Y → R X . Using this, we get ι ( dx ∧ · · · ∧ dx d ) = (det e G ) − ( Hx q − ) dy ∧ · · · ∧ dy d where e G = g , . . . g ,d ... ... g d. . . . g d,d . Note that this matrix is invertible since e G ≡ I mod p · Mat( R X ). We do not know how wecompute the determinant of this matrix further.Now the following theorem follows from the construction and Proposition 1.5. We preserve the notation of paragraph , and let us denote by ( dx • ) ∨ thedual basis of dx • := dx ∧ · · · ∧ dx d in ω − X , and the same for dx ′• and ( dx ′• ) ∨ . Let M be a left D ( m ) X ′ -module, and N be a right D ( m ) X ′ -module. Recall two isomorphisms of Berthelot [Be2, 2.4.3,2.4.4]: µ M : ω X ⊗ O X F ∗ M ∼ −→ F ♭ ( ω X ′ ⊗ O X ′ M ) ,ν N : ω − X ⊗ O X F ♭ N ∼ −→ F ∗ ( ω − X ′ ⊗ O X ′ N ) . Let m ∈ M , m ′ ∈ N , and f ∈ O X . Then we get that µ M ( dx • ⊗ ( f ⊗ m )) = ( dx ′• ⊗ m ) ⊗ ( Hx − ( q − · f ) ,ν N (cid:0) ( dx • ) ∨ ⊗ ( m ′ ⊗ ( Hx − ( q − · f )) (cid:1) = f ⊗ ( dx ′• ) ∨ ⊗ m ′ by using the notation of (1.4.1) . 2. Explicit calculation of Frobenius isomorphisms In this section, we will give first applications of the theorem in the previous section. The mainresult of this section is the calculation of the Frobenius structure of push-forwards. We will fix two situations for the basis R often used in this paper.i The ring R is complete discrete valuation ring as in Notation.ii We moreover assume that the s -th absolute Frobenius isomorphism F sk lifts to an au-tomorphism Spf( R ) ∼ −→ Spf( R ) which is denoted by σ . In this case k is automaticallyperfect. 10or a scheme X over k , we recall X ( s ) := X ⊗ k,F s ∗ k k . Let X be a smooth formal scheme overSpf( R ), and let X the special fiber. Suppose that X ( s )0 can be lifted to a smooth formal scheme X ′ over Spf( R ). In the situation 1, we are able to consider Frobenius pull-backs even if there areno lifting of the relative Frobenius morphism X → X ( s )0 (cf. [Be2, 2.2.3]). Thus we are able todiscuss the commutativity of Frobenius pull-backs with several cohomological operations such aspush-forwards or duals etc . In the situation 2, moreover, we are able to define F - D † X , Q -modules(cf. [Be2, 4.5.1]). We will review the second ingredient to calculate Frobenius isomorphisms, which areresults of Garnier [Ga]. See [Ab] for another aspect of Garnier’s result.We will consider the situation 2.1.1. Let X be a smooth affine formal scheme over R whosespecial fiber is denote by X , and suppose given a system of local coordinates { x , . . . , x d } . Wedenote by { ∂ , . . . , ∂ d } the corresponding differential operators. For a positive integer s , we let F s ∗ k : k → k be the s -th absolute Frobenius homomorphism. Let X ′ be a smooth affine formalscheme over R which is a lifting of X × σ s k . The relative Frobenius morphism X → X ⊗ F s ∗ k k can be lifted to a morphism F X : X → X ′ by the universal property of smoothness since X isassumed to be affine. We sometimes denote F X by F . We also fix a system of local coordinates { x ′ , . . . , x ′ d } of X ′ such that F ∗ ( x ′ ) = x q using the universal property once again. We denotethe corresponding differential operators by { ∂ ′ , . . . , ∂ ′ d } .Garnier constructed in [Ga] a special differential operator H ∈ b D ( s ) X called the Dwork operator with the following properties:i Suppose a primitive q -th root of unity is contained in R . Then H i = X ζ q =1 X k ≥ ( ζ − k x ki ∂ [ k ] i , H := Y ≤ i ≤ d H i . These are global sections of b D ( s ) X . The operator H i is called the Dwork operator corre-sponding to x i . (cf. [Ga, Proposition 4.5.2])ii The operator H is a projector from O X to O X ′ . Precisely, we have H = H in b D ( s ) X , andits action on O X is O X ′ -linear. (cf. [Ga, Proposition 2.5.1])iii For 0 ≤ k < q , we get that Hx − k in an element of b D ( s ) X . We have P ≤ k Throughout [Ga], the residue field k is assumed to be perfect (cf. [ loc. cit. Let X be as in the previous paragraph. Let Y be another smooth affine formal schemeover R possessing a system of local coordinates { y , . . . , y d ′ } . We also assume that we have asmooth lifting Y ′ of the relative Frobenius with a system of local coordinates { y ′ , . . . , y ′ d ′ } andmorphism F Y : Y → Y ′ such that F ∗ Y ( y ′ i ) = y qi for any i . We fix one non-negative integer j and denote X ⊗ R j , Y ⊗ R j , X ′ ⊗ R j , Y ′ ⊗ R j by X , Y , X ′ , Y ′ respectively. Suppose given amorphism of special fibers f : X → Y . Consider the following diagram. X F X / / f (cid:15) (cid:15) X ′ f ′ (cid:15) (cid:15) Y F Y / / Y ′ Here F X and F Y are reductions of F X and F Y , and f and f ′ are liftings of f and f ′ . Ingeneral, we are not able to take f and f ′ so that the diagram is commutative. However, we cantake F X and F Y locally with respect to X . To see this, it suffices to treat the case where f is a closed immersion and smooth morphism individually, and in both cases, the verification isstraightforward.Let M be a quasi-coherent O X ′ -module. Since X and Y are affine schemes, we will identifyquasi-coherent sheaves and its global sections. We list up conventions of identifications used todescribe sections of certain sheaves as follows. F ♭X ( M ) ∼ = M ⊗ O X ′ O ∨ X D ( m ) Y → X ∼ = O X ⊗ O Y D ( m ) Y (2.3.1) F ∗ X ( M ) ∼ = O X ⊗ O X ′ M D ( m ) Y ← X ∼ = ( O X ⊗ O Y ( D ( m ) Y ⊗ O Y ω − Y )) ⊗ O X ω X For example we have an identification F ♭X F ∗ Y D ( m ) Y ′ ← X ′ ∼ = O Y ⊗ O Y ′ (( O X ′ ⊗ O Y ′ D ( m ) Y ′ ⊗ O Y ′ ω − Y ′ ) ⊗ O ′ X ω X ′ ) ⊗ O X ′ O ∨ X , and for f ∈ O Y , g ∈ O Y ′ , P ∈ D ( m ) Y ′ and 0 ≤ k < q , the section f ⊗ (( g ⊗ P ⊗ ( dy ′• ) ∨ ) ⊗ dx ′• ) ⊗ Hx − k on the right side of the equality equally means a section on the left side by the identification. We preserve the notation of the previous paragraph. The inverse of theisomorphism of Berthelot [Be2, 3.4.2 (i)] χ : F ♭X F ∗ Y D ( m ) Y ′ ← X ′ ∼ −→ D ( m + s ) Y ← X can be described in the following way using the identification of (2.3.1) . Let f ∈ O X , and P ∈ D ( m ) Y ′ , then we get χ ( f ⊗ (1 ⊗ P ⊗ ( dy ′• ) ∨ ⊗ dx ′• ) ⊗ Hx − k ) = x q − k − ⊗ ( P ◦ · Hy − ( q − · f ) ⊗ ( dy • ) ∨ ⊗ dx • . roof. Recall that we are identifying quasi-coherent sheaves and its global sections. By using ν − in Theorem 1.7, we get an isomorphism O Y ⊗ O Y ′ ( D ( m ) Y ′ ⊗ ω − Y ′ ) → ( D ( m ) Y ′ ⊗ O Y ′ O ∨ Y ) ⊗ ω − Y . The theorem is saying that this sends f ⊗ P ⊗ ( dy ′• ) ∨ to ( P ⊗ ( Hy − ( q − · f )) ⊗ ( dy • ) ∨ . By using µ − , we get ( D ⊗ O X ′ ω X ′ ) ⊗ O X ′ O ∨ X → ( O X ⊗ O X ′ D ) ⊗ ω X , where D := f ′∗ (( D ( m ) Y ′ ⊗ O ∨ Y ) ⊗ ω − Y ). For a section D of D , this homomorphism sends ( D ⊗ dx ′• ) ⊗ Hx − k to ( x q − k − ⊗ D ) ⊗ dx • . At last, there exists the following isomorphism O X ⊗ O X ′ f ′∗ (( D ( m ) Y ′ ⊗ O ∨ Y ) ⊗ ω − Y ) ∼ −−→ ( ∗ ) f ∗ ( O Y ⊗ ( D ( m ) Y ′ ⊗ O ∨ Y ) ⊗ ω − Y ) ∼ −→ f ∗ D ( m + s ) Y ⊗ ω − Y . Here the first isomorphism follows from the commutativity of the diagram. According to Gar-nier’s calculation, this sends f ⊗ ⊗ (( Q ⊗ Hy − k ) ⊗ ( dy • ) ∨ ) to f ⊗ ( Q ◦ Hy − k ) ⊗ ( dy • ) ∨ . Combiningthese, we get the proposition. (cid:4) Remark. — We will describe shortly the way to calculate χ when the diagram in paragraph2.3 is not commutative. In this case, suppose the integer m satisfies the inequality p m > e/ ( p − e was the absolute ramification index of R . Under this condition, we may use the Taylorisomorphism of [Be2, 2.1.5] to compare F Y ◦ f and f ′ ◦ F X . This isomorphism can be describedin the following way. Let f, f ′ : X → Y be two morphisms of smooth schemes whose reductionsover k are the same morphisms, and suppose that Y possesses a system of local coordinates { y , . . . , y d } . Let T := X k ≥ ( f ′∗ ( y ) − f ∗ ( y )) { k } ( m ) ⊗ ∂ h k i ( m ) in f ∗ D ( m ) Y . This is defined since we have the assumption on m . Let M be a D ( m ) Y -module. Thenwe can check that the isomorphism τ f,f ′ : f ′∗ M ∼ −→ f ∗ M sends 1 ⊗ m to T ⊗ m for a section m of M .Now, when the diagram is not commutative, the calculation of χ goes well exactly in thesame way except for ( ∗ ) in the proof of the proposition. We use this calculation of Taylorisomorphism to compute ( ∗ ). We use the same notation as paragraph 2.3. As the first application of Proposition 2.4,we will calculate the isomorphism(2.5.1) F ∗ Y f ′ + M ∼ −→ f + F ∗ X M concretely. This result is used in [AM] to calculate the Frobenius structure of geometric Fouriertransform defined by Noot-Huyghe explicitly.Let M be a D ( m ) X ′ -module. In the proof of [Be2, 3.4.4], the isomorphism ξ : F ∗ Y ( D ( m ) Y ′ ← X ′ ⊗ L D ( m ) X ′ M ) := f − O Y ⊗ f ′− O Y ′ ( D ( m ) Y ′ ← X ′ ⊗ L D ( m ) X ′ M ) ∼ −→ ( D ( m + s ) Y ← X ⊗ L D ( m + s ) X F ∗ X M )is defined. Using the projection formula, the isomorphism (2.5.1) is nothing but Rf ∗ ( ξ ). Proposition. — Let ξ := H ( ξ ) . For P ∈ D ( m ) Y ′ and m ∈ M , we have ξ (cid:0) y l ⊗ (1 ⊗ P ⊗ ( dy ′• ) ∨ ⊗ dx ′• ) ⊗ m (cid:1) = (cid:0) ⊗ ( P ◦ · Hy − ( q − · y l ) ⊗ ( dy • ) ∨ ⊗ dx • (cid:1) ⊗ ( x q − ⊗ m ) . roof. Let us review the definition of ξ . Tensoring both sides of ξ with f − F ♭Y D ( m ) Y ′ , it isequivalent to defining an isomorphism(2.5.2) D ( m ) Y ′ ← X ′ ⊗ D ( m ) X ′ M ∼ −→ f − F ♭Y D ( m ) Y ′ ⊗ f − D ( m + s ) Y ( D ( m + s ) Y ← X ⊗ D ( m + s ) X F ∗ X M )by [Be2, 2.5.6]. We get an isomorphism D ( m ) Y ′ ← X ′ ⊗ D ( m ) X ′ M ∼ −→ F ♭X D ( m ) Y ′ ← X ′ ⊗ D ( m + s ) X F ∗ X M ∼ −→ f − F ♭Y D ( m ) Y ′ ⊗ F ∗ Y F ♭X D ( m ) Y ′ ← X ′ ⊗ F ∗ X M where the first isomorphism is by [Be2, 2.5.7] and the second by [Be2, 2.5.6]. Combining thisisomorphism with χ , we get the isomorphism (2.5.2). By Proposition 2.4, we see that (2.5.2)sends (1 ⊗ P ⊗ ( dy ′• ) ∨ ⊗ dx ′• ) ⊗ m to(1 ⊗ Hy − l ) ⊗ (cid:0) x q − k − ⊗ ( P ◦ · Hy − ( q − · y l ) ⊗ ( dy • ) ∨ ⊗ dx • (cid:1) ⊗ ( x k ⊗ m )where k and l denote any element in N d such that ≤ q − 1, and the proposition follows. (cid:4) We consider the situation 2.1.1. As another application of the explicit description, wewill show a proper base change type result. For the most familiar statement, see paragraph 5.7.Let m be an integer such that p m > e/ ( p − 1) and i be a non-negative integer. Consider thefollowing cartesian diagram of smooth schemes over R i X ′ g ′ / / f ′ (cid:15) (cid:15) h AAAAAAAA X f (cid:15) (cid:15) Y ′ g / / Y where h = g ◦ f ′ . Let Z be one of X , X ′ , Y , Y ′ . We denote by Z the reduction of Z over k .We assume that Z ( s )0 possesses a smooth lifting e Z over R i . Lemma. — We assume that g is smooth. Then there is a canonical equivalence of functors (2.6.1) g ! ◦ f + ∼ = f ′ + ◦ g ′ ! : D b qc ( D ( m ) X ) → D b qc ( D ( m ) Y ′ ) . This equivalence is compatible with raising levels and Frobenius pull-backs.Proof. Let us first construct the following homomorphism:(2.6.2) f ′− D ( m ) Y ′ → Y ⊗ L h − D ( m ) Y g ′− D ( m ) Y ← X → D ( m ) Y ′ ← X ′ ⊗ L D ( m ) X ′ D ( m ) X ′ → X . For short, we denote D ( m ) by D . There exists a canonical homomorphism of rings g − D Y → D Y ′ .Thus, we get a homomorphism h − D Y ⊗ h − O Y ω X ′ /Y ′ α −→ f ′− D Y ′ ⊗ O Y ′ ω X ′ /Y ′ of ( f ′− D Y ′ , g ′− D X )-bimodules where ω X ′ /Y ′ := ω X ′ ⊗ f ′− ω Y ′ . We have the canonical section 1 ⊗ D X ′ → X ∼ = O X ′ ⊗ f − O X f − D X . This section induces a homomorphism f ′− D Y ′ ⊗ O Y ′ ω X ′ /Y ′ β −→ ( f ′− D Y ′ ⊗ O Y ′ ω X ′ /Y ′ ) ⊗ D X ′ D X ′ → X . Combining these, we get the homomorphism β ◦ α : h − D Y ⊗ h − O Y ω X ′ /Y ′ → ( f ′− D Y ′ ⊗ O Y ′ ω X ′ /Y ′ ) ⊗ D X ′ D X ′ → X . This induces the homomorphism(2.6.3) f ′− D Y ′ → Y ⊗ h − D Y g ′− D Y ← X → D Y ′ ← X ′ ⊗ D X ′ D X ′ → X . f ′− D Y ′ → Y is a flat right h − D Y -module, we get a quasi-isomorphism(2.6.4) f ′− D Y ′ → Y ⊗ L h − D Y g ′− D Y ← X ∼ −→ f ′− D Y ′ → Y ⊗ h − D Y g ′− D Y ← X . From now on, we omit f ′− and so on, but we consider sheaves are on X ′ . Let us show that(2.6.5) D ( m ) Y ′ ← X ′ ⊗ L D ( m ) X ′ D ( m ) X ′ → X ∼ −→ D ( m ) Y ′ ← X ′ ⊗ D ( m ) X ′ D ( m ) X ′ → X , or in other words H i ( D ( m ) Y ′ ← X ′ ⊗ L D ( m ) X ′ D ( m ) X ′ → X ) = 0 for i = 0. When m = 0, the argument isstandard using the Spencer resolution D (0) X ′ ⊗ Θ • X ′ /X of D (0) X ′ → X where Θ X ′ /X denotes the relativetangent bundle of X ′ → X (cf. [Be2, 4.3.1]). Let us see the general case. Since the verificationis local, we may assume that the schemes are affine and s = m . We get D ( m ) Y ′ ← X ′ ⊗ L D ( m ) X ′ D ( m ) X ′ → X ∼ = F ∗ Y ′ F ♭X (cid:0) D (0) e Y ′ ← e X ′ ⊗ L D (0) e X ′ D (0) e X ′ → e X (cid:1) by [Be2, 2.5.6, 3.4.2]. This reduces the verification to the m = 0 case, and the claim follows.Combining (2.6.3), (2.6.4), (2.6.5), we get the desired homomorphism (2.6.2).By construction, (2.6.2) is compatible with raising levels. Let us show the compatibility ofFrobenius. We denote by relative Frobenius morphisms of the special fiber X etc. by F X etc .Let us show that the following diagram is commutative where homomorphisms are isomorphisms.(2.6.6) F ∗ Y ′ (cid:0) D ( m ) e Y ′ → e Y ⊗ L D ( m ) e Y D ( m ) e Y ← e X ⊗ L D ( m ) e X F ♭X D ( m ) e X (cid:1) / / (cid:15) (cid:15) α ) ) β , , F ∗ Y ′ (cid:0) D ( m ) e Y ′ ← e X ′ ⊗ L D ( m ) e X ′ D ( m ) e X ′ → e X ⊗ L D ( m ) e X F ♭X D ( m ) e X (cid:1) (cid:15) (cid:15) (cid:0) D ( m + s ) Y ′ → Y ⊗ L D ( m + s ) Y D ( m + s ) Y ← X (cid:1) ⊗ L D ( m + s ) X F ∗ X F ♭X D ( m ) e X / / (cid:0) D ( m + s ) Y ′ ← X ′ ⊗ L D ( m + s ) X ′ D ( m + s ) X ′ → X (cid:1) ⊗ L D ( m + s ) X F ∗ X F ♭X D ( m ) e X Let M be a left D ( m ) e X -complex. By taking ⊗ L D ( m + s ) X F ∗ X M , (2.6.6) induces the following commu-tative diagram: F ∗ Y ′ (cid:0) D ( m ) e Y ′ → e Y ⊗ L D ( m ) e Y D ( m ) e Y ← e X ⊗ L D ( m ) e X M (cid:1) / / (cid:15) (cid:15) F ∗ Y ′ (cid:0) D ( m ) e Y ′ ← e X ′ ⊗ L D ( m ) e X ′ D ( m ) e X ′ → e X ⊗ L D ( m ) e X M (cid:1) (cid:15) (cid:15) (cid:0) D ( m + s ) Y ′ → Y ⊗ L D ( m + s ) Y D ( m + s ) Y ← X (cid:1) ⊗ L D ( m + s ) X F ∗ X M / / (cid:0) D ( m + s ) Y ′ ← X ′ ⊗ L D ( m + s ) X ′ D ( m + s ) X ′ → X (cid:1) ⊗ L D ( m + s ) X F ∗ X M , which implies the compatibility of Frobenius.Let us prove the commutativity of (2.6.6). Since these complexes are concentrated in degree0, the problem is local on X ′ . Thus, we may assume that any scheme possesses a system of localcoordinates, and the following diagram is commutative. e X ′ / / (cid:15) (cid:15) e X (cid:15) (cid:15) X ′ F X ′ rrrrrr g ′ / / f ′ (cid:15) (cid:15) X F X ssssss f (cid:15) (cid:15) e Y ′ / / e YY ′ F Y ′ rrrrrr g / / Y F Y ssssss { e t i } ≤ i ≤ d X (resp. { t i } ≤ i ≤ d X , { e s j } ≤ j ≤ d Y , { s j } ≤ j ≤ d Y ) be a system of local coordinatesof e X (resp. X , e Y , Y ) such that F ∗ X ( e t i ) = t qi , F ∗ Y ( e s j ) = s qj . As usual, we use the notation dt • := dt ∧ · · · ∧ dt d X and so on. Let H s j be the Dwork operator (cf. subsection 2.2) in D ( m + s ) Y ′ corresponding to s j , and we put H s := Q d Y j =1 H s j . Let x , . . . , x d be a local coordinate of Y ′ over Y . This can be seen also as a local coordinate of X ′ over X . We denote by H i the Dworkoperator in D ( m + s ) X ′ (resp. D ( m + s ) Y ′ ) corresponding to x i . We put H x := Q di =1 H i . Using theconvention of (2.3.1), letΞ := 1 ⊗ (cid:0) (1 ⊗ P ) ⊗ (1 ⊗ ⊗ ( d e t • ) ∨ ⊗ d e s • ) ⊗ ( Q ⊗ φ ) (cid:1) be a section of F ∗ Y ′ (cid:0) D ( m ) e Y ′ → e Y ⊗ D ( m ) e Y D ( m ) e Y ← e X ⊗ D ( m ) e X F ♭X D ( m ) e X (cid:1) . To see the commutativity of (2.6.6),it suffices to show α (Ξ) = β (Ξ). Let S := Q ≤ i ≤ d x q − i H i x − ( q − i . By using Proposition 2.5, α (Ξ) = (cid:0) ⊗ P ◦ · H s s − ( q − · H x x − ( q − · f ⊗ ( dt • ) ∨ ⊗ ds • (cid:1) ⊗ ( s q − x q − ⊗ ⊗ ( Q ⊗ φ )= (cid:0) ⊗ P ◦ · H s s − ( q − · f ⊗ ( dt • ) ∨ ⊗ ds • (cid:1) ⊗ t S · ( s q − ⊗ ⊗ ( Q ⊗ φ ) β (Ξ) = (cid:0) ⊗ P ◦ · H s s − ( q − · f ⊗ ( dt • ) ∨ ⊗ ds • (cid:1) ⊗ ( s q − ⊗ ⊗ ( Q ⊗ φ )To show that these two quantities are equal, it suffices to see that the image of t S by thehomomorphism D ( m + s ) X ′ → D ( m + s ) X ′ → X is 1 ⊗ 1. To show this, it suffices to see that t S (1) = 1 in O X ′ . Since the claim is stable under base change, we may assume that ζ ∈ R i . By definition of H , it suffices to show that x − ( q − i q − X ζ q =1 X k ≥ ( ζ − k ∂ [ k ] i ( x k + q − i ) = 1for any 1 ≤ i ≤ d . The sum is equal to q − X ζ q =1 X k ( ζ − k (cid:18) k + q − k (cid:19) = q − X ζ q =1 X k (1 − ζ ) k (cid:18) − qk (cid:19) = q − X ζ q =1 ζ − q = 1 , and the commutativity of (2.6.6) follows.Finally, let us complete the proof of (2.6.1). We have g ! ◦ f + ( M ) := D Y ′ → Y ⊗ L g − D Y g − Rf ∗ D Y ← X ⊗ L D X M ∼ = Rf ′∗ (cid:0) f ′− D Y ′ → Y ⊗ L h − D Y g ′− ( D Y ← X ⊗ L D X M ) (cid:1) ∼ −→ Rf ′∗ (cid:0) D Y ′ ← X ′ ⊗ L D X ′ D X ′ → X ⊗ L g ′− D X g ′− M (cid:1) ∼ = f ′ + ◦ g ′ ! ( M ) , where we used the flat base change in the first isomorphism. Thus, we get the isomorphism. (cid:4) Consider the situation in 2.1.1. Let us define the Tate twist (cf. [BeP, 2.3.8 (i)]). Let X and Y are two smooth formal schemes, and suppose there exist smooth liftings X ′ and Y ′ of X ( s )0 and Y ( s )0 where X and Y are special fibers of X and Y as usual. Let ∗ be one of X , X ′ , Y , Y ′ , and A ( ∗ ) be either D b coh ( D †∗ , Q ) or LD −→ b Q , qc ( b D ( • ) ∗ ). Let G : A ( X ) → A ( Y ) , G ′ : A ( X ′ ) → A ( Y ′ )be Q -linear functors. Now, suppose given a equivalence of functorsΨ : F ∗ Y ◦ G ′ ∼ −→ G ◦ F ∗ X . G, G ′ , Ψ) (we often abbreviate this as ( G, Ψ) or even G if no confusion can arise) iscalled a cohomological functor with Frobenius isomorphism . The natural transform Ψ is calledthe Frobenius isomorphism of the triple. Given ( G, G ′ , Ψ) and an integer d , we define its Tatetwist Ψ( d ) of the Frobenius isomorphism by Ψ( d ) := q − d · Ψ. We often denote by G ( d ) the triple( G, G ′ , Ψ( d )) for simplicity.Now, we consider the situation 2.1.2. For a D † X , Q -module M , we denote a D † X , Q -moduleby M σ the base change of M by using σ . Let M be an F - D † X , Q -module (resp. complex). Bydefinition, this is a D † X , Q -module (resp. complex) equipped with an isomorphism (1) Φ : F ∗ M σ ∼ −→ M . For any integer d , we define an F - D † X , Q -module (resp. complex) M ( d ) called the Tatetwist of M in the following way. The underlying D † X , Q -module (resp. complex) is the same asthat of M . We denote by Φ ′ the isomorphism F ∗ M ( d ) σ → M ( d ) induced by the Frobeniusstructure of M . The Frobenius structure Φ( d ) : F ∗ M ( d ) σ → M ( d ) of M ( d ) is by definition q − d Φ ′ . Now, let ( G, Ψ) be a cohomological functor with Frobenius isomorphism. Then we getthat G ( M ) is naturally equipped with Frobenius structure, and we get for any integer d that G ( d )( M ) ∼ = G ( M )( d ) ∼ = G ( M ( d )).Let ( M , Φ) and ( N , Ψ) be two F - D † X , Q -modules, and ϕ : M → N be a homomorphismof D † X , Q -modules (where we do not consider the Frobenius structures). Consider the followingdiagrams where the left diagram is that of modules (or sheaves of modules): M α / / γ (cid:15) (cid:15) M β (cid:15) (cid:15) M δ / / M , F ∗ M σ F ∗ ϕ σ / / Φ (cid:15) (cid:15) F ∗ N σ Ψ (cid:15) (cid:15) M ϕ / / N . First, pay attention to the left diagram. Let n be a rational number. We say that the diagram is commutative up to multiplication by n if n · ( β ◦ α ) = δ ◦ γ holds. Now, changing the attention tothe right diagram, suppose that the diagram is commutative up to multiplication by q d . Thenwe get that ϕ defines a homomorphism M ( d ) → N as F - D † X , Q -modules. 3. Extraordinary pull-back and duality In this section, we prove a commutation result of the extraordinary pull-back functor and theduality functor. The result can be seen as a part of a “Poincar´e duality” in the theory ofarithmetic D -modules. For the explanation of this interpretation, see § 5. By applying thecommutation result, we get D X ,Z ( O X , Q ( † Z )) ∼ = O X , Q ( † Z )( − d )where X is a smooth formal scheme, Z is a divisor of its special fiber, and d is the dimensionof X . Combining this result with a result of Caro, we are able to compare duality functorsof arithmetic D -modules and that of overconvergent isocrystals with Frobenius structures in aprecisely way. At the last part of this section, we compare the rigid cohomologies and thepush-forwards in the theory of arithmetic D -modules. (1) The definition of Frobenius structure here is slightly different from that of [Be2, 4.5.1] in the sense that in loc. cit. , Φ is an isomorphism M ∼ −→ F ∗ M σ . Since Φ is an isomorphism, it causes no difference. We adopted ourdefinition to make it easier to see the compatibility with the definition of Frobenius structure of F -isocrystals.See also [ loc. cit. , Remarque 4.5.1]. .1. We consider the situation 2.1.1. Let X be a smooth formal scheme, and Z be a divisorof its special fiber X . In this situation, we say that ( X , Z ) is a d-couple (2) . Let ( Y , W ) beanother d-couple. A morphism of d-couples f : ( X , Z ) → ( Y , W ) is a morphism of special fibers f : X → Y such that f ( X \ Z ) ⊂ Y \ W , and f − ( W ) is a divisor. A strict morphism ofd-couples f is a morphism e f : X → Y whose reduction on the special fiber is a morphism ofd-couples. We say that the morphism f (resp. e f ) is the realization of the (resp. strict) morphismof d-couples f .For a d-couple ( X , Z ), let us review the definition of the dual functor D X ,Z : D b perf ( D † X , Q ( † Z )) → D b perf ( D † X , Q ( † Z )) . We note that there exists the canonical equivalence of categories D b perf ( D † X , Q ( † Z )) ∼ −→ D b coh ( D † X , Q ( † Z ))by [NH2, 3.2.3]. Let d be the dimension of X . For a perfect D † X , Q ( † Z )-complex C , we definethe functor D X ,Z by D X ,Z ( C ) := R H om D † X , Q ( † Z ) ( C , D † X , Q ( † Z )) ⊗ O X ω − X [ d ] . For fundamental properties of this functor, see [Vi]. Here, we only note that this functorcommutes with Frobenius pull-backs, and induces an equivalence between the derived categoriesof perfect complexes (with or without Frobenius structure).Let ( X , Z ) and ( Y , W ) be d-couples, and let f : X → Y be a morphism such that f ( X \ Z ) ⊂ Y \ W . Assume that f is smooth. Then f − ( W ) is a divisor, and in particular, f induces a morphism of d-couples f : ( X , Z ) → ( Y .W ). The functor f !0 : D b coh ( b D ( m ) Y , Q ( W )) → D b coh ( b D ( m ) X , Q ( f − ( W ))) is defined in [Be2, 3.2.3 (ii)] and [BeI, 3.4.6]. By taking the inductivelimit as [BeI, 4.3.3], we have the functor f !0 : D b coh ( D † Y , Q ( † W )) → D b coh ( D † X , Q ( † f − ( W ))). Wedefine the functor f ! : D b coh ( b D ( m ) Y , Q ( W )) → D b coh ( b D ( m ) X , Q ( Z ))by ( Z ) ◦ f !0 , where ( Z ) denotes the functor tensoring with b D ( m ) X , Q ( Z ). By taking the inductivelimit, we also get a functor f ! : D b coh ( D † Y , Q ( † W )) → D b coh ( D † X , Q ( † Z )). Let m be an integer such that p m > e/ ( p − 1) (cf. [Be2, A.4]). We denote B ( m ) X ( Z ) and b B ( m ) X ( Z ) (cf. [Be1, 4.2.4]) by B ( m ) X and b B ( m ) X , b B ( m ) X b ⊗ b D ( m ) X by e D ( m ) X , and lim −→ m e D ( m ) X (= D † X ( † Z )) by e D † X . We put e ω X := b B ( m ) X b ⊗ O X ω X . We denote by D ∗ qc ( e D ( m ) X ) ( ∗ ∈ {− , b } ) the full subcategoryof D ∗ ( b D ( m ) X ) consisting of quasi-coherent complexes (cf. [BeI, 3.2.1]), and D ∗ Q , qc ( e D ( m ) X ) by thecategory obtained by localizing D ∗ qc ( e D ( m ) X ) with respect to isogenies (cf. [BeI, 3.3.2]). Finally,we denote by D ∗ ( X ) ( ∗ ∈ { + , − , b } ) the derived category of R -modules on X . Lemma. — Let X be a smooth formal scheme. Let M be a complex in D b perf ( e D ( m ) X , Q ) g , and N be a complex in D b Q , qc ( e D ( m ) X ) g . Then the complex M b ⊗ L b B ( m ) X , Q N is bounded.Proof. We will use the notation of SGA6 Exp. I. Since D b perf ⊂ D ftd by 0.3, we may assumethat parf-amp( M ) ⊂ [0 , a ]. Let n be an integer such that H i ( N ) = 0 for i < n . It suffices to (2) This “d” stands for divisor. H i ( M b ⊗ L b B ( m ) X , Q N ) = 0 for i < n − 1. Since this is local, and we may assume that X is affine. We will assume X to be affine in the following.For a positive integer r , we say that a finitely generated e D ( m ) X -module P is r -nearly projectiveif there exists a e D ( m ) X -module Q , an integer b , and a short exact sequence 0 → P ⊕ Q → ( e D ( m ) X ) ⊕ b → R → π r R = 0. For any finitely generated projective e D ( m ) X , Q -module P ′ ,there exists an integer r and an r -nearly projective e D ( m ) X -module P such that P ⊗ Q ∼ = P ′ .This shows that there exists a complex P • of r -nearly projective e D ( m ) X -modules concentrated in[0 , a ] such that P • ⊗ Q is quasi-isomorphic to M . Thus, it suffices to show that for any r -nearlyprojective e D ( m ) X -module P and N ∈ D b qc ( e D ( m ) X ) such that H i ( N ) = 0 for i < n , we get ̟ r H i ( P b ⊗ L b B ( m ) X N ) = 0for i < n − 1. Since P is r -nearly projective, ̟ r H i ( P ⊗ L b B ( m ) X N ) = 0 for any i < n . This showsthat ̟ r H i ( P j ⊗ L B ( m ) Xj N ) = 0for any j and i < n − 1, where P j := P ⊗ R R j . Now, it remains to take R l ←− X ∗ , but since thisfunctor is a right derived functor, we get the claim. (cid:4) Now, let us state a key proposition in this section. Let X be a smooth formal scheme,and X ′ be a smooth lifting of X ( s )0 . Proposition. — Let M be a complex in D b perf ( e D ( m ) X , Q ) g , and N be a complex in D b Q , qc ( e D ( m ) X ) g .We denote D X ,Z by D . Then there exists the following quasi-isomorphism in D + ( X )Ψ : R H om e D ( m ) X , Q ( b B ( m ) X , Q , D ( M ) b ⊗ L b B ( m ) X , Q N ) ∼ −→ R H om e D ( m ) X , Q ( M , N ) . Let F be a complex in D b perf ( e D ( m ) X ′ , Q ) g , and G be a complex in D b Q , qc ( e D ( m ) X ′ ) g . Consider thefollowing diagram: R H om e D ( m ) X ′ , Q ( b B ( m ) X ′ , Q , D ( F ) b ⊗ L b B ( m ) X ′ , Q G ) Ψ / / ∼ (cid:15) (cid:15) R H om e D ( m ) X ′ , Q ( F , G ) ∼ (cid:15) (cid:15) R H om e D ( m + s ) X , Q ( b B ( m + s ) X , Q , D ( F ∗ F ) b ⊗ L b B ( m + s ) X , Q F ∗ G ) Ψ / / R H om e D ( m + s ) X , Q ( F ∗ F , F ∗ G ) where the vertical homomorphisms are canonical isomorphisms of complexes which are defined bythe theorem of Frobenius descent [Be2, 4.1.3] . This diagram is commutative up to multiplicationby q d where d denotes the dimension of X . Remark. — We note that the complex R H om e D ( m ) X , Q ( b B ( m ) X , Q , D ( M ) b ⊗ L b B ( m ) X , Q N ) makes sensethanks to Lemma 3.2.The proof of the proposition will be given in paragraph 3.8, and we will start preparations ofthe proof from the next paragraph. Unless otherwise stated, M , N , F , G are not the sheavesin the proposition. 19 .4. First, we will prove the following small lemma. Lemma. — Let M be a complex in D b Q , coh ( e D ( m ) X ) and N be one in D b Q , qc ( e D ( m ) X ) . Then, thereis a canonical isomorphism Hom D b Q , qc ( e D ( m ) X ) ( M , N ) ∼ −→ Hom e D ( m ) X , Q ( M , N ) . Proof. For F and G in D b Q , qc ( e D ( m ) X ), we define R H om D b Q , qc ( e D ( m ) X ) ( F , G ) := R H om e D ( m ) X ( F , G ) ⊗ Q . There exists the canonical homomorphism R H om D b Q , qc ( e D ( m ) X ) ( M , N ) → R H om e D ( m ) X , Q ( M ⊗ Q , N ⊗ Q ) , and it suffices to show that this is an isomorphism. Since the problem is local, we may assumethat X is affine. It suffices to show the claim in the case where M is projective. Since M isa direct factor of ( e D ( m ) X ) ⊕ n for some n , we are reduced to showing the case M = e D ( m ) X . In thiscase, the lemma is straightforward. (cid:4) To compare Frobenius pull-backs, we need to construct a certain isomorphism in D − ( X ).Let N be a complex in D b perf ( e D ( m ) X ′ , Q ) d ( e.g. e ω X ′ , Q by (3.6.1)), and M be a complex in D − ( e D ( m ) X , Q ) g .The isomorphism we will construct is the following:(3.5.1) N ⊗ L e D ( m ) X ′ , Q M ∼ −→ F ♭ N ⊗ L e D ( m + s ) X , Q F ∗ M . Let Y be a smooth formal scheme, and let F be a complex in D b perf ( e D ( m ) Y , Q ) d , and E be a complexin D − ( e D ( m ) Y , Q ) g . First, there exists an isomorphism D d ◦ D d ( F ) ∼ = F where D d ( F ) := R H om e D ( m ) Y , Q ( F , e D ( m ) Y , Q ) ⊗ ω Y by [Vi, II, 3.6]. Using this, we get F ⊗ L e D ( m ) Y , Q E ∼ = R H om e D ( m ) Y , Q ( D d ( F ) , e D ( m ) Y , Q ⊗ ω Y ) ⊗ L e D ( m ) Y , Q E ∼ = R H om e D ( m ) Y , Q ( D d ( F ) , ω Y ⊗ E )where the second isomorphism is by [Ca3, 2.1.17 (i)]. Now, we define (3.5.1) in the followingway: N ⊗ L e D ( m ) X ′ , Q M ∼ = R H om e D ( m ) X ′ , Q ( D d ( N ) , ω X ′ ⊗ M ) ∼ −→ R H om e D ( m + s ) X , Q ( F ♭ D d ( N ) , F ♭ ( ω X ′ ⊗ M )) ∼ = R H om e D ( m + s ) X , Q ( D d ( F ♭ N ) , ω X ′ ⊗ F ∗ M ) ∼ = F ♭ N ⊗ L e D ( m + s ) X , Q F ∗ M . Here the second isomorphism follows by the theorem of Frobenius descent.20 .6. We will construct the homomorphism Ψ in the proposition step by step. Let Θ X be thetangent sheaf on X , and we put Θ i X := V i Θ X . First, note that there exists the Spencerresolution(3.6.1) e D ( m ) X , Q ⊗ Θ • X → b B ( m ) X , Q . This can be seen in exactly the same way as the proof of [Be2, 4.3.3]. Indeed, B ( m ) X , Q ⊗ D ( m ) X , Q is flatover D ( m ) X , Q . Since B ( m ) X ⊗ D ( m ) X is noetherian, the p -adic completion e D ( m ) X is flat over B ( m ) X ⊗ D ( m ) X .Thus, e D ( m ) X , Q is flat over D (0) X , Q . It remains to show that e D ( m ) X , Q ⊗ D (0) X , Q O X , Q ∼ = b B ( m ) X , Q , whose proofis straightforward. This shows that b B ( m ) X , Q is perfect as a complex.Let M be a bounded e D ( m ) X , Q -complex. We have the following quasi-isomorphisms R H om e D ( m ) X , Q ( b B ( m ) X , Q , M ) ∼ = R H om e D ( m ) X , Q ( b B ( m ) X , Q , e D ( m ) X , Q ) ⊗ L e D ( m ) X , Q M ∼ = e ω X , Q ⊗ L e D ( m ) X , Q M [ − d ]where the first isomorphism holds by [Ca3, 2.1.17], and the second by (3.6.1) and [BeI, (3.5.5.1)].Now the point where we need to use the explicit computation of Frobenius isomorphism of § Lemma. — Let M be a complex in D b ( e D ( m ) X ′ , Q ) . Consider the following diagram in D b ( X ) . (3.6.2) R H om e D ( m ) X ′ , Q ( b B ( m ) X ′ , Q , M ) ∼ / / ∼ (cid:15) (cid:15) e ω X ′ , Q ⊗ L e D ( m ) X ′ , Q M [ − d ] ∼ (cid:15) (cid:15) R H om e D ( m + s ) X , Q ( b B ( m + s ) X , Q , F ∗ M ) ∼ / / e ω X , Q ⊗ L e D ( m + s ) X , Q F ∗ M [ − d ] Here the right vertical homomorphism is (3.5.1) composed with the canonical isomorphism F ♭ ω X ′ ∼ = ω X , and the left vertical homomorphism is induced by the functor F ∗ . Then this diagram iscommutative up to multiplication by q d .Proof. Let G be the functor H om e D ( m + s ) X , Q ( F ∗ b B ( m ) X ′ , Q , • ), and G ′ be the functor H om e D ( m + s ) X , Q ( b B ( m + s ) X , Q , • ).By (3.6.1), R i G and R i G ′ vanishes for i > d . We define a functor H to be R d G , and H ′ tobe R d G ′ . The canonical isomorphism F ∗ b B ( m ) X ′ , Q ∼ −→ b B ( m + s ) X , Q induces an isomorphism of functors H ∼ −→ H ′ . By [Ha, I, 7.4], we get the following commutative diagram of functors. RG ∼ / / ∼ (cid:15) (cid:15) LH [ − d ] ∼ (cid:15) (cid:15) RG ′ ∼ / / LH ′ [ − d ]We note that flat e D ( m + s ) X , Q -modules belong to the set P of loc. cit. For a flat e D ( m + s ) X , Q -module P , we get a canonical isomorphism H ( P ) ∼ = F ∗ e ω X ′ ⊗ e D ( m + s ) X , Q P and H ′ ( P ) ∼ = e ω X ⊗ e D ( m + s ) X , Q P ,which induces canonical isomorphisms of functors LG ∼ = F ♭ e ω X ′ ⊗ L e D ( m + s ) X , Q , and LG ′ ∼ = e ω X ⊗ L e D ( m + s ) X , Q .21onsider the following diagram: R H om e D ( m ) X ′ , Q ( b B ( m ) X ′ , Q , M ) / / (cid:15) (cid:15) (cid:9) e ω X ′ ⊗ L e D ( m ) X ′ M [ − d ] (cid:15) (cid:15) R H om e D ( m + s ) X , Q ( F ∗ b B ( m ) X ′ , Q , F ∗ M ) / / (cid:15) (cid:15) (cid:9) LH ( F ∗ M )[ − d ] / / (cid:15) (cid:15) a (cid:13) F ♭ e ω X ′ ⊗ L e D ( m + s ) X F ∗ M [ − d ] (cid:15) (cid:15) R H om e D ( m + s ) X , Q ( b B ( m ) X , Q , F ∗ M ) / / LH ′ ( F ∗ M )[ − d ] / / e ω X ⊗ L e D ( m + s ) X F ∗ M [ − d ]where all the arrows are isomorphisms, and (cid:9) denotes that the marked square is commutative.Thus, to show that the big diagram is commutative up to some multiplication is equivalentto showing that the diagram a (cid:13) is commutative up to the same multiplication. Since thehomomorphisms of the diagram a (cid:13) are induced by a diagram of functors between H , H ′ , F ∗ e ω X ′ ⊗ , e ω X ⊗ , it suffices to show the commutativity up to the same multiplication for thisdiagram of functors. Thus the problem is local.We may assume that X ′ is affine, and possesses a system of local coordinates { y , . . . , y d } .Moreover, we can take a system of local coordinates { x , . . . , x d } of X and a lifting F : X → X ′ of relative Frobenius morphism such that F ∗ ( y i ) = x qi . Under this situation, let us show that thediagram is commutative up to multiplication by q d . From now on, we do not make any differencebetween quasi-coherent modules and its global sections. It suffices to show the commutativityin the case where M is flat over b D ( m ) X ′ , Q .Let F ∗ : Θ X → F ∗ Θ X ′ be the canonical homomorphism. We have the following homo-morphism e D ( m + s ) X , Q ⊗ Θ k X → F ∗ ( e D ( m ) X ′ , Q ⊗ Θ k X ′ ) sending P ⊗ ( dy i ∧ · · · ∧ dy i k ) to P · (1 ⊗ qx q − i . . . qx q − i k ( dx i ∧ · · · ∧ dx i k )). This defines, in fact, a homomorphism of complexes(3.6.3) e D ( m + s ) X , Q ⊗ Θ • X → F ∗ ( e D ( m ) X ′ , Q ⊗ Θ • X ′ )by the proof of [Be2, 4.3.5]. It suffices to show that the diagram of modulesΩ d X ′ ⊗ O X ′ M / / (cid:15) (cid:15) ω X ′ ⊗ b D ( m ) X ′ , Q M (cid:15) (cid:15) Ω d X ⊗ O X F ∗ M / / ω X ⊗ b D ( m ) X , Q F ∗ M is commutative up to multiplication by q d , where the left vertical homomorphism is induced by(3.6.3). Since the right vertical homomorphism sends ( dy ∧ · · · ∧ dy d ) ⊗ m to x q − . . . x q − d · ( dx ∧ · · · ∧ dx d ) ⊗ (1 ⊗ m ) by using Proposition 1.5 and Theorem 1.7, we get the claim, andconclude the proof of the lemma. (cid:4) We have the following lemma whose proof is similar to that of [Ca3, 2.1.27], and we leaveit to the reader. Lemma. — Let N be a complex in D − qc ( e D ( m ) X ) d , and M and M ′ be two complexes in D − qc ( e D ( m ) X ) g . Then there is a canonical isomorphism N b ⊗ L e D ( m ) X ( M b ⊗ L b B ( m ) X M ′ ) ∼ = ( N b ⊗ L b B ( m ) X M ) b ⊗ L e D ( m ) X M ′ . .8. Proof of Proposition 3.3 Here, we use the notation in the proposition. We apply Lemma 3.6 to D ( M ) b ⊗ L b B ( m ) X , Q N . Whenwe omit bases of tensor products, they are taken over b B ( m ) X , Q . Then we get isomorphisms R H om e D ( m ) X , Q ( b B ( m ) X , Q , D ( M ) b ⊗ L N ) ∼ = e ω X , Q ⊗ L e D ( m ) X , Q ( D ( M ) b ⊗ L N )[ − d ] ∼ = ( e ω X , Q ⊗ L D ( M )) ⊗ L e D ( m ) X , Q N [ − d ] ∼ = R H om e D ( m ) X , Q ( M , e D ( m ) X , Q ) ⊗ L e D ( m ) X , Q N ∼ = R H om e D ( m ) X , Q ( M , N )where the first isomorphism is the one in paragraph 3.6, the second isomorphism is defined byLemma 3.7, and we used the fact that M is a perfect complex in the last isomorphism (cf. [Ca3,2.1.12, or 2.1.17]). This is nothing but Ψ that we are looking for. The second, third, and thelast isomorphisms are compatible with Frobenius. Thus the statement of Frobenius follows byLemma 3.6. (cid:4) Let X be a smooth formal scheme. We denote by e D b perf ( e D † X , Q ) the full subcategory of D b perf ( e D † X , Q ) consisting of a complex M such that there exists a complex M ′ in D b perf ( e D ( m ) X , Q )for some m and an isomorphism e D † X , Q ⊗ e D ( m ) X , Q M ′ ∼ = M . Lemma. — Assume that X is quasi-compact. For any complex M in D b coh ( e D ( m ) X , Q ) , thereexists an integer m ′ ≥ m such that e D ( m ′ ) X , Q ⊗ e D ( m ) X , Q M is in D b perf ( e D ( m ′ ) X , Q ) . In particular, we havethe canonical equivalence of categories e D b perf ( e D † X , Q ) ∼ −→ D b perf ( e D † X , Q ) ∼ −→ D b coh ( e D † X , Q ) . Proof. Let us see the first claim. Since X is quasi-compact, the problem is local, and we mayassume that X is affine. Since X is affine, we can take M to be a bounded complex suchthat each term is a coherent e D ( m ) X , Q -module. For m ′ ≥ m , we denote M ( m ′ ) := e D ( m ′ ) X , Q ⊗ M and M † := e D † X , Q ⊗ M . Now, there exists a bounded finite locally projective e D † X , Q -complex L anda quasi-isomorphism of complexes ϕ : L → M † since e D † X , Q has finite Tor-dimension by theresult of Noot-Huyghe in [NH2]. For a sufficiently large m ′ , this complex can be descended tolevel m ′ . Namely there exists a bounded locally finite projective e D ( m ) X , Q -complex L ( m ′ ) and ahomomorphism of complexes ϕ ′ : L ( m ′ ) → M ( m ′ ) such that e D † X , Q ⊗ ϕ ′ ∼ = ϕ . The homomorphism ϕ ′ may not be a quasi-isomorphism, but since the complexes are bounded and each term iscoherent, there exists m ′′ ≥ m ′ such that the homomorphism e D ( m ′′ ) X , Q ⊗ ϕ ′ becomes a quasi-isomorphism, which concludes the proof. The latter statement follows from [BeI, 4.2.4]. (cid:4) Remark. — We do not know if e D b perf ( e D † X , Q ) and D b perf ( e D † X , Q ) coincide or not in general. Let f : ( X , Z ) → ( Y , W ) be a morphism of d-couples whose realization issmooth. We assume that X ( s )0 and Y ( s )0 can be lifted to smooth formal schemes X ′ and Y ′ . Letus denote by d ≥ the relative dimension of X over Y . Then there is a canonical equivalence ofcohomological functors from e D b perf ( D † Y , Q ( † W )) to e D b perf ( D † X , Q ( † Z )) with Frobenius isomorphisms (3.10.1) ( D X ,Z ◦ f ! )( d )[2 d ] ∼ −→ f ! ◦ D Y ,W . roof. Let M † be a complex in e D b perf ( e D † Y , Q ). Then by definition, there exists M in D b perf ( e D ( m ) Y , Q )for some m and e D † Y , Q ⊗ e D ( m ) Y , Q M ∼ = M † . First, let us define the homomorphism. By Proposition3.3, we get a homomorphism Hom e D ( m ) Y , Q ( M , M ) ∼ ←− Hom e D ( m ) Y , Q ( b B ( m ) Y , Q , D ( M ) b ⊗ M ). By Lemma 3.4and the functoriality of the extraordinary pull-back functor f ! , we getHom e D ( m ) Y , Q ( b B ( m ) Y , Q , D ( M ) b ⊗ M ) → Hom e D ( m ) X , Q ( f ! b B ( m ) Y , Q , f ! ( D ( M ) b ⊗ M )) . This homomorphism is compatible with Frobenius pull-backs by the functoriality of the isomor-phism [Be2, 3.2.4]. We get f ! b B ( m ) Y , Q ∼ = b B ( m ) X , Q [ d f ]. Moreover, f ! ( D ( M ) b ⊗ M ) ∼ = f ! ◦ D ( M ) b ⊗ f ! M [ − d f ].This isomorphism is also compatible with Frobenius pull-backs. Thus we getHom e D ( m ) X , Q ( f ! b B ( m ) Y , Q , f ! ( D ( M ) b ⊗ M )) ∼ = Hom e D ( m ) X , Q ( b B ( m ) X , Q [ d f ] , f ! D ( M ) b ⊗ f ! M [ − d f ]) . Now, using the proposition once again, we get an isomorphismHom e D ( m ) X , Q ( b B ( m ) X , Q [ d f ] , f ! D ( M ) b ⊗ f ! M [ − d f ]) ∼ −→ Hom e D ( m ) X , Q ( D f ! M , f ! D ( M )[ − d f ]) . Composing all of them, we obtain a homomorphismHom e D ( m ) Y , Q ( M , M ) → Hom e D ( m ) X , Q ( D f ! M , f ! D ( M )[ − d f ]) . The image of the identity is the homomorphism we wanted. By using [BeI, 4.3.3, 4.3.11], we getthe following diagram.Hom e D ( m ) Y ′ , Q ( M , M ) / / F ∗ Y (cid:15) (cid:15) Hom e D ( m ) X ′ , Q ( D f ′ ! ( M ) , f ′ ! D ( M )[ − d f ]) F ∗ X (cid:15) (cid:15) Hom e D ( m ) Y , Q ( F ∗ Y M , F ∗ Y M ) / / Hom e D ( m ) X , Q ( D f ! ( F ∗ Y M ) , f ! D ( F ∗ Y M )[ − d f ])This diagram is commutative up to multiplication by q − d Y · q d X = q d f by the commutativityof Proposition 3.3. Thus we obtain the homomorphism D ◦ f ! ( M )( d f )[2 d f ] → f ! ◦ D ( M ). Bytensoring with e D † X , Q , we get D ◦ f ! ( M † )( d f )[2 d f ] → f ! ◦ D ( M † )by using [BeI, 3.4.6 (iii)] and [Vi, I.5.4]. By construction, this does not depend on the choice of M . It remains to show that this homomorphism is an isomorphism when f is smooth.It suffices to show the equality for M = e D † Y , Q , and we can forget about Frobenius pull-backs.We get R H om e D † X , Q ( e D † X → Y , Q , e D † X , Q )[ d f ] ∼ = e D † Y ← X , Q . Indeed, R H om e D † X , Q ( e D † X → Y , Q , e D † X , Q )[ d f ] ∼ = R H om D † X , Q ( D † X , Q ⊗ Θ • X / Y , D † X , Q ) [ d f ] ∼ = Ω • X / Y ⊗ e D † X , Q ∼ = e D † Y ← X , Q . Thus, we get D ◦ f ! ( e D † Y , Q )[2 d f ] ∼ = D ( e D † X → Y , Q )[ d f ] ∼ = e D † Y ← X , Q ⊗ ω − X [ d f ] ∼ = f ! e D † Y , Q ⊗ ω − Y ∼ = f ! ◦ D ( e D † Y , Q ) . We can see that this isomorphism coincides with the homomorphism we have constructed, andthe theorem follows. (cid:4) emark. — We may be able to weaken the assumption of the theorem. The theorem shouldhold only by assuming that X \ Z → Y \ W is smooth. Moreover, we may be able to see thetheorem as a solution of a part of “Cauchy-Kovalevskaya type problem”. Classically, this obser-vation first appeared in Kashiwara’s thesis [Ka], and the problem was interpreted in terms of thelanguage of D -modules. We expect that the analogous theorem also holds in our setting: if themorphism f is “non-characteristic” to a coherent F - D † X , Q -module, then we get the isomorphism(3.10.1). Let X be a smooth formal scheme. Let d be the dimension of X , and Z be a divisor of the special fiber of X . Then we get a canonical isomorphism D X ,Z ( O X , Q ( † Z )) ∼ −→ O X , Q ( † Z )( − d ) where d denotes the dimension of X .Proof. Apply Theorem 3.10 in the case where Y = Spf( R ). (cid:4) Remark. — The question to calculate D ( O X , Q ) was posed by Caro in [Ca5, 4.3.3], saying that“ En effet, lorsque X = P , on retrouve l’isomorphisme canonique: D P ,T ( O P ( † T ) Q ) → O P ( † T ) Q .Je n’ai pas de contre-exemple mais la compatibilit´e `a Frobenius de ce dernier isomorphisme meparaˆıt inexacte. ” Let X be a smooth formal scheme of dimension d , Z be a divisor of the special fiber X of X . For an overconvergent F -isocrystal M on X , we denote by M ∨ the dual overconvergent F -isocrystal of M . Corollary. — Let sp : X K → X be the specialization map, and let M be an overconvergent F -isocrystal on X \ Z . Then, ( D X ,Z (sp ∗ ( M )) ∼ = sp ∗ ( M ∨ )( − d ) . Proof. Apply Corollary 3.11 to [Ca3, 2.3.37]. (cid:4) Remark. — This corollary completes the comparison of L -functions of isocrystals and arith-metic D -modules [Ca2, 3.3.1]. Namely, we get L ( Y, E, t ) = L ( Y , sp ∗ E, q d X · t )using the notation of loc. cit . However, in loc. cit. the definition of the Frobenius structure ofthe push-forward is modified in order to make the relative duality compatible with Frobenius(cf. [Ca2, 1.2.11]), and the definition may not be the same as that of Berthelot. Still, we willshow that this duality is compatible with Frobenius in the next section (cf. Corollary 4.14), andthe Frobenius structure of the push-forward is in fact the same as that of Berthelot. Now, we will compare the rigid cohomologies and the push-forwards of arithmetic D -modules with Frobenius structure when varieties can be lifted to smooth formal schemes. If wedo not consider Frobenius structure, they coincide up to shifts of degree, which is a result ofBerthelot (cf. [BeI, 4.3.6.3]). If we consider Frobenius structure, we need a Tate twist. Thistwist naturally appears in the philosophy of six functors (cf. paragraph 5.9).Let us fix the notation. We consider the situation 2.1.2. Let X be a smooth formal schemeand let p : X → S := Spf( R ) be the structural morphism. Let X be the special fiber of X as usual, X ′ be a lifting of X ( s )0 , and X K be the Raynaud generic fiber. We denote by25p : X K → X the specialization map of topoi. Let M be an F - D † X , Q -module. We define the“rigid cohomology” of M in the following way. Let p K : X K → Spm( K ) be the structuralmorphism. We define H i rig ( X , M ) := R i p K ∗ (Ω • X K ⊗ O X K sp ∗ M ) . We define the Frobenius structure in the following way. There exists an isomorphism ϕ : Ω • X ′ K ⊗ O X ′ M σ ∼ −→ Ω • X K ⊗ O X F ∗ M σ ∼ −→ Ω • X K ⊗ O X M in D + ( X ) where the first isomorphism follows from [Be2, 4.3.5] and we used the Frobeniusstructure of M in the second isomorphism. Thus, we get an isomorphism H i rig ( X , M ) ∼ −→ H i rig ( X ( s )0 , M σ ) ∼ −→ ϕ H i rig ( X , M ) , where the first isomorphism is the base change homomorphism. This is the induced Frobeniusstructure on the cohomology.Suppose moreover that X is smooth proper and let Z be a divisor of X . When M is anoverconvergent F -isocrystal on X \ Z along Z , the rigid cohomology of sp ∗ ( M ) is isomorphicto the usual rigid cohomology of M . We preserve the notation. We suppose that X is purely of dimension d .Let M be a coherent F - D † X , Q -module. Then we get H i p + M ∼ = H i + d rig ( X , M )( d ) . Proof. Let M ′ be a coherent D † X ′ , Q -module, and consider the following diagram of complexesin D b ( X ). F ∗ (Ω • X ′ ⊗ O X ′ M ′ )[ d ] / / (cid:15) (cid:15) F ∗ D † S ′ ← X ′ , Q ⊗ L D † X ′ , Q M ′ ∼ = F ∗ ( ω X ′ ⊗ L D † X ′ , Q M ′ ) (cid:15) (cid:15) Ω • X ⊗ O X F ∗ M ′ [ d ] / / D † S ← X , Q ⊗ L D † X , Q F ∗ M ′ ∼ = ω X ⊗ L D † X , Q F ∗ M ′ where the horizontal arrows are induced by [BeI, 4.2.1.1]. We need to see that this diagramis commutative up to multiplication by q d . Indeed, this diagram is nothing but Lemma 3.6 bytaking into account the proof of [Be2, 4.3.5].Now to know the Frobenius actions on the cohomologies, apply M ′ to be M σ . We only needto take Rp ∗ to the four sheaves in the diagram with this M ′ , and we get the theorem. (cid:4) (i) We can also compare in the relative situations. Namely, when we aregiven a smooth morphism of smooth formal schemes X → Y , we are able to compare therelative rigid cohomology and the push-forward as arithmetic D -module. Required methods areexactly the same, so we leave the precise formulation and calculation to the readers.(ii) In [NH1, 6.2], Noot-Huyghe cited a calculation of Baldassarri-Berthelot [BB]. However,the definition of Frobenius structures in [NH1] and [BB] are not the same, and we need a Tatetwist here. Precisely, Noot-Huyghe used cohomological functors of the arithmetic D -moduletheory to define the Frobenius structure. On the other hand Baldassarri-Berthelot used therelative rigid cohomologies to define the Frobenius structure on the Fourier transform. Thus weneed to add the Tate twist ( N ) on the right hand side of the isomorphism in [NH1, 6.2], namely F π ( O † Y, Q )[2 − N ] ∼ = H † NX ( O † Y ∨ , Q )( N )26sing the notation of loc. cit .(iii) D. Caro pointed out to the author in personal communications that we need a suitableTate twist in [Ca4, Proposition 2.3.12]. If we put this Tate twist, this proposition can be seen asa generalization of our theorem. He also pointed out that some modifications might be neededin loc. cit. Theorem 3.3.4, in whose proof he used the proposition. 4. Complementary results In this section, we will prove three complementary results; 1) commutation of the dual functorand the tensor product, 2) the K¨unneth formula, and 3) the compatibility of the relative dualityisomorphism by Virrion with Frobenius. The first commutation result is another application ofTheorem 3.14, and the proofs of 2) and 3) are independent from the other part of this paper.Although the K¨unneth formula for arithmetic D -modules seems to be well-known to experts, wecould not find any appropriate reference. We think that this would be a good occasion to includethe proof. The compatibility of relative duality is needed to establish the Poincar´e duality. Commutation of the dual functor and tensor product4.1. We consider the situation 2.1.1. Let X be a smooth scheme over Spec( R i ) for some i , andlet Z be a divisor. We put b B ( m ) X , Q := b B ( m ) X , Q ( Z ), and e D ( m ) X , Q := b B ( m ) X , Q b ⊗ O X b D ( m ) X . We denote thedual functor with respect to e D ( m ) X , Q by D . First, we get the following lemma. Lemma. — Let M be a complex in D b perf ( e D ( m ) X , Q ) , and N be a coherent e D ( m ) X , Q -module whichis also coherent as a b B ( m ) X , Q -module. We denote the dimension of X by d . Then, we have thefollowing isomorphism κ : D ( M ) ⊗ L b B ( m ) X , Q N ∼ = R H om e D ( m ) X , Q ( M , e D ( m ) X , Q ⊗ L b B ( m ) X , Q N ) ⊗ ω − X [ d ] of complexes in D b ( e D ( m ) X , Q ) . Here, the right module structure of e D ( m ) X , Q ⊗ L b B ( m ) X , Q N is defined bythat of e D ( m ) X , Q , and the left structure by [Be2, 1.1.7] . Moreover, this isomorphism is compatiblewith Frobenius.Proof. By [Be1, 4.4.2], N is a locally projective b B ( m ) X , Q -module, and we do not need to takethe derived tensor products. Let M ′ be a right e D ( m ) X , Q -module. Then M ′ ⊗ b B ( m ) X , Q N possessesa right e D ( m ) X , Q -module structure. Indeed let U be an affine open formal subscheme of X . Let M ′ := Γ( U , M ′ ), N := Γ( U , N ), B := Γ( U , b B ( m ) X , Q ), and D := Γ( U , e D ( m ) X , Q ). Then it sufficesto define a right D -module structure on M ′ ⊗ B N . For a ⊗ b ∈ M ′ ⊗ B N and P ∈ D , it sufficesto define ( a ⊗ b ) · P . Take S ⊂ M ′ to be the finite Γ( U , e D ( m ) X , Q )-submodule generated by a . Then S ⊗ N is naturally a D -module considering [Be1, 4.4.7] and [Be2, 1.1.7]. This defines ( a ⊗ b ) · P .Now back to the proof, by using [Vi, I, 1.2.2], we get an isomorphism κ of complexes in D b ( b B ( m ) X , Q ). It suffices to show that this isomorphism is an isomorphism of e D ( m ) X , Q -complexes. Fora bi- e D ( m ) X , Q -module I , we have a canonical homomorphism H om e D ( m ) X , Q ( M , I ) ⊗ b B ( m ) X , Q N → H om e D ( m ) X , Q ( M , I ⊗ b B ( m ) X , Q N ) . priori , this is a homomorphism of b B ( m ) X , Q -modules. By the argument above, both sides of thehomomorphism possess the right e D ( m ) X , Q -module structures. To finish the proof, is suffices to seethat the homomorphism is e D ( m ) X , Q -linear. The verification is straightforward. (cid:4) Now, consider the situation 2.1.2. Let X be a smooth formal scheme over Spf( R ), and Z be a divisor of the special fiber. Let N ′ be a coherent F - D † X , Q ( † Z )-module which is also coherentas an O X , Q ( † Z )-module. By abuse of language, we say that N ′ is a convergent F -isocrystaloverconvergent along Z (cf. Notation 0.5). We put N ′ := sp ∗ ( N ′ ), which is an overconvergent F -isocrystal in the usual sense. Then thanks to Corollary 3.12, we get the following isomorphismsof bimodules. We omit subscripts Q and denote by ( Z ) instead of ( † Z ) in the next equality tosave the space. R H om O X ( Z ) ( D X ,Z ( N ′ ) , D † X ( Z ))(4.2.1) ∼ = D † X ( Z ) ⊗ O X ( Z ) R H om O X ( Z ) (sp ∗ ( N ′∨ )( − d ) , O X ( Z )) ∼ = D † X ( Z ) L ⊗ †O X ( Z ) N ′ ( d )Here the right module structure of the first module is defined by [Be2, 1.1.7] using the rightmodule structure of D † X , Q ( † Z ), and the left structure by using that of D † X , Q ( † Z ). This iscompatible with Frobenius, which means that the following diagram of canonical isomorphismsis commutative. We again omit Q and denote by ( Z ) in the following. F ♭ R H om O X ′ ( Z ) ( D X ,Z ( N ′ ) , D † X ′ ( Z )) (cid:15) (cid:15) / / F ♭ (cid:0) D † X ′ ( Z ) L ⊗ †O X ′ ( Z ) N ′ ( d ) (cid:1) (cid:15) (cid:15) R H om O X ( Z ) ( D X ,Z ( F ∗ N ′ ) , F ♭ D † X ′ ( Z )) / / F ♭ D † X ′ ( Z ) L ⊗ †O X ( Z ) F ∗ N ′ ( d )Here the homomorphisms are the canonical ones except for the right vertical homomorphism,which is q − d times the canonical homomorphism.Let M be an object in F - D b perf ( D † X , Q ( † Z )). We get the following isomorphisms compatiblewith Frobenius structures: D X ,Z ( M ) L ⊗ †O X , Q ( † Z ) N ′ ∼ = R H om D † X , Q ( † Z ) ( M , D † X , Q ( † Z ) L ⊗ †O X , Q ( † Z ) N ′ ) ⊗ ω − X [ d ](4.2.2) ∼ = R H om D † X , Q ( † Z ) (cid:0) M , R H om O X , Q ( † Z ) ( D X ,Z ( N ′ ) , D † X , Q ( † Z )) (cid:1) ⊗ ω − X ( − d )[ d ] ∼ = R H om D † X , Q ( † Z ) (cid:0) M L ⊗ †O X , Q ( † Z ) D X ,Z ( N ′ ) , D † X , Q ( † Z ) (cid:1) ⊗ ω − X ( − d )[ d ] ∼ = D X ,Z ( M L ⊗ †O X , Q ( † Z ) D X ,Z ( N ′ ))( − d )where the first isomorphism by Lemma 4.1, the second by (4.2.1), and the third by using [Ca3,2.1.34]. Now, we get the following proposition. Let X be a smooth formal scheme over Spf( R ) , and Z be a divisorof its special fiber. Let M be a coherent F - D † X , Q ( † Z ) -module, and N be an overconvergent F -isocrystal along Z . Then we get (4.3.1) ( D X ,Z ( M ) L ⊗ †O X , Q ( † Z ) D X ,Z ( N ))( d ) ∼ = D X ,Z ( M L ⊗ †O X , Q ( † Z ) N ) which is compatible with Frobenius structures. roof. Take N ′ := D X ,Z ( N ), and the isomorphism (4.2.2) induces the isomorphism we arelooking for. (cid:4) Remark. — We are not able to expect the isomorphism (4.3.1) in general. For example,consider the closed immersion i : { } ֒ → b A R . Then taking M = N = i + K , we do not have suchan isomorphism. The K¨unneth formula4.4. Now, we will show the K¨unneth formula. The ideas used to show the formula is due to P.Berthelot. Let S be a Z ( p ) -scheme, and X , Y , T be smooth formal schemes over S . Considerthe following commutative diagram:(4.4.1) Z q Y ' ' NNNNNNN q X w w ppppppp p (cid:15) (cid:15) X p X ' ' NNNNNNN Y p Y w w ppppppp T where Z = X × T Y . Suppose that p X and p Y are smooth. Let B T , (resp. B X , B Y ) be a commutative O T -algebra (resp. q ∗ X B T -algebra, q ∗ Y B Y -algebra)endowed with an action of D ( m ) T (resp. D ( m ) X , D ( m ) Y ) compatible with that of O T (resp. O X , O Y ). For a B X -module F and a B Y -module G , we put F ⊠ B T G := q ∗ X F ⊗ p ∗ B T q ∗ Y G . Let B Z := B X ⊠ B T B Y . We put e D ( m ) ∗ := B ∗ ⊗ D ( m ) ∗ where ∗ ∈ { X, Y, Z } . We note that p ∗ e D ( m ) T is asub- O Z -algebra of e D ( m ) Z . We get the following lemma. Lemma. — We preserve the notation.(i) There exists the canonical isomorphism e D ( m ) Z ∼ = q ∗ X e D ( m ) X ⊗ p ∗ e D ( m ) T q ∗ Y e D ( m ) Y . (ii) There exists the canonical isomorphism e D ( m ) T ← Z ∼ = q ∗ X e D ( m ) T ← X ⊗ p ∗ e D ( m ) T q ∗ Y e D ( m ) T ← Y . Proof. The natural homomorphisms q ∗ X e D ( m ) X → e D ( m ) Z and q ∗ Y e D ( m ) Y → e D ( m ) Z induces the homo-morphism q ∗ X e D ( m ) X ⊗ p ∗ e D ( m ) T q ∗ Y e D ( m ) Y → e D ( m ) Z . To see that this is an isomorphism, we may assumethat T possesses a system of local coordinate and X and Y possesses a system of local coordinateover T . Then proof is straightforward, and we leave the reader for the detail. Let us see (ii).We know the following isomorphisms ω Z/T ∼ = q ∗ X ω X/T ⊗ O Z q ∗ Y ω Y/T . We get e D ( m ) T ← Z ∼ = p ∗ e D ( m ) T ⊗ O Z ω Z/T ∼ = (cid:16) q ∗ X p ∗ X e D ( m ) T ⊗ p ∗ e D ( m ) T q ∗ Y p ∗ Y e D ( m ) T (cid:17) ⊗ O Z ( q ∗ X ω X/T ⊗ q ∗ Y ω Y/T ) ∼ = q ∗ X e D ( m ) T ← X ⊗ p ∗ e D ( m ) T q ∗ Y e D ( m ) T ← Y . (cid:4) .6 Lemma. — Let M be a left flat e D ( m ) X -module, and N be a left flat e D ( m ) Y -module. Thenwe get that M ⊠ B T N is a flat left e D ( m ) Z -module.Proof. In the case where T = Spf( R ), the verification is left to the readers. To see the lemma,since the verification is local, we may assume that T is affine. Let i : Z = X × T Y ֒ → W := X × Y be the canonical inclusion. Since T is separated, this is a closed immersion. By the T = Spf( R )case, we get that M ⊠ S N is a flat e D ( m ) W -module. We put e D ( m ) Z → W := i ∗ e D ( m ) W , which is a( e D ( m ) Z , i − e D ( m ) W )-module as usual. Let F be a right e D ( m ) Z -module. Then we get F ⊗ e D ( m ) Z ( M ⊠ B T N ) ∼ = ( F ⊗ e D ( m ) Z e D ( m ) Z → W ) ⊗ i − e D ( m ) W i − ( M ⊠ S N ) . Since e D ( m ) Z → W is flat over e D ( m ) Z and M ⊠ S N is flat over e D ( m ) W , we get the lemma. (cid:4) Using this preparation, we get the following K¨unneth formula. We preserve the notation. Let M (resp. N ) be a complex in D − qc ( e D ( m ) X ) (resp. D − qc ( e D ( m ) Y ) ). Then we get a canonical isomorphism in D − qc ( e D ( m ) T )(4.7.1) p + ( M ⊠ L B T N ) ∼ = p X + ( M ) ⊗ L B T p Y + ( N ) . Proof. Let F (resp. G ) be a quasi-coherent e D ( m ) X -module (resp. e D ( m ) Y -module). Then by Lemma4.5, we get a canonical isomorphism of p − e D ( m ) T -modules(4.7.2) e D ( m ) T ← Z ⊗ e D ( m ) Z ( F ⊠ B T G ) ∼ = ( e D ( m ) T ← X ⊗ e D ( m ) X F ) ⊠ B T ( e D ( m ) T ← Y ⊗ e D ( m ) Y G ) . Let L • be a flat resolution of F as a e D ( m ) X -module, and M • be a flat resolution of G as a e D ( m ) Y -module. Then we get that L i ⊠ B T M j is a flat e D ( m ) Z -module for any i and j by Lemma4.6. Thus we get e D ( m ) T ← Z ⊗ L e D ( m ) Z ( F ⊠ L B T G ) ∼ = e D ( m ) T ← Z ⊗ e D ( m ) Z ( L • ⊠ L B T M • ) ∼ = ( e D ( m ) T ← X ⊗ e D ( m ) X L • ) ⊠ B T ( e D ( m ) T ← Y ⊗ e D ( m ) Y M • ) ∼ = ( e D ( m ) T ← X ⊗ L e D ( m ) X F ) ⊠ L B T ( e D ( m ) T ← Y ⊗ L e D ( m ) Y G ) . By using the K¨unneth formula for quasi-coherent sheaves, we get the proposition. (cid:4) Now, let us consider Frobenius. Suppose T is endowed with a quasi-coherent m -PD-ideal( a , b , α ), and p ∈ a . With this hypothesis, we are able to consider Frobenius pull-back even ifthere are no liftings of relative Frobenius morphisms. Lemma. — Suppose that X and Y be the reductions of X and Y respectively, and X ′ , Y ′ be liftings of X ( s )0 , Y ( s )0 . We take Z ′ := X ′ × T Y ′ . Then the isomorphism (4.7.1) is compatiblewith Frobenius isomorphisms. Moreover, if a is m -PD-nilpotent, it is compatible with Frobeniusisomorphism even if there are no liftings.Proof. The verification uses only standard arguments, so we leave the details to the readers. (cid:4) .9. Finally, by taking inverse limit and inductive limit, we get the following K¨unneth formulafor D † -modules. Proposition. — Consider the following diagram Z q Y ' ' OOOOOOO q X w w ooooooo p (cid:15) (cid:15) X p X ' ' OOOOOOO Y p Y w w ooooooo T where T is a smooth formal scheme, p X and p Y are smooth, and Z := X × T Y . Let D be a divisor of the special fiber of T , D X (resp. D Y ) be a divisor of the special fiber of X (resp. Y ) such that D X ⊃ p − X ( D ) (resp. D Y ⊃ p − Y ( D ) ). Let M (resp. N ) be a complex in LD −→ b Q , qc ( b D ( • ) X ( D X )) (resp. LD −→ b Q , qc ( b D ( • ) Y ( D Y )) ). Then we get the canonical isomorphism p + ( M L ⊠ †O T ( † D ) N ) ∼ = p X + ( M ) L ⊗ †O T ( † D ) p Y + ( N ) in LD −→ b Q , qc ( b D ( • ) T ( D )) . This isomorphism is compatible with the Frobenius isomorphisms. To get the proposition directly, we can also proceed as follows. Considerthe following cartesian diagram. X × T Y i / / p (cid:15) (cid:15) (cid:3) X × Y q (cid:15) (cid:15) T i T / / T × T We see easily that q + ( M ⊠ L N ) ∼ = q + ( M ) ⊠ L O T × T q + ( N ). Let S := Spf( R ). By using thisand [Ca1, 2.1.9], we get i T + p + i ! ( M L ⊠ † S N ) ∼ = i T + i ! T ( q X + M L ⊠ † S q Y + N ) ∼ = i T + ( q X + M L ⊗ †O T ( † D ) q Y + N ) . Taking H i ! T , we get what we want. Compatibility of Frobenius pull-backs with relative duality4.11. We will show that the relative duality homomorphism by Virrion is compatible withFrobenius pull-back.First, let us fix the situation. We consider the situation 2.1.1. Let f : X → Y be a proper morphism of smooth formal schemes, and W be a divisor of the special fiber of Y such that Z := f − ( W ) is a divisor. Under this situation, Virrion [Vi2] defined the trace homomorphism(4.11.1) Tr + ,f : f + ω X , Q [ d X ] → ω Y , Q [ d Y ]where d X and d Y denotes the dimension of X and Y . Using the trace map, for an object E in D b perf ( D † X , Q ( † Z )), she also constructed the relative duality isomorphism(4.11.2) χ : D Y ,W f + ( E ) ∼ −→ f + D X ,Z ( E )in D b perf ( D † Y , Q ( † W )) (see also [Ca2, 1.2.7]). Now, we assume that there exist liftings X ′ and Y ′ of X ( s )0 and Y ( s )0 where X and Y are special fibers as usual. We also assume that thereexists a lifting f ′ : X ′ → Y ′ of the morphism X ( s )0 → Y ( s )0 induced by f . These assumptionsautomatically hold when we consider the situation 2.1.2.31 .12. Before stating the theorem, we will prepare a commutative diagram, which is needed inthe proof of the compatibility. We freely use the notation of [Ha]. Let X , X ′ , Y , Y ′ be locallynoetherian schemes. Suppose we are given the following commutative diagram. X ′ f ′ / / u ′ (cid:15) (cid:15) Y ′ u (cid:15) (cid:15) X f / / Y We assume that all the morphisms are proper, u is finite flat, and all the schemes admit dualizingcomplexes (cf. [Ha, V, § u ∗ f ′∗ f ′△ u △ Tr f ′ / / Tr u ′ (cid:15) (cid:15) u ∗ u △ Tr u (cid:15) (cid:15) f ∗ f △ Tr f / / idHere, Tr u ′ denotes the composition u ∗ f ′∗ f ′△ u △ ∼ = f ∗ u ′∗ u ′△ f △ f ∗ ◦ Tr u ′ ◦ f △ −−−−−−−→ f ∗ f △ . We remind thatthis diagram consists of homomorphisms in the category of complexes by [Ha, VII, 2.1]. Let u · denotes the functor u ′ in [Ha, VI, 4.1] to avoid confusions with the morphism u ′ . We note that u · ∼ = u ♭ ∼ = u ! in the derived category since u is finite and flat. We have the canonical homomor-phism id → u · u ∗ . Let c : f ′∗ f ′△ u △ → u · u ∗ f ′∗ f ′△ u △ → u · f ∗ f △ where the second morphism is thatinduced by the left vertical morphism Tr u ′ in the diagram above. Let c ′ : u △ → u · u ∗ u △ → u · where the second homomorphism is the trace map. Taking u · to the above diagram, we get thefollowing commutative diagram of complexes.(4.12.1) f ′∗ f ′△ u △ Tr f ′ ◦ u △ / / c (cid:15) (cid:15) u △ c ′ (cid:15) (cid:15) u · f ∗ f △ u · ◦ Tr f / / u · The morphism id → u · u ∗ ∼ = u ♭ u ∗ is nothing but the adjunction homomorphism, and c ′ is theidentity in the derived category. We preserve the assumptions and notations of paragraph . Then therelative duality isomorphism (4.11.2) is compatible with Frobenius pull-back.Proof. Recall that d X (resp. d Y ) was the dimension of X (resp. Y ). We let d to be the relativedimension, namely d := d X − d Y . We denote by f ( m )+ the push-forward of level m . We will use thepush-forward for right modules as in [Vi2, III]: recall that f ( m )+ ( M ) := Rf ∗ ( M ⊗ L b D ( m ) X , Q b D ( m ) X → Y , Q )for a coherent right b D ( m ) X , Q -module M . It suffices to show that the trace homomorphism (4.11.1)is compatible with Frobenius pull-backs by [Ca6, 1.5]. By the result of Caro [Ca6, 2.2.7], weknow the compatibility in the case where f is a closed immersion. Thus, we are reduced toshowing the case where f is smooth using the standard factorization X → X × Y → Y andthe transitivity of the trace map [Vi2, III, 5.5]. In the following, we assume f to be smooth.Let i be a non-negative integer, and X and Y be the reductions of X and Y over R i . Itsuffices to show the compatibility for these X and Y , namely we need to prove the commutativity32f the following diagram. f ( m + s )+ ω X [ d X ] Tr + ,f / / (cid:15) (cid:15) ω Y [ d Y ] (cid:15) (cid:15) F ♭Y f ′ ( m )+ ω X ′ [ d X ] F ♭Y Tr + ,f ′ / / F ♭Y ω Y ′ [ d Y ]Since f is smooth and the relative dimension is d , we get that for any point y in Y , the dimensionof f − ( y ) is equal to d . Since f is proper, R i f ∗ ( F ) = 0 for any quasi-coherent sheaf F and i > d (cf. for example [Kl, I Lemma (3)]). Thus, by the definition of f + , the canonical homomorphism R d f ∗ ω X ։ H d f ( m )+ ω X is surjective, and H i f ( m )+ ω X = 0 for i > d , in other words H i ( f ( m )+ ω X , Q [ d X ]) = 0 for i > − d Y .Thus, we get Hom( f ( m )+ ω X [ d X ] , ω Y [ d Y ]) ∼ = Hom( H d f ( m )+ ω X , ω Y ) . This implies that to give the trace map f ( m )+ ω X [ d X ] → ω Y [ d Y ] is equivalent to giving thehomomorphism H d f ( m )+ ω X → ω Y . We can retrieve the trace map by the composition f ( m )+ ω X [ d X ] → H d f ( m )+ ω X [ d Y ] → ω Y [ d Y ] . Note that we have the following commutative diagram by [Vi2, III, 5.4]. R d f ∗ ω X / / Tr f HHHHHHHHH H d f ( m )+ ω X Tr + ,f y y ttttttttt ω Y It suffices to see that the following diagram is commutative. H d f ( m + s )+ ω X / / (cid:15) (cid:15) ω Y (cid:15) (cid:15) F ♭Y H d f ′ ( m )+ ω X ′ / / F ♭Y ω Y ′ This shows that the problem is local with respect to Y . Assume that there exist liftings F X : X → X ′ and F Y : Y → Y ′ of relative Frobenius morphisms such that the two morphisms f ′ ◦ F X , F Y ◦ f : X → Y ′ coincide. Under this particular situation, the theorem is reduced to showing thefollowing diagram R d f ∗ ω X Tr f / / (cid:15) (cid:15) ω Y (cid:15) (cid:15) F ♭Y R d f ′∗ ω X ′ F ♭Y Tr f ′ / / F ♭Y ω Y ′ is commutative, where the left vertical homomorphism is the base change homomorphism. Thisis nothing but (4.12.1). In particular, the theorem holds in the case where f is finite ´etale.Let x be a point of X (which may not be closed). For a right D ( m ) X -module F and an integer i , we put H ( m ) ,ix ( F ) := lim −→ x ∈ U j U ∗ R i Γ ( m ) U ∩{ x } ( F )33here U runs over open neighborhoods of x , j U : U ֒ → X is the inclusion, and Γ ( m ) denotes thelevel m local cohomology functor defined in [BeI, 4.4.4]. We note that(4.13.1) lim −→ m ′ H ( m ′ ) ,ix ( F ) ∼ = H ix ( F )by [Be1, 1.5.4], and H ix ( ω X ) can be seen as a quasi-coherent right D ( m ) X -module. The Frobeniusisomorphism H ( m + s ) ,ix ( ω X ) ∼ −→ F ♭X H ( m ) ,ix ( ω X ′ ) induces the isomorphism φ ix : H ix ( ω X ) ∼ −→ F ♭X H ix ( ω X ′ ) . This induces the following isomorphism of Cousin complexes (cf. [Ha, IV, § ω X / / ∼ (cid:15) (cid:15) H Z /Z ( ω X ) / / ∼ P φ x (cid:15) (cid:15) H Z /Z ( ω X ) / / ∼ P φ x (cid:15) (cid:15) . . .F ♭X ω X ′ / / F ♭X H Z /Z ( ω X ′ ) / / F ♭X H Z /Z ( ω X ′ ) / / . . . We denote by C • X the Cousin complex of ω X .Let y be a point of codimension i in Y , and let x be a closed point of the fiber f − ( y ) in X . Forclosed subsets of schemes, let us endow with the reduced induced scheme structure. Then since f is smooth, there exists an open subscheme U of X , such that f ′ : Z := { x }∩ U → W := { y }∩ V is finite ´etale where V := f ( U ), and W is smooth. Consider the following commutative diagram. Z i / / f ′ (cid:15) (cid:15) U f (cid:15) (cid:15) W i ′ / / V The trace map Tr + ,f ′ : f ′ ( m )+ ω Z → ω W can be identified with the usual trace map Tr f ′ by theisomorphism f ′∗ ∼ = f ′ + since f ′ is finite ´etale. Since H ( m ) ,iZ ( ω X ) ∼ = i ( m )+ ( ω Z ) and H ( m ) ,iW ( ω Y ) ∼ = i ′ ( m )+ ( ω W ), the functor i ′ ( m )+ ◦ Tr + ,f ′ induces the homomorphism(4.13.2) H f ( m )+ H ( m ) ,d + ix ( ω X ) → H ( m ) ,iy ( ω Y )by taking inductive limit over V . Note that since f ′ is finite ´etale, this trace map is compatiblewith Frobenius by the result of the first part of this proof. By taking the inductive to (4.13.2)over m and using the identification (4.13.1), we get a homomorphismTr ix : H f ( m )+ H d + ix ( ω X ) ∼ = lim −→ m ′ H f ( m )+ H ( m ′ ) ,d + ix ( ω X )(4.13.3) → lim −→ m ′ H f ( m ′ )+ H ( m ′ ) ,d + ix ( ω X ) ( ) −−−−−→ lim −→ m ′ H ( m ′ ) ,iy ( ω Y ) ∼ = H iy ( ω Y ) . This homomorphism is compatible with Frobenius as well since (4.13.2) is. The composition(4.13.4) f ∗ H d + ix ( ω X ) → H f ( m )+ H d + ix ( ω X ) Tr ix −−→ H iy ( ω Y )is the usual trace map by construction. 34here exists the surjection D ( m ) X → D ( m ) X → Y sending 1 to 1 ⊗ 1. This gives us a flat resolution L • → D ( m ) X → Y such that L := D ( m ) X and L i = 0 for i > 0. The double complex f ∗ ( C • X ⊗ D ( m ) X L • )induces the spectral sequence E a,b = f ∗ H − b ( H aZ a /Z a +1 ( ω X ) ⊗ L D ( m ) X → Y ) ⇒ H a + b f ( m )+ ω X . Note that E a,b ∼ = H − b f ( m )+ H aZ a /Z a +1 ( ω X ). The trace map (4.13.4) E d + i, → H iZ i /Z i +1 ( ω Y )induces the homomorphism of complexes f ∗ ( C d + iX ⊗ L • ) → H iZ i /Z i +1 ( ω Y ) = C iY , and this inducesthe homomorphism of double complexes(4.13.5) f ∗ ( C d + • X ⊗ L • ) → C • Y . This homomorphism defines the homomorphism γ : f + ω X [ d X ] → ω Y [ d Y ]. Let us show that γ = Tr + ,f . The canonical homomorphism C • X → C • X ⊗ L induces the homomorphism ofdouble complexes(4.13.6) f ∗ C • X → f ∗ ( C d + • X ⊗ L • ) . Let I E a,b := ( f ∗ C aX if b = 00 if b = 0 II E a + d,b := ( C aY if b = 00 if b = 0 . Then we get the trivial spectral sequences I E a,b ⇒ R a + b f ∗ ω X and II E a,b ⇒ II E n where II E d := ω Y and 0 otherwise. The homomorphisms (4.13.5) and (4.13.6) induce the homomorphisms I E a,b → E a,b → II E a,b of spectral sequences. We get the following homomorphisms of complexesof E -terms of these spectral sequences.(4.13.7) f ∗ H d − Z d − /Z d ( ω X ) / / (cid:15) (cid:15) f ∗ H dZ d /Z d +1 ( ω X ) / / (cid:15) (cid:15) f ∗ H d +1 Z d +1 /Z d +2 ( ω X ) (cid:15) (cid:15) / / . . . Rf ∗ ω X (cid:15) (cid:15) E d − , / / (cid:15) (cid:15) E d, / / P Tr x (cid:15) (cid:15) E d +1 , P Tr x (cid:15) (cid:15) / / . . . f ( m )+ ω Xγ (cid:15) (cid:15) / / H Z /Z ( ω Y ) / / H Z /Z ( ω Y ) / / . . . ω Y [ − d ]Here the right homomorphisms are the homomorphisms of complexes of corresponding spectralsequences. To show that γ is the trace map, it suffices to show that H − d Y ( γ ) is the trace map.Consider the homomorphisms R d f ∗ ω X → H d f ( m )+ ω X H d γ −−−→ ω Y induced by the E d -terms of the homomorphism of the spectral sequences. The composition isthe usual trace map Tr f since (4.13.4) is (cf. [Ha, VI, 4.2] for the construction of the classicaltrace map). Moreover the first homomorphism is the natural map which is surjective. Thus thesecond homomorphism is nothing but the trace map of Virrion.Since it suffices to see the Frobenius compatibility for H d γ , this shows that it suffices to showthe Frobenius compatibility for the lower homomorphism of complexes of the diagram (4.13.7).Thus it is reduced to showing the Frobenius compatibility for the homomorphism X Tr ix : E d + i, → H iZ i /Z i +1 ( ω Y )for any i . It is enough to show the compatibility for Tr ix for each x ∈ X and i , which we havealready verified at (4.13.3). (cid:4) .14. We preserve the assumptions and notations from paragraph 4.11. Let M ′ be an objectin D b coh ( D † X ′ , Q ( † Z )), and N ′ be an object in D b coh ( D † Y ′ , Q ( † W )). We put M := F ∗ X M and N := F ∗ X N ′ . Since F ∗ Y induces an equivalence between D b coh ( D † Y ′ , Q ( † W )) and D b coh ( D † Y , Q ( † W )),we get R Hom D † Y , Q ( † W ) ( f + ( M ) , N ) ∼ −→ R Hom D † Y , Q ( † W ) ( F ∗ Y f + ( M ′ ) , F ∗ Y N ′ ) ∼ −→ R Hom D † Y ′ , Q ( † W ) ( f ′ + ( M ′ ) , N ′ )where the first isomorphism is induced by the isomorphism of functors F ∗ Y ◦ f ′ + ∼ = f + ◦ F ∗ X . Inthe same way, we get an isomorphism R Hom D † X , Q ( † Z ) ( M , f ! N ) ∼ −→ R Hom D † X , Q ( † Z ) ( M ′ , f ! N ′ ). Corollary. — We preserve the assumptions and notations. The adjoint isomorphism is com-patible with Frobenius, in other words, the following diagram is commutative. R Hom D † Y , Q ( † W ) ( f + ( M ) , N ) (cid:15) (cid:15) / / R Hom D † X , Q ( † Z ) ( M , f ! N ) (cid:15) (cid:15) R Hom D † Y , Q ( † W ) ( f ′ + ( M ′ ) , N ′ ) / / R Hom D † X , Q ( † Z ) ( M ′ , f ! N ′ ) Here the horizontal isomorphism are the adjoint formula isomorphism [Vi, IV, 4.2] , and thevertical isomorphisms are those we have just defined.Proof. We only need to check the compatibility with Frobenius of the isomorphisms used in theproof of [Vi2, IV, 4.1]. The compatibility of the isomorphism [Vi2, IV, 1.1 (i)] is nothing but[Ca3, 2.1.19]. The compatibility of [Vi2, IV, 3.4] is Theorem 4.13. (cid:4) 5. Cohomological operations in arithmetic D -modules In this last section, we will collect results on six operations in the theory of arithmetic D -modules with Frobenius structures in the liftable case. Before starting, recall the notations andterminologies of paragraph 3.1.In this section, any (formal) scheme is assumed to be of finite type over its basis. In this section, we consider the situation 2.1.1 if we do not consider the Frobenius structure,and 2.1.2 if we use modules with Frobenius structure. Let f : ( X , Z ) → ( Y , W ) be a morphismof d-couples. We put Z ′ := f − ( Z ) ⊂ Z , which is a divisor by the definition of morphisms ofd-couples. For a coherent ( F -) D † Y , Q ( † W )-complex M , recall that f ! M := D † X , Q ( † Z ) ⊗ D † X , Q ( † Z ′ ) f !0 ( M )in ( F -) LD −→ b Q , qc ( b D ( • ) X ( Z )) (cf. paragraph 3.1). Also recall that we denote by D X ,Z the dual functorwith respect to ( F -) D † X , Q ( † Z )-modules. Let M be a coherent ( F -) D † Y , Q ( † W )-module (or perfectcomplex), and suppose that f ! ◦ D Y ,W ( M ) is a perfect complex. Then we put f + ( M ) := ( D X ,Z ◦ f ! ◦ D Y ,W )( M )in ( F -) D b coh ( D † X , Q ( † Z )). When the realization of f is smooth, this functor is defined for anyperfect ( F -) D † Y , Q ( † W )-complexes by [BeI, 4.3.3]. If Berthelot’s conjecture (cf. [BeI, 5.3.6]) isvalid, this functor is defined for any holonomic F - D † Y , Q ( † W )-complexes.36 .2. Let f : ( X , Z ) → ( Y , W ) be a morphism of d-couples such that the realization is proper.Let Z ′ := f − ( W ) ⊂ Z . We denote by f ,Z ′ , + the proper push-forward from ( F -) LD −→ b Q , qc ( b D ( • ) X ( Z ′ ))to ( F -) LD −→ b Q , qc ( b D ( • ) Y ( W )). Let M be a coherent D † X , Q ( † Z )-complex. We denote by j + M theunderlying D † X , Q ( † Z ′ )-complex of M . We define f + ( M ) := f ,Z ′ , + ( j + M )in LD −→ b Q , qc ( b D ( • ) Y ( W )). Let M be a perfect ( F -) D † X , Q ( † Z )-complex such that D X ,Z ( M ) is acoherent ( F -) D † X , Q ( † Z ′ )-complex. In this case, we say that M is f ! -admissible. Then we define f ! ( M ) := ( D Y ,W ◦ f + ◦ D X ,Z )( M )in ( F -) D b coh ( D † Y , Q ( † W )). When Z ′ = Z , any perfect complex is f ! -admissible. If Berthelot’sconjecture is valid, any holonomic module is f ! -admissible. Another example we have in mind isthe geometric Fourier transform [NH1] (see also Definition A.2.3) or that with compact support(cf. Definition A.2.9). Let ( X , Z ) be a d-couple. For coherent D † X , Q ( † Z )-modules M and N , we denote M L ⊗ †O X , Q ( † Z ) N simply by M L ⊗ † N . This is an object in LD −→ b Q , qc ( b D ( • ) X ( Z )). Now, let f bea morphism of d-couples. We have defined functors L ⊗ † , D X ,Z , and f + , f ! , f + , f ! . Thesefunctors are expected to fit in the framework of six functors if we consider the category of holo-nomic complexes. Let us explain this shortly. Consider the category d-couples such that themorphisms consist of strict morphisms of d-couples (cf. paragraph 3.1) whose realizations areproper. For a d-couple ( X , Z ), we consider the category of holonomic F - D † X , Q ( † Z )-complexesdenoted by C ( X ,Z ) . If the Berthelot conjecture holds, the category C ( X ,Z ) is stable under sixoperations. Philosophically, considering C ( X ,Z ) means to consider a “good category” of coeffi-cients on X \ Z . See [BeI, 5.3.6] for some explanations. The following theorems are stating thatfundamental relations of these functors hold in this framework. We use the notation of paragraph 5.2. We get the following theorem. Theorem. — Assume that M is f ! -admissible. Then there exists a canonical homomorphism (5.4.1) f ! ( M ) → f + ( M ) compatible with Frobenius pull-backs. Moreover, when Z ′ = Z , this homomorphism is an iso-morphism.Proof. By using Theorem 4.13, we get D Y ,W ◦ f Z ′ , + ∼ = f Z ′ , + ◦ D X ,Z ′ . The extension of scalar D † X , Q ( † Z ) ⊗ D † X , Q ( † Z ′ ) induces the functor D X ,Z ′ ◦ j + → j + ◦ D X ,Z . Using the isomorphism D X ,Z ◦ D X ,Z ∼ = id compatible with the Frobenius isomorphisms by [Vi, II, 3.5], and combiningthese morphisms of functors we get the homomorphism (5.4.1). The latter assertion is nowclear. (cid:4) Now, we will show the Poincar´e duality theorem for arithmetic D -module theory in thestyle of SGA4 Exp. XVIII Th´eor`eme 3.2.5. 37et f : ( X , Z ) → ( Y , W ) be a strict morphism of d-couples such that the realization isproper. Moreover, we assume that f − ( W ) = Z . There exists the following isomorphism thanksto [Vi2, IV, 7.4].(5.5.1) R Hom D † X , Q ( † Z ) ( M , f ! N ) ∼ −→ R Hom D † Y , Q ( † W ) ( f + M , N )We get that this isomorphism is compatible with Frobenius by Corollary 4.14. Remark. — The isomorphism (5.5.1) should hold without assuming that f − ( W ) = Z if wereplace f + by f ! . For this, we need to assume the Berthelot conjecture. In the following, wefreely use this conjecture. Let us sketch a proof. It suffices to show the case where f : ( X , Z ) → ( X , Z ′ ) such that the realization is the identity and Z ′ ⊂ Z . We can see easily that it sufficesto show that for M ∈ C ( X ,Z ′ ) and N ∈ C ( X ,Z ) , the homomorphism induced by scalar extension R Hom D † X , Q ( † Z ′ ) ( M , f + N ) → R Hom D † X , Q ( † Z ) ( f ! M , N )is an isomorphism. We can reduce the verification to the following two cases; when the supportof M is contained in Z , and when M is a D † X , Q ( † Z )-module. To see the former case, use thetheorem of Berthelot-Kashiwara [BeI, 5.3.3]. To see the latter case, it suffices to show that thehomomorphism R Hom D † X , Q ( † Z ′ ) ( O X , Q ( † Z ′ ) , D X ,Z ′ ( M ) L ⊗ † f + N ) → R Hom D † X , Q ( † Z ) ( O X , Q ( † Z ) , D X ,Z ( M ) L ⊗ † N )is an isomorphism. Using the Spencer resolution, it suffices to show that the canonical ho-momorphism D X ,Z ′ ( M ) L ⊗ † f + N → D X ,Z ( M ) L ⊗ † N is an isomorphism. The verification iseasy.To complete the Poincar´e duality we need to calculate f ! in the case where f is smooth.Namely, we get the following. Theorem. — Let f : ( X , Z ) → ( Y , W ) be a morphism of d-couples such that the realizationis smooth . Then there is a canonical isomorphism of cohomological functors with Frobeniusisomorphisms f ! ∼ = f + ( d )[2 d ] : D b coh ( D † Y , Q ( † W )) → D b coh ( D † X , Q ( † Z )) where d denotes the relative dimension of f .Proof. Put the functor D X ,Z to the both sides of the isomorphism of Theorem 3.10. Using theinvolutivity [Vi, II.3.5] of D X ,Z , we get the claim. (cid:4) By using the comparison between dual functor of arithmetic D -modules and that ofisocrystals, we can prove a purity type result. Namely we have Theorem (Purity) . — Let ( X , Z ) → ( Y , W ) be a morphism of d-couple. Moreover, supposethat the realization of f is a closed immersion . Let M be a convergent F -isocrystal on Y overconvergent along W . Then f + ( M ) is defined, and we get f ! ( M ) ∼ = f + ( M )( − d )[ − d ] where d denotes the codimension of X in Y .Proof. We know that f ∗ (sp ∗ ( M ) ∨ ) ∼ = ( f ∗ sp ∗ ( M )) ∨ . Together with the comparison theoremof duality functors Corollary 3.12 and the compatibility of pull-backs [Ca5, 4.1.8], the theoremfollows. (cid:4) .7. Consider the following cartesian diagram of d-couples.( X ′ , Z ′ ) g ′ / / f ′ (cid:15) (cid:15) (cid:3) ( X , Z ) f (cid:15) (cid:15) ( Y ′ , W ′ ) g / / ( Y , W )Here, cartesian means it is cartesian as a diagram of underlying formal schemes and X ′ \ Z ′ ∼ =( X \ Z ) × ( Y \ W ) ( Y ′ \ W ′ ). Now, we get the following base change theorem. Theorem (Proper base change) . — We preserve the notation. We get a canonical equivalenceof functors g ! ◦ f + ∼ = f ′ + ◦ g ′ ! : LD −→ b Q , qc ( b D ( • ) X ( Z )) → LD −→ b Q , qc ( b D ( • ) Y ′ ( W ′ )) . This isomorphism is compatible with Frobenius pull-backs.Proof. Using the standard factorization, it suffices to show the theorem in the cases where g isa closed immersion and a smooth morphism. When g is smooth, we get the theorem by Lemma2.6. When g is a closed immersion, this is a result of Caro [Ca1, 2.2.18]. (cid:4) Remark. — Let M be an object in D b coh ( D † X , Q ( † Z )). When g + ◦ f ! ( M ) and f ′ ! ◦ g ′ + ( M ) aredefined, the above equivalence and the isomorphism D ◦ D ∼ = id induces an isomorphism g + ◦ f ! ( M ) ∼ = f ′ ! ◦ g ′ + ( M ) . We preserve the notation. Let M and N be perfect ( F -) D † Y , Q ( † W )-complexes. Weassume that D Y ,W ( M ) L ⊗ † D Y ,W ( N ) is also perfect ( F -) D † Y , Q ( † W )-complex. Then, we definethe twisted tensor product of M and N denoted by M e ⊗ L † N to be D Y ,W ( D Y ,W ( M ) L ⊗ † D Y ,W ( N )) . One of the reasons we introduce this twisted tensor product is the following. For M and N in LD −→ b Q , qc ( b D ( • ) Y ), we get that(5.8.1) f ! ( M L ⊗ † N )[ d f ] ∼ = f ! M L ⊗ † f ! N where d f := dim( X ) − dim( Y ). This is compatible with the Frobenius structures. However,if we replace f ! by f + , the equality (5.8.1) does not hold in general. Nevertheless, if we alsoreplace L ⊗ † by e ⊗ L † , the equality holds in turn. Namely, (5.8.1) induces an isomorphism f + ( M e ⊗ L † N )[ − d f ] ∼ = f + M e ⊗ L † f + N if the both sides are defined. A consequence of Lemma 4.3 is the following. Proposition. — Let ( X , Z ) be a d-couple, and M and N be coherent F - D † X , Q ( † Z ) -complexes.Assume further that M is an overconvergent isocrystal along the divisor Z . Then we get M e ⊗ L † N ∼ = ( M L ⊗ † N )( d ) where d denotes the dimension of X . .9. Finally, let us compare the rigid cohomology with the push-forward as arithmetic D -modules.Let X be a proper smooth formal scheme of dimension d , Z be a divisor of the special fiberof X , U be the complement, and U be its special fiber. We denote by f : ( X , Z ) → (Spf( R ) , ∅ )the morphism of d-couples induced by the structural morphism of X . Let M be a coherent D † X , Q ( † Z )-module which is an overconvergent isocrystal along Z . Suppose that it is coherent asa D † X , Q -module.By Corollary 3.14, we get the canonical isomorphism H i f + M ∼ = H d + i rig ( U , sp ∗ ( M ))( d ) . To see the relation for cohomologies with compact supports, we use the Poincar´e duality ofrigid cohomology. In the curve case, Poincar´e duality is proven in [Cr]. In the general case, wecould not find any literature explicitly stating the Poincar´e duality with Frobenius structure.However in [St, 8.3.14], the coupling is defined, and in [Ke], the perfectness of the couple isproven. Thus we get the following isomorphism H i rig ( U , M ) ∨ ∼ = H d − i rig ,c ( U , M ∨ ( d ))for an overconvergent F -isocrystal M on the smooth variety U over k . Using this, we get H i f ! M ∼ = ( H − i f + D X ,Z ( M )) ∨ ∼ = (cid:0) H d − i rig ( U , sp ∗ ( M ) ∨ ( − d ))( d ) (cid:1) ∨ ∼ = H d + i rig ,c ( U , sp ∗ ( M ))( d ) . Here the second isomorphism follows from Corollary 3.12. Summing up, we get H i f + M ∼ = H d + i rig ( U , sp ∗ ( M ))( d ) , H i f ! M ∼ = H d + i rig ,c ( U , sp ∗ ( M ))( d ) . In particular, we note that there exist canonical isomorphisms H i f + f + ( K ) ∼ = H i rig ( U /K ) , H i f ! f + ( K ) ∼ = H i rig ,c ( U /K )compatible with Frobenius isomorphism. A. Comparison theorem between Fourier transform and Fouriertransform with compact support. C. Huyghe Introduction Let V be a discrete valuation ring of inequal characteristics (0 , p ), containing an element π ,the π of Dwork, satisfying the equation π p − = − p , S = Spf ( V ) the formal spectrum of V , X the formal affine line over S , X ∨ the dual affine line. Let us introduce Y and Y ∨ two copiesof the formal projective line over R compactifying X and X ∨ and denote ∞ Y (resp. ∞ Y ∨ )the complementary divisors. In [Be1] Berthelot constructed sheaves of arithmetic differentialoperators with overconvergent singularities along a divisor ( e.g. D †Y ( ∞ Y ) in our situation). Forseveral reasons these sheaves have to be thought as sheaves over the open subset which is thecomplementary of the considered divisor, thus, in our case, over the formal affine line. In [Hu0]one constructed the Fourier transform of D †Y ( ∞ Y )-modules, using the Dwork exponential moduleas kernel and we checked the compatibility with the so-called na¨ıve Fourier transform. The aimof this note is to define the compact support Fourier transform in dimension 1 and to prove thatit coincides with the Fourier transform without compact support introduced in loc. cit. .This result is part from the unpublished part 4.4 of [Hu0], where it is proven in dimension N .The dimension 1 case is technically easier and sufficient for [AM] while giving a good idea of theproof in dimension N . That’s why we restrict to this case here. The extra work in dimension N consists into proving a generalization of the division lemma A.2.7 and to deal with longercomplexes of length N + 1. 40 .1. PreliminariesA.1.1. Denote K the fraction field of V . For l ∈ Z , | l | is the usual archimedean absolute valueof l . The p -adic valuation of an element a of a p -adically separated ring is v p ( a ). For a ∈ K , | a | p = p − v p ( a ) . If F is a sheaf of abelian groups over a topological space we denote F Q = Q ⊗ Z F . The product Z = Y × Y ∨ = b P S × b P S is endowed with the ample divisor ∞ = ∞ Y × b P S S b P S ×∞ Y ∨ .When needed, we will use [ u , u ] and [ v , v ] as homogeneous coordinates over the two copiesof b P S , u = 0 and v = 0 are equations of the infinite divisor over each copy of b P S , x = u /u and y = v /v will be coordinates over the affine plane complementary of the ∞ divisor over Z .These two coordinates x and y over X = b A S and X ∨ = b A S should be considered as dual to eachother. A.1.2. Let V be a smooth formal scheme over S , endowed with a relative divisor D (meaningthat D induces a divisor of the special fiber), and U = V\ D . Then the direct image by special-ization of the constant overconvergent F -isocrystal over U , the special fiber of U , is a sheaf over V denoted by O V ( † D ) (4.4 of [Be1]). If V is affine and if f is an equation of D over V , we havethe following descriptionΓ( V , O V ( † D )) = (X l ∈ N a l f l , a l ∈ Γ( V , O V , Q ) | and ∃ C ′ > | v p ( a l ) − lC ′ → + ∞ if l → + ∞ ) . Consider the ring D †V of arithmetic differential operators over V . Suppose that x , . . . , x n are coordinates on V , denote ∂ x i the corresponding derivations for i ∈ { , . . . , n } , and ∂ [ k i ] x i = ∂ k i x i k i ! , and ∂ [ k ] = ∂ [ k ] x · · · ∂ [ k n ] x n , then we have the following descriptionΓ( V , D †V ) = X k ∈ N n a k ∂ [ k ] , a k ∈ Γ( V , O V , Q ) and ∃ C > | v p ( a k ) − | k | C → + ∞ if | k | → + ∞ . In 4.2 of [Be1], Berthelot introduces also rings of differential operators with overconvergentsingularities, which are sheaves of O V ( † D )-modules. Suppose that x , . . . , x n are coordinates on V and that the divisor D is defined by the equation f = 0 on V , then we have the followingdescriptionΓ( V , D †V ( D )) = X l ∈ N ,k ∈ N n a l,k f l ∂ [ k ] , a l,k ∈ Γ( V , O V , Q )and ∃ C > | v p ( a l,k ) − l + | k | C → + ∞ if l + | k | → + ∞ . All these sheaves are weakly complete, in the sense that they are inductive limit of sheavesof p -adic complete rings as described in the following subsection. Let us stress here upon thefact that these sheaves are always sheaves of K vector spaces (even if there is no Q in theirnotation). We made this convention to avoid too heavy notations. Note also that this won’t bethe case with the sheaves that we will introduce in the following subsections.41 .1.3. In this subsection, we use notations of A.1.2. Let us fix m ∈ N , elements x , . . . , x n ∈ Γ( V , O V ) which are coordinates on V and f ∈ Γ( V , O V ) such that D T V = V ( f ).Let us introduce here some coefficients. First we define the application ν m : Z → N by thefollowing way. If k < 0, we set ν m ( k ) = 0. Let k ∈ N , and q and r be the quotient andthe remainder of the division of k by p m +1 . If r = 0, we set ν m ( k ) = q , otherwise we set ν m ( k ) = q + 1. We extend this application to Z r , by ν m (( k , . . . , k r )) = ν m ( k ) + · · · + ν m ( k r ).We denote also q ( m ) k the quotient of the division of a positive integer k by p m , and ∂ h k i ( m ) = q ( m ) k ! · · · q ( m ) k n ! ∂ [ k ] x · · · ∂ [ k n ] x n . If the choice of m is clear, we will omit it in the notation.Berthelot defines a sheaf b B ( m ) V by setting locallyΓ( V , b B ( m ) V ) = (X l a l f l , a l ∈ Γ( V , O V ) | v p ( a l ) ≥ ν m ( l ) and v p ( a l ) − ν m ( l ) → + ∞ if l → + ∞ ) . Then there are canonical injective morphisms b B ( m ) V ⊂ b B ( m +1) V and O V ( † D ) = lim −→ m b B ( m ) V , Q . Berthelot defines also sheaves of rings differential operators D ( m ) V ( D ) and their p -adic com-pletion b D ( m ) V ( D ) over V byΓ( V , D ( m ) V ( D )) = X l ∈ N ,k ∈ N n a l,k f l ∂ h k i ( m ) , a l,k ∈ Γ( V , O V ) | v p ( a l,k ) ≥ ν m ( l ) , where the sums are finite andΓ( V , b D ( m ) V ( D )) = X l ∈ N ,k ∈ N n a l,k f l ∂ h k i ( m ) , a l,k ∈ Γ( V , O V ) | v p ( a l,k ) ≥ ν m ( l )and v p ( a l,k ) − ν m ( l ) → + ∞ if | k | + l → + ∞ . Then there are canonical injective morphisms b D ( m ) V , Q ( D ) ⊂ b D ( m +1) V , Q ( D ) and D †V ( D ) = lim −→ m b D ( m ) V , Q ( D ) . If D = ∅ the previous sheaves are simply denoted D ( m ) V and b D ( m ) V . A.1.4. We finally recall the following inequalities for | k | and | l | elements of N n , l and r in N . | k | p − − n log p ( | k | + 1) − n ≤ v p ( k !) ≤ | k | p − | k | p m ( p − − n log p ( | k | + 1) − n pp − ≤ v p ( q ( m ) k !) ≤ | k | p m ( p − | l | p m +1 ≤ ν m ( l ) ≤ | l | p m +1 + n ≤ ν m ( l ) − ν m ( | l | ) ≤ n ≤ v p (cid:18)(cid:18) lr (cid:19)(cid:19) ≤ p (log p ( l ) + 1) . .1.5. By definition an induced D †V ( D )-module is a D †V ( D ) of the type D †V ( D ) ⊗ O V ( † D ) E , where E is a coherent O V ( † D )-module and where the D †V ( D )-module structure comes from theone of D †V ( D ). A.1.6. Coming back to the situation of the introduction, let p and p be the two projections Z → Y and Z → Y ∨ , and ∞ ′ = p − ∞ Y ∨ .Let us now recall how to describe the structure of the category of left D †Y ( ∞ Y )-modules,resp. D †Z ( ∞ )-coherent modules, resp. D †Z ( ∞ ′ )-coherent modules.Let us first start with coherent D †Y ( ∞ Y )-modules (see [Hu1]). First note thatΓ( Y , O Y ( † ∞ Y )) = (X l b l x l , b l ∈ K, and ∃ C > , η < | | b l | p < Cη l ) and set A ( K ) † = X k ∈ N ,l ∈ N a l,k x l ∂ [ k ] x , a l,k ∈ K, and ∃ C > , η < | | a k,l | p < Cη l + k , the weak completion of the Weyl algebra. It is a coherent algebra and we have ([Hu1]) A.1.7 Theorem. — The functor Γ( Y , . ) (resp. R Γ( Y , . )) establishes an equivalence of cate-gories between the category of left coherent D †Y ( ∞ Y )-modules (resp. D bcoh ( D †Y ( ∞ Y ))) and thecategory of left coherent A ( K ) † -modules (resp. D bcoh ( A ( K ) † ).In particular, A ( K ) † ≃ Γ( Y , D †Y ( ∞ Y )) and every coherent D †Y ( ∞ Y ) admits globally over Y a resolution by globally projective and finitely generated D †Y ( ∞ Y )-modules. This resolution canbe taken finite since A ( K ) † has finite cohomological dimension ([NH2]).Consider now the situation over Z for m fixed. In theorem 3 of [Hu1], we proved that theelements ∂ h k i ( m ) x are global sections over b P S of the sheaves D ( m ) b P S , so that ∂ h k i ( m ) x ∂ h k i ( m ) y ∈ Γ( Z , D ( m ) Z ) . Moreover, an easy computation (2.1 of [Hu1]) shows that elements p ν m ( l ) x l for l ≥ b B ( m ) b P S , implying that p ν m ( l ) x l y l ∈ Γ( Z , b B ( m ) Z ) . Define b E ( m ) = X k ∈ N ,l ∈ N a l,k x l y l ∂ h k i x ∂ h k i y , a l,k ∈ V, and v p ( a l,k ) ≥ ν m ( l ) | v p ( a l,k ) − ν m ( l ) → + ∞ if | l | + | k | → + ∞ . b D ( m ) Z is a sheaf of p -adically complete algebras, wesee that b E ( m ) ⊂ Γ( Z , b D ( m ) Z ( ∞ )) . Consider the weak completion of the Weyl algebra in 2 variables A ( K ) † = X k ∈ N ,l ∈ N a l,k x l y l ∂ [ k ] x ∂ [ k ] y , a l,k ∈ K, and ∃ C > , η < | | a k,l | < Cη | l | + | k | , which is coherent from [Hu1]. It is easy to see that A ( K ) † = lim −→ m b E ( m ) Q . We endow A ( K ) † with the inductive limit topology coming from this filtration.Because the divisor ∞ over Z is ample, we can apply 4.5.1 of [Hu3] which tells us thatΓ( Z , D † Z ( ∞ )) ≃ A ( K ) † . Moreover we have the following theorem (4.5.1 of [Hu3] and 5.3.4 of [Hu2]). A.1.8 Theorem. — The functor Γ( Z , . ) (resp. R Γ( Z , . )) establishes an equivalence of cat-egories between the category of left coherent D †Z ( ∞ )-modules (resp. D bcoh ( D †Z ( ∞ ))) and thecategory of left coherent A ( K ) † -modules (resp. D bcoh ( A ( K ) † )).Consider now the scheme Z = Y ×Y ∨ endowed with the divisor ∞ ′ = p − ( ∞ Y ∨ ) and D †Z ( ∞ ′ )the ring of arithmetic differential operators with overconvergent coefficients along ∞ ′ . In orderto deal with coherent D †Z ( ∞ ′ )-modules, denote B ( K ) † = X l ∈ N ,k ∈ N a l ,k y l ∂ [ k ] x ∂ [ k ] y , a l ,k ∈ K, and ∃ C > , η < | | a l ,k | p < Cη l + | k | , which we endow with the induced topology of A ( K ) † . Consider b F ( m ) = b E ( m ) T B ( K ) † . Asbefore we observe that b F ( m ) ⊂ Γ( Z , b D ( m ) Z ( ∞ ′ )) , which leads to the following inclusionlim −→ m b F ( m ) Q = B ( K ) † ⊂ Γ( Z , D †Z ( ∞ ′ )) . We apply 2.3.3 of [NH1] to see that this is actually an equality.We will also use the following division lemma (4.3.4.2 of [NH1]). A.1.9 Theorem. — i. For any P ∈ A ( K ) † there exists a unique ( Q, R ) ∈ A ( K ) † × B ( K ) † such that P = Q ( − ∂ y + πx ) + R. ii. The maps P Q and P R are continuous. More precisely, if P ∈ b E ( m ) Q , then Q ∈ b E ( m +2) Q , and R ∈ b E ( m +2) Q T B ( K ) † . 44 .2. Fourier transformsA.2.1. Kernel of the Fourier transform. In dimension 1, the duality pairing X ×X ∨ → b A S is given by t xy where t is the global coordinate on the right-hand side. It extends to δ : Z = Y × Y ∨ → b P S , by the formula t − x − y − on neighborhoods of ∞ . Let L π bethe realization over b P S of the overconvergent Dwork F -isocrystal. It is given by a connection ∇ (1) = − πdt . We define K π to be the D †Z , Q ( ∞ )-module associated to the overconvergent F -isocrystal δ ∗ ( L π ). The module K π is thus isomorphic to O Z ( † ∞ ) with a connection on Y × Y ∨ defined by ∇ (1) = − π ( xdy + ydx ) . A.2.2. Explicit descriptions of cohomological operations. Let us consider the followingdiagram Z = Y × Y ∨ p ( ( QQQQQQQQQ p w w nnnnnnnnnn Y Y ∨ . For M, N two O Z ( † ∞ )-modules, denote M e ⊗ N = M ⊗ L O Z ( † ∞ ) N [ − . Note that the sheaves p − D †Y ( ∞ Y ) (resp. p − D †Y ∨ ( ∞ Y ∨ )), are canonically subsheaves of ringsof D †Z ( ∞ ). Cohomological operations involve sheaves D †Z→Y ( ∞ ), respectively D †Y←Z ( ∞ ), whichare left (resp. right) coherent D †Z ( ∞ )-modules and right (resp. left) p − D †Y ( ∞ Y )-modules, whichcan be explicitely described in our case. The module structures over D †Y←Z ( ∞ ) are obtained fromthese of D †Z→Y ( ∞ ) by twisting by the adjoint operator (1.3 of [Be2]). In particular underlyingabelian groups of both sheaves are the same. Because the sheaves O Z ( † ∞ ) ⊗ O Z ω Z and O Z ( † ∞ ) ⊗ p − O Y p − ω Y are free, the twisted actions are easy to describe globally. For example, for P ∈ D †Y←Z ( ∞ ),the right action of ∂ x over P is equal to the left action by − ∂ x over P seen as an element of D †Z→Y ( ∞ ).From 4.2.1 of [NH1], we know that these sheaves admit a free resolution, as D †Z ( ∞ )-modules0 / / D †Z ( ∞ ) / / D †Z ( ∞ ) / / P (cid:31) / / P ∂ y . For the sheaf D †Y←Z ( ∞ ), consider the map P ∂ y P . Actually, if we endow D †Z ( ∞ ) withthe canonical structure of right (resp. left)- p − D †Y ( ∞ Y )-module this complex is a resolution of D †Z→Y ( ∞ ) (resp. D †Y←Z ( ∞ )) which is D †Z ( ∞ ) × D †Y ( ∞ Y ) (resp. D †Y ( ∞ Y )) × D †Z ( ∞ )-linear).Moreover, thanks to A.1.8 the sheaves D †Z→Y ( ∞ ) and D †Y←Z ( ∞ ) are acyclic for the globalsection functor.Moreover, both sheaves D †Y←Z ( ∞ ) and D †Z→Y ( ∞ ) can be considered as subsheaves of D †Z ( ∞ ).For example, D †Z→Y ( ∞ ) is a D †Z ( ∞ )-coherent module, and its global sections over Z are thesections of A ( K ) † with no term in ∂ x .This is the same for D †Y←Z ( ∞ ) once we have twisted the two actions of D †Z ( ∞ ) over itselfby the adjoint operator. 45or M ∈ D bcoh ( D †Y ( ∞ Y )) we state as usual p !1 ( M ) = D †Z→Y ( ∞ ) ⊗ L p − D †Y ( ∞ Y ) p − M [1] ∈ D bcoh ( D †Z ( ∞ ))and for N ∈ D bcoh ( D †Z ( ∞ )) p ( N ) = R p ∗ (cid:16) D †Y ∨ ←Z ( ∞ ) ⊗ L D †Z ( ∞ ) N (cid:17) , whose cohomology sheaves are not coherent in general. Fourier transform can now be defined. A.2.3 Definition. — For M ∈ D bcoh ( D †Y ( ∞ Y )) F ( M ) = p ( p !1 M e ⊗ K π ) . Let us recall the fundamental results of [NH1]. A.2.4 Theorem. — i. (4.3.4 of [NH1]) there is a canonical isomorphism D †Y ∨ ( ∞ Y ∨ )[ − → F ( D †Y ( ∞ Y )) , ii. (5.3.1 of loc. cit. ) If M ∈ D bcoh ( D †Y ( ∞ Y )), then F ( M ) ∈ D bcoh ( D †Y ∨ ( ∞ Y ∨ )). A.2.5. To define Fourier transform with compact support, we will need to work with D †Z ( ∞ ′ )-modules. In particular, we will use the cohomological functor p ′ : D bcoh ( D †Z ( ∞ ′ )) → D bcoh ( D †Y ∨ ( ∞ Y ∨ )),which preserves coherence, since p is proper and thanks to the fact that ∞ ′ = p − ( ∞ ).We will also use the scalar restriction functor ρ : D b ( D †Z ( ∞ )) → D b ( D †Z ( ∞ ′ )). A.2.6. For M in D bcoh ( D †Z ( ∞ )), we denote the dual D Z ( M ) = R H om D †Z ( ∞ ) ( M, D †Z ( ∞ ) ⊗ O Z ω − Z [2]) ∈ D bcoh ( D †Z ( ∞ )) , since the sheaf D †Z ( ∞ ) has finite cohomological dimension ([NH2]), and the corresponding dualfunctor D ′Z ( M ), for M ∈ D bcoh ( D †Z ( ∞ ′ )), (resp. D Y and D Y ∨ for D †Y ( ∞ Y ), resp. D †Y ∨ ( ∞ Y ∨ )-modules).The following division lemma will be crucial. A.2.7 Lemma. — Let U = D + ( u ) × D + ( v ) or U = D + ( u ) × D + ( v ) and x ′ = 1 /x .i. The elements ( − x ′ ∂ y + π ) and − ∂ y + πx generate the same left ideal of D †Z ( ∞ )( U ).ii. For any P ∈ D †Z ( ∞ )( U ) there exists ( Q, R ) ∈ D †Z ( ∞ )( U ) × D †Z ( ∞ ′ )( U ) such that P = Q ( − x ′ ∂ y + π ) + R. iii. The previous decomposition is not unique. But, if Q ( − ∂ y + πx ) ∈ D †Z ( ∞ ′ )( U ), then Q ∈ D †Z ( ∞ ′ )( U ).iv. If Q ∈ D †Z ( ∞ )( U ) and Q ( − ∂ y + πx ) = 0, then Q = 0. Remark . Analogous statements hold for the left multiplication by − ∂ y + πx , or ∂ y + πx ,considering right ideals generated by these elements.46 roof. Recall that x ∈ Γ( Z , O Z ( † ∞ )). The first assertion comes from the equality x ( − x ′ ∂ y + π ) = − ∂ y + πx. The open set D + ( u ) × D + ( v ) will be endowed with coordinates x ′ = 1 /x and t = y , and D + ( u ) × D + ( v ) with coordinates x ′ and t = 1 /y . Denote D † = D †Z ( ∞ )( U ), and D ′† = D †Z ( ∞ ′ )( U ), which contains the algebras b F ( m ) Q described in A.1.6.By considering the right O Z -module structure on D †Z ( ∞ ), we see that an element Q ∈ D † can be written Q = X k ∈ N ,l ∈ Z a l,k ∂ [ k ] x ′ ∂ [ k ] t x ′ l t l , such that a l,k ∈ K and ∃ C > , M > , | v p ( a l,k ) ≥ max {− l , } + max {− l , } + | k | C − M, (A.2.1)(note that on D + ( u ) × D + ( v ) coefficients a ( l ,l ) ,k are equal to 0 if l < Q ∈ D ′† if and only if ∀ k ∈ N , ∀ l | l < , a l,k = 0. We can also write Q this way Q = X l ∈ Z Q l x ′ l , with Q l = X k =( k ,k ∈ N × N l ∈ Z a ( l ,l ) ,k ∂ [ k ] x ′ ∂ [ k ] t t l ∈ Γ( U , D †Z ( ∞ ′ )) . Let us observe that the condition(A.2.1) is equivalent to the fact that there exist m > Q l ∈ Γ( U , b D ( m ) Z , Q ( ∞ ′ ))and elements T l of Γ( U , b D ( m ) Z ( ∞ ′ )) such that Q l = u l p ν m ( − l ) T l , for some u l ∈ K satisfying v p ( u l ) → + ∞ if | l | → + ∞ . (A.2.2)We define R ′′ = X l ∈ N Q l x ′ l ∈ Γ( U , b D ( m ) Z , Q ( ∞ ′ )) , so that we can write down Q = X l < u l p ν m ( | l | ) T l x | l | + R ′′ , (A.2.3)with T l ∈ Γ( U , b D ( m ) Z , Q ( ∞ )) and v p ( u l ) → + ∞ if | l | → + ∞ .We will first prove (iii). Suppose that Q ( − ∂ y + πx ) ∈ Γ( U , D †Z ( ∞ ′ )) . Because of (A.2.3), wecan suppose that Q = X l < Q l x ′ l . Now we are reduced to prove that Q = 0, that we can do in restriction to D + ( u ) × D + ( v v ),on which we choose x ′ and y as coordinates.Let us compute Q ( − ∂ y + πx ) = X l < ( − Q l ∂ y + πQ l +1 ) x ′ l . Q ( − ∂ y + πx ) is an element of Γ( U , D †Z ( ∞ ′ )) if and only if Q − ∂ y = 0 , and(A.2.4) ∀ l ≤ − , − Q l ∂ y + πQ l +1 = 0 . (A.2.5)Let us decompose Q − = X k β k ( y ) ∂ [ k ] x ′ ∂ [ k ] y , with β k ( y ) ∈ O Z , Q ( D + ( v u u )), then we compute Q − ∂ y = X k | k ≥ k β ( k ,k − ( y ) ∂ [ k ] x ′ ∂ [ k ] y , which is null if and only if ∀ k, β k = 0, i.e. Q − = 0. Thus by descending induction, one seesfrom (A.2.5) that ∀ l ≤ − , Q l = 0 , and Q = 0.Let us prove now (ii), the existence of the decomposition. Recall that ∂ y ∈ Γ( Z , D †Z ( ∞ ′ ))and ( − ∂ y + πx ) ∈ Γ( Z , D †Z ( ∞ )). Since these two elements commute we have the followingequalities p ν m ( | l | ) x | l | = p ν m ( | l | ) π | l | (( − ∂ y + πx ) + ∂ y ) | l | = p ν m ( | l | ) π | l | | l | ! ∂ [ | l | ] y + | l | X r =1 r − X s =0 ( − r − s p ν m ( | l | ) π | l |− s (cid:18) | l | r (cid:19)(cid:18) r − s (cid:19) ( | l | − − s )! ∂ [ | l |− − s ] y x s ( − ∂ y + πx )= p ν m ( | l | ) π | l | | l | ! q ( m +2) | l | ! ∂ h| l |i ( m +2) y + | l | X r =1 r − X s =0 ( − r − s p ν m ( | l | ) π | l |− s (cid:18) | l | r (cid:19)(cid:18) r − s (cid:19) ( | l | − − s )! q ( m +2) | l |− − s ! ∂ h| l |− − s i ( m +2) y x s ( − ∂ y + πx ) . Denote R l = p ν m ( | l | ) π | l | | l | ! q ( m +2) | l | ! ∂ h| l |i ( m +2) y ,S l = | l | X r =1 r − X s =0 ( − r − s p ν m ( | l | ) π | l |− s (cid:18) | l | r (cid:19)(cid:18) r − s (cid:19) ( | l | − − s )! q ( m +2) | l |− − s ! ∂ h| l |− − s i ( m +2) y x s ,c l ( r, s ) = ( − r − s p ν m ( | l | ) π | l |− s (cid:18) | l | r (cid:19)(cid:18) r − s (cid:19) ( | l | − − s )! q ( m +2) | l |− − s ! . By definition, we have the following relation p ν m ( | l | ) x | l | = S l ( − ∂ y + πx ) + R l . (A.2.6)Then, from estimations A.1.4 we see that v p p ν m ( | l | ) π | l | | l | ! q ( m +2) | l | ! ≥ p − p − p m +2 ( p − | l | − log p ( | l | + 1) − → ∞ if | l | → + ∞ , R l ∈ b E ( m +2) for | l | big enough. We also see that ∀ r ≤ | l | − , s ≤ r,v p ( c l ( r, s )) ≥ | l | p m +1 − | l | − sp − − log p ( | l | + 1) + | l | − − sp − − − | l | − − sp m +2 ( p − ≥ (cid:18) p − p − p m +2 ( p − (cid:19) | l | − log p ( | l | + 1) − → ∞ if | l | → + ∞ , which proves that S l ∈ b E ( m +2) for | l | big enough. As a consequence S l and resp. R l areelements of Γ( U , b D ( m +2) Z ( ∞ )) for | l | big enough (resp. Γ( U , b D ( m +2) Z ( ∞ ′ ))).Let Q ∈ D † . We can use the description given in (A.2.3). Since | u l | → | l | → + ∞ , weobserve that Q ′ = X l < u l T l S l ∈ Γ( U , b D ( m +2) Z , Q ( ∞ )) , and R ′ = X l < u l T l R l ∈ Γ( U , b D ( m +2) Z , Q ( ∞ ′ )) . Moreover, we have the following equalities Q = X l < u l p ν m ( | l | ) T l x | l | + R ′′ = X l < u l T l ( S l ( − ∂ y + πx ) + R l ) + R ′′ = Q ′ x ( − x ′ ∂ y + π ) + R ′ + R ′′ , which shows (ii) of the lemma.Now, let Q ∈ D † such that Q ( − ∂ y + πx ) = 0. From (iii), we know that in fact Q ∈ D ′† . Asin the previous case, we restrict ourselves to U = D + ( u ) × D + ( v v ) and we decompose Q = X l ∈ N Q l x ′ l , The recursive formula (A.2.5) still holds and we get πQ = 0 , and ∀ l ≥ , πQ l +1 = Q l ∂ y . By induction, this proves that Q l = 0 for all l ≥ Q = 0. (cid:4) The key lemma to define the Fourier transform with compact support is the following: A.2.8 Lemma. — Let M ∈ D bcoh ( D †Y ( ∞ Y ))), then ρ ∗ D Z ( p !1 M e ⊗ K π ) ∈ D bcoh ( D †Z ( ∞ ′ )) . Proof. It is enough to prove this lemma in the case of a single D †Y ( ∞ Y ) coherent module M .Such a module admits a resolution by direct factors of free modules of finite rank (A.1.7). It isthus enough to prove this lemma in the case of D †Y ( ∞ Y ) itself.If F is an O Z -coherent module, we denote e F = F ⊗ O Z O Z ( † ∞ ) . e T Z /S (resp. e T Z / Y ) are free O Z ( † ∞ )-modules of basis ∂ x , ∂ y (resp. ∂ y ).Let us reformulate lemma 4.2.4 of [NH1]. Let K • the complex of induced D †Z ( ∞ )-modulesin degrees − → D †Z ( ∞ ) ⊗ O †Z ( ∞ ) ^ e T Z / Y d → D †Z ( ∞ ) → , where d ( P ⊗ ∂ y ) = P · ( ∂ y + πx ) . A.2.8.1 Lemma. — In the derived category D bcoh ( D †Z ( ∞ )) this complex K • is equal to thecomplex p !1 D †Y ( ∞ Y ) e ⊗ O †Z ( ∞ ) K π [1] , which is nothing but the complex p ∗ D †Y ( ∞ Y ) ⊗ K π (in degree 0).The augmentation map ε ′ : D †Z ( ∞ ) → p ∗ D †Y ( ∞ Y ) ⊗ K π is given by ε ′ ( ∂ y ) = − πx ⊗ ε ′ ( ∂ x ) = ( ∂ x − πy ) ⊗ . In the rest of the proof, we identify the left induced D †Z ( ∞ )-module D †Z ( ∞ ) ⊗ O †Z ( ∞ ) Λ e T Z / Y with D †Z ( ∞ ). Then R H om D †Z ( ∞ ) ( K • , D †Z ( ∞ ))[2] is represented by the following complex ofright D †Z ( ∞ )-modules, whose terms are in degrees − − → D †Z ( ∞ ) d ′ → D †Z ( ∞ ) → , (A.2.7)such that d ′ ( P ) = ( ∂ y + πx ) P . Finally, we see that L • = D ( K • ) is represented in D bcoh ( D †Z ( ∞ ))by the following complex in degrees − − → D †Z ( ∞ ) d ′′ → D †Z ( ∞ ) → , (A.2.8)such that d ′′ ( P ) = P ( − ∂ y + πx ).Consider now the canonical map D †Z ( ∞ ′ ) → D †Z ( ∞ ) . D †Z ( ∞ )( − ∂ y + πx ) . Over D + ( u ) both sheaves D †Z ( ∞ ′ ) and D †Z ( ∞ ) coincide. Let us study the situation over D + ( u ).From the previous lemma A.2.7, we see that this map is surjective over D + ( u ) × D + ( v ) and D + ( u ) × D + ( v ) and that over these open subsets the following complex is exact (using notationsof A.2.7) 0 → D †Z ( ∞ ′ ) d ′′ → D †Z ( ∞ ′ ) → ρ ∗ (cid:16) D †Z ( ∞ ) . D †Z ( ∞ )( − ∂ y + πx ) (cid:17) → , where d ′′ ( P ) = P ( − x ′ ∂ y + π ), showing that ρ ∗ D Z ( K • ) ∈ D bcoh ( D †Z ( ∞ ′ )) , hence ρ ∗ D Z ( p !1 M e ⊗ K π ) ∈ D bcoh ( D †Z ( ∞ ′ )) for any M ∈ D bcoh ( D †Z ( ∞ )). (cid:4) Finally, this leads us to the following definition. A.2.9 Definition. — For M ∈ D bcoh ( D †Y ( ∞ Y )) F ! ( M ) = p ′ D ′Z ρ ∗ D Z ( p !1 M e ⊗ K π ) ∈ D bcoh ( D †Z ( ∞ ′ )) . Note that from the previous lemma A.2.8, we know that F ! ( M ) ∈ D bcoh ( D †Y ∨ ( ∞ Y ∨ )).50 .3. Comparison theoremA.3.1 Proposition. — Let M ∈ D bcoh ( D †Z ( ∞ )), there is a canonical map: F ! ( M ) → F ( M ). Proof. Let E be a coherent D †Z ( ∞ )-module, such that ρ ∗ E is a coherent D †Z ( ∞ ′ )-module. Thereare canonical maps D ′Z ( ρ ∗ E ) → D Z ( E ) and also canonical maps p ′ D ′Z ( E ) → p D Z ( E ) . Applying this to E = ρ ∗ D Z ( ∞ )( p !1 M e ⊗ K π ) gives a canonical map F ! ( M ) → p D Z D Z ( p !1 M e ⊗ K π ) . And we apply the biduality theorem 3.6 of [Vi], to see that the RHS can be identified with F ( M ). (cid:4) A.3.2 Theorem. — Let M ∈ D bcoh ( D †Z ( ∞ )), there is a canonical isomorphism: F ! ( M ) ≃F ( M ).Since F and F ! are both way-out functors, we are reduced to the case of a single module M .Thanks to A.1.7, we can suppose that M = D †Y ( ∞ Y ). A.3.3. Computation of F ! ( D †Y ( ∞ Y )) . Let us consider U one of the open sets D + ( u ) × D + ( v ) or D + ( u ) × D + ( v ) of Z , denote by x ′ and t coordinates over U . Then, over U , we havethe following resolution of the sheaf D †Y ∨ ←Z ( ∞ ′ ) by right D †Z ( ∞ ′ )-modules0 / / D †Z ( ∞ ′ ) / / D †Z ( ∞ ′ ) / / D †Y ∨ ←Z ( ∞ ′ ) / / P (cid:31) / / ∂ x ′ P and exactly the same resolution for the sheaf D †Z ( ∞ ), replacing D †Z ( ∞ ′ ) by D †Z ( ∞ ). This provesthat there is a canonical isomorphism of right D †Z ( ∞ )-modules D †Y ∨ ←Z ( ∞ ′ ) ⊗ D †Z ( ∞ ′ ) D †Z ( ∞ ) ≃ D †Y ∨ ←Z ( ∞ ) . In fact, this isomorphism is also right D †Y ∨ ( ∞ Y ∨ )-linear, property that we can check over theopen set D + ( u ) × D + ( v ) where the previous isomorphism coincides with identity.Now we have the following lemma. A.3.3.1 Lemma. — Let M be a coherent D †Z ( ∞ )-module such that ρ ∗ M is a coherent D †Z ( ∞ ′ )-module, then there is a canonical isomorphism of coherent D †Z ( ∞ )-modules D †Z ( ∞ ) ⊗ D †Z ( ∞ ′ ) ρ ∗ M ≃ M. Proof. The canonical map of the statement is a morphism of coherent D †Z ( ∞ )-modules, whichis an isomorphism over D + ( u ) × D + ( v ) and thus an isomorphism (cf 4.3.7 of [Be1]). (cid:4) With the hypothesis of the lemma, we get p M = Rp ∗ (cid:16) D †Y ∨ ←Z ( ∞ ) ⊗ D †Z ( ∞ ) M (cid:17) ≃ Rp ∗ (cid:16) D †Y ∨ ←Z ( ∞ ′ ) ⊗ D †Z ( ∞ ′ ) D †Z ( ∞ ) ⊗ D †Z ( ∞ ) M (cid:17) ≃ Rp ∗ (cid:16) D †Y ∨ ←Z ( ∞ ′ ) ⊗ D †Z ( ∞ ′ ) ρ ∗ ( M ) (cid:17) ≃ p ′ M. M = K • [ − 1] of A.2.8.1, we see that F ! ( D †Y ( ∞ Y )) = p ′ D ′Z ρ ∗ D Z ( K • [ − ≃ D Y p ′ ρ ∗ D Z ( K • [ − ≃ D Y p D Z ( K • [ − A.3.3.2 Lemma. — There is a canonical isomorphism D †Y ∨ ( ∞ Y ∨ )[1] ≃ p D Z ( K • ) . Proof. Using (A.2.8), we know that D †Y ∨ ←Z ( ∞ ) ⊗ L D †Z ( ∞ ) D Z ( K • ) is represented by the followingcomplex with terms in degrees − − / / D †Y ∨ ←Z ( ∞ ) / / D †Y ∨ ←Z ( ∞ ) / / P (cid:31) / / P ( − ∂ y + πx ) . Denote by E the − D †Y ∨ ←Z ( ∞ ) is a rightcoherent D †Z ( ∞ )-module, it is acyclic for the functor Γ( Z , . ). It is also the case for E by the longcohomology exact sequence and we can then compute Γ( Z , E ) as the cokernel of the map0 / / Γ( Z , D †Y ∨ ←Z ( ∞ )) / / Γ( Z , D †Y ∨ ←Z ( ∞ )) / / P (cid:31) / / P ( − ∂ y + πx ) . In particular, we get an element 1 ∈ Γ( Z , E ), allowing us to consider a morphism ϕ : D †Y ∨ ( ∞ Y ∨ ) → R p ∗ E , that sends P to P · 1, where 1 ∈ R p ∗ E .On the other hand, from the previous lemma A.3.3.1, we know that p D Z ( K • [ − E , is a coherent D †Y ( ∞ Y )-module. By A.1.8, it is enough to prove that the morphism inducedon global sections of both sheaves is an isomorphism, to see that ϕ is an isomorphism.As D †Z ( ∞ )-coherent module, D †Y ∨ ←Z ( ∞ ) is acyclic for the functor Γ. Using the resolu-tion given in A.2.2, we identify Γ( Z , D †Y ∨ ←Z ( ∞ )) with A ( K ) † /∂ x A ( K ) † . Finally, we get thefollowing isomorphismsΓ( Z , D †Y ∨ ←Z ( ∞ )) . Γ( Z , D †Y ∨ ←Z ( ∞ ))( − ∂ y + πx ) ≃ A ( K ) † . ∂ x A ( K ) † + A ( K ) † ( − ∂ y + πx ) ≃ B ( K ) † . ∂ x B ( K ) † ≃ A ( K ) † ≃ Γ( Y ∨ , D †Y ∨ ( ∞ Y ∨ )) . But R Γ( Z , E ) is isomorphic to Γ( Z , E ) placed in degree 0, and also to R Γ( Z , R p ∗ E ). Becausethe cohomology sheaves of Rp ∗ E are acyclic for Γ( Y ∨ , . ), the spectral sequence attached tocomposite functors Γ( Y ∨ , . ) and p ∗ degenerates, proving that R i p ∗ E = 0 for i = 0. Finally theprevious computation gives that p ∗ E , and thus p D Z ( K • )[ − 1] is isomorphic to D †Y ∨ ( ∞ Y ∨ ) (indegree 0) and the lemma. 52e finally get F ! ( D †Y ( ∞ Y )) ≃ D Y p D Z ( K • [ − ≃ D Y ( D †Y ∨ ( ∞ Y ∨ )[2]) ≃ D †Y ∨ ( ∞ Y ∨ )[ − ≃ F ( D †Y ( ∞ Y )) (cf. A.2.4) . It remains to show that the canonical map F ! ( D †Y ( ∞ Y )) → F ( D †Y ( ∞ Y )) maps 1 to 1. For thiswe observe that the canonical map p ′ D ′Z ρ ∗ D Z ( K • )[ − → p D Z D Z ( K • )[ − K • and the map of functors p ′ D ′Z → p D Z . Then identification of D Z D Z ( K • ) also maps 1 to 1, and this finally gives us theisomorphism F ! ( D †Y ( ∞ Y )) ≃ F ( D †Y ( ∞ Y )) . (cid:4) References [Ab] Abe, T.: Coherence of certain overconvergent isocrystals without Frobenius structures on curves , to appearin Math. 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Tomoyuki Abe:Institute for the Physics and Mathematics of the Universe (IPMU)The University of Tokyo5-1-5 Kashiwanoha, Kashiwa, Chiba, 277-8583, Japane-mail: [email protected] Christine Noot-Huyghe:Institut de Recherche Math´ematique Avanc´eeUMR 7501, Universit´e de Strasbourg et CNRS7 rue Ren´e Descartes, 67000 Strasbourg, Francee-mail: [email protected]@math.unistra.fr