[Page 1]

Comparison theorem between Fourier transform

and Fourier transform with compact support

Christine Huyghe

Abstract. We prove a comparison theorem between Fourier transform without support and and Fourier transform with compact support in the context of arithmetic 𝒟-modules.

Abstract. Nous démontrons un théorème de comparaison entre la transformation de Fourier à support compact et la transformation de Fourier sans support, pour les 𝒟-modules arithmétiques.

Contents

1 Preliminaries  2
2 Fourier transforms  7
3 Comparison theorem  13

Introduction

Let p be a prime number, V a discrete valuation ring of inequal characteristics (0, p), containing an element π, the π of Dwork, satisfying the equation π^{p−1} = −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 copies of the formal projective line over 𝒮 compactifying 𝒳 and 𝒳^{∨} and denote ∞_{𝒴} (resp. ∞_{𝒴^{∨}} ) the complementary divisors. In [Ber96] Berthelot constructed sheaves of arithmetic differential operators with overconvergent singularities along a divisor (e.g. 𝒟_{𝒴}^{†}(∞_{𝒴})) in our situation). For several reasons these sheaves have to be thought as sheaves over the open subset which is the complementary of the considered divisor, thus, in our case, over the formal affine line. In [Huy95a] one constructed the Fourier transform of 𝒟_{𝒴}^{†}(∞_{𝒴})-modules, using the Dwork exponential module as kernel and we checked the compatibility with the so-called naïve Fourier transform. The aim of this article is to define the compact support Fourier transform in dimension 1 and to prove that it coincides with the Fourier transform without compact support introduced in loc. cit.. This result, already known in the complex case, is thus the p-adic analogous of the comparison theorem 2.4.1 of Katz-Laumon for the Fourier transform of l-adic sheaves ([KL85]).

This result was originally part from the unpublished part 4.4 of [Huy95a], where it is proven in dimension N. Recently Abe and Marmora ([AM11]) gave a proof of the product formula for p-adic epsilon factors, using this comparison theorem between Fourier and Fourier with compact support for curves and making necessary its publication, at least in the case of dimension 1. The extra work in dimension N consists into proving a generalization of the division lemma 2.4.1 and to deal with longer complexes of length N + 1.

Compared with the original proof, our redaction takes into account recent progress in the understanding of the sheaves of Berthelot, which allows us to simplify the exposition of the proof. The main difference with the complex case, which makes things technically heavier, is that we have to work with sheaves with overconvergent coefficients at infinity over the formal projective line, instead of working directly with sheaves over the affine formal line. We surmount this difficulty thanks to the equivalence of categories given in 1.5.1. One key point of the proof is the fact that the Fourier transform with compact support preserves coherence (2.4.2), which relies on a division lemma (2.4.1). Then, to establish the comparison theorem, by (1.5.1), we may reduce the statement to the case of the sheaf 𝒟_{𝒴}^{†}(∞_{𝒴})) for which we can give an explicit computation.

1. Preliminaries

1.1. Notations.

Denote K the fraction field of V. For l ∈ Z, |l| is the usual archimedean absolute value of l. The p-adic valuation of an element a of a p-adically separated ring is v_{p}(a). For a ∈ K, let us state

|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^{∨} = ℙ_{𝒮}^{1}×ℙ_{𝒮}^{1} is endowed with the ample divisor ∞ = ∞_{𝒴} ×ℙ_{𝒮}^{1} ⋃ ℙ_{𝒮}^{1}×∞_{𝒴^{∨}} . When needed, we will use [u_{0}, u_{1}] and [v_{0}, v_{1}] as homogeneous coordinates over the two copies

of ℙ_{𝒮}^{1}, u_{0} = 0 and v_{0} = 0 are equations of the infinite divisor over each copy of ℙ_{𝒮}^{1}, x = u_{1}/u_{0} and y = v_{1}/v_{0} will be coordinates over the affine plane complementary of the ∞ divisor over 𝒵. These two coordinates x and y over 𝒳 = 𝔸_{𝒮}^{1} and 𝒳^{∨} = 𝔸_{𝒮}^{1} should be considered as dual to each other.

1.2. Sheaves of arithmetic differential operators.

Let 𝒱 be a smooth formal scheme over 𝒮, endowed with a relative divisor D (meaning that D induces a divisor of the special fiber), and 𝒰 = 𝒱\D. Then the direct image by specialization 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 [Ber96]). If 𝒱 is affine and if f is an equation of D over 𝒱, we have the following description

Γ(𝒱, 𝒪_{𝒱} (^{†}D)) = \{∑_{l∈N} \frac{a_{l}}{f^{l}} , a_{l} ∈ Γ(𝒱, 𝒪_{𝒱,Q} ) | and ∃C′ > 0 | v_{p}(a_{l}) − \frac{l}{C′} → +∞ if l → +∞\} .

Consider the ring 𝒟_{𝒱}^{†} of arithmetic differential operators over 𝒱. Suppose that x_{1}, …, x_{n} are coordinates on 𝒱, denote ∂_{x_{i}} the corresponding derivations for i ∈ {1, …, n}, and

∂_{x_{i}}^{[k_{i}]} = \frac{∂_{x_{i}}^{k_{i}}}{k_{i}!} , and \underline{∂}^{[\underline{k}]} = ∂_{x_{1}}^{[k_{1}]} ⋯ ∂_{x_{n}}^{[k_{n}]},

then we have the following description

Γ(𝒱, 𝒟_{𝒱}^{†} ) = \{ ∑_{\underline{k}∈N^{n}} a\underline{k}\underline{∂}^{[\underline{k}]}, a\underline{k} ∈ Γ(𝒱, 𝒪_{𝒱,Q} ) and ∃C > 0 | v_{p}(a\underline{k}) − \frac{|\underline{k}|}{C} → +∞ if |\underline{k}| → +∞\} .

In 4.2 of [Ber96], Berthelot introduces also rings of differential operators with overconvergent singularities, which are sheaves of 𝒪_{𝒱} (^{†}D)-modules. Suppose that x_{1}, …, x_{n} are coordinates on 𝒱 and that the divisor D is defined by the equation f = 0 on 𝒱, then we have the following description

Γ(𝒱, 𝒟_{𝒱}^{†} (D)) = \{ ∑_{l∈N,\underline{k}∈N^{n}} \frac{a_{l,\underline{k}}}{f^{l}} \underline{∂}^{[\underline{k}]}, a_{l,\underline{k}} ∈ Γ(𝒱, 𝒪_{𝒱,Q} )

and ∃C > 0 | v_{p}(a_{l,\underline{k}}) − \frac{l + |\underline{k}|}{C} → +∞ if l + |\underline{k}| → +∞\} .

All these sheaves are weakly complete, in the sense that they are inductive limit of sheaves of p-adic complete rings as described in the following subsection. Let us stress here upon the fact that these sheaves are always sheaves of K vector spaces (event if there is no Q in their notation). We made this convention to avoid too heavy notations. Note also that this won’t be the case with the sheaves that we will introduce in the following subsections.

1.3. Sheaves of arithmetic differential operators with overconvergent coefficients.

In this subsection, we use notations of 1.2. Let us fix m ∈ N, elements x_{1}, …, x_{n} ∈ Γ(𝒱, 𝒪_{𝒱} ) which are coordinates on 𝒱 and f ∈ Γ(𝒱, 𝒪_{𝒱} ) such that D ⋂ 𝒱 = V (f).

Let us introduce here some coefficients. First we define the application ν_{m} : Z → N by the following way. If k < 0, we set ν_{m}(k) = 0. Let k ∈ N, and q and r be the quotient and the 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_{1}, …, k_{r})) = ν_{m}(k_{1}) + ⋯ + ν_{m}(k_{r}).

We denote also q_{k}^{(m)} the quotient of the division of a positive integer k by p^{m}, and

\underline{∂}^{⟨\underline{k}⟩_{(m)}} = q_{k_{1}}^{(m)}! ⋯ q_{k_{n}}^{(m)}!∂_{x_{1}}^{[k_{1}]} ⋯ ∂_{x_{n}}^{[k_{n}]}.

If the choice of m is clear, we will omit it in the notation.

Berthelot defines a sheaf ℬ̂_{𝒱}^{(m)} by setting locally

Γ(𝒱, ℬ̂_{𝒱}^{(m)}) = \{∑_{l} \frac{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 ℬ̂_{𝒱}^{(m)} ⊂ ℬ̂_{𝒱}^{(m+1)} and

𝒪_{𝒱} (^{†}D) = \underrightarrow{lim}_{m} ℬ̂_{𝒱,Q}^{(m)}.

Berthelot defines also sheaves of rings differential operators 𝒟_{𝒱}^{(m)}(D) and their p-adic completion 𝒟̂_{𝒱}^{(m)}(D) over 𝒱 by

Γ(𝒱, 𝒟_{𝒱}^{(m)}(D)) = \{ ∑_{l∈N,\underline{k}∈N^{n}} \frac{a_{l,\underline{k}}}{f^{l}} \underline{∂}^{⟨\underline{k}⟩_{(m)}}, a_{l,\underline{k}} ∈ Γ(𝒱, 𝒪_{𝒱} ) | v_{p}(a_{l,\underline{k}}) ≥ ν_{m}(l)\} ,

where the sums are finite and

Γ(𝒱, 𝒟̂_{𝒱}^{(m)}(D)) = \{ ∑_{l∈N,\underline{k}∈N^{n}} \frac{a_{l,\underline{k}}}{f^{l}} \underline{∂}^{⟨\underline{k}⟩_{(m)}}, a_{l,\underline{k}} ∈ Γ(𝒱, 𝒪_{𝒱} ) | v_{p}(a_{l,\underline{k}}) ≥ ν_{m}(l)

and v_{p}(a_{l,\underline{k}}) − ν_{m}(l) → +∞ if |\underline{k}| + l → +∞\} .

Then there are canonical injective morphisms 𝒟̂_{𝒱,Q}^{(m)}(D) ⊂ 𝒟̂_{𝒱,Q}^{(m+1)}(D) and

𝒟_{𝒱}^{†} (^{†}D) = \underrightarrow{lim}_{m} 𝒟̂_{𝒱,Q}^{(m)}(D).

If D = ∅ the previous sheaves are simply denoted 𝒟_{𝒱}^{(m)} and 𝒟̂_{𝒱}^{(m)}.

1.4. Inequalities.

We finally recall the following inequalities for |\underline{k}| and |\underline{l}| elements of N^{n}, l and r in N.

\frac{|\underline{k}|}{p − 1} − nlog_{p}(|\underline{k}| + 1) − n ≤ v_{p}(\underline{k}!) ≤ \frac{|\underline{k}|}{p − 1}
\frac{|\underline{k}|}{p^{m}(p − 1)} − nlog_{p}(|\underline{k}| + 1) − n\frac{p}{p − 1} ≤ v_{p}(q(m) ; \underline{k}!) ≤ \frac{|\underline{k}|}{p^{m}(p − 1)}
\frac{|\underline{l}|}{p^{m+1}} ≤ ν_{m}(\underline{l}) ≤ \frac{|\underline{l}|}{p^{m+1}} + n
0 ≤ ν_{m}(\underline{l}) − ν_{m}(|\underline{l}|) ≤ n
0 ≤ v_{p} (\binom{l}{r}) ≤ p(log_{p}(l) + 1).

1.5. An equivalence of categories.

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 the one of 𝒟_{𝒱}^{†} (D).

Coming back to the situation of the introduction, let p_{1} and p_{2} be the two projections 𝒵 → 𝒴 and 𝒵 → 𝒴^{∨}, and ∞′ = p_{2}^{−1}∞_{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 [Huy95b]). First note that

Γ(𝒴, 𝒪_{𝒴} (^{†}∞_{𝒴})) = {∑_{l} b_{l}x^{l}, b_{l} ∈ K, and ∃C > 0, η < 1 | |b_{l}|_{p} < Cη^{l}}

and set

A_{1}(K)^{†} = \{ ∑_{k∈N,l∈N} a_{l,k}x^{l}∂_{x}^{[k]}, a_{l,k} ∈ K, and ∃C > 0, η < 1 | |a_{\underline{k},\underline{l}}|_{p} < Cη^{l+k}\} ,

the weak completion of the Weyl algebra. It is a coherent algebra and we have ( [Huy95b])

Theorem 1.5.1. The functor Γ(𝒴, .) (resp. RΓ(𝒴, .)) establishes an equivalence of categories between the category of left coherent 𝒟_{𝒴}^{†}(∞_{𝒴})-modules (resp. D_{coh}^{b}(𝒟_{𝒴}^{†}(∞_{𝒴}))) and the category of left coherent A_{1}(K)^{†}-modules (resp. D_{coh}^{b}(A_{1}(K)^{†}).

In particular, A_{1}(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 can be taken finite since A_{1}(K)^{†} has finite cohomological dimension ([NH07]).

Consider now the situation over 𝒵 for m fixed. In theorem 3 of [Huy95b], we proved that the elements ∂_{x}^{⟨k_{1}⟩_{(m)}} are global sections over ℙ_{𝒮}^{1} of the sheaves 𝒟_{ℙ_{𝒮}^{1}}^{(m)}, so that

∂_{x}^{⟨k_{1}⟩_{(m)}}∂_{y}^{⟨k_{2}⟩_{(m)}} ∈ Γ(𝒵, 𝒟_{𝒵}^{(m)}).

Moreover, an easy computation (2.1 of [Huy95b]) shows that elements p^{ν_{m}(l_{1})}x^{l_{1}} for l_{1} ≥ 0 are global sections of ℬ̂_{ℙ_{𝒮}^{1}}^{(m)}, implying that

p^{ν_{m}(\underline{l})}x^{l_{1}}y^{l_{2}} ∈ Γ(𝒵, ℬ̂_{𝒵}^{(m)}).

Define

Ê^{(m)} = \{ ∑_{\underline{k}∈N^{2},\underline{l}∈N^{2}} a_{\underline{l},\underline{k}}x^{l_{1}}y^{l_{2}}∂_{x}^{⟨k_{1}⟩}∂_{y}^{⟨k_{2}⟩}, a_{\underline{l},\underline{k}} ∈ V, and v_{p}(a_{\underline{l},\underline{k}}) ≥ ν_{m}(\underline{l})

| v_{p}(a_{\underline{l},\underline{k}}) − ν_{m}(\underline{l}) → +∞ if |\underline{l}| + |\underline{k}| → +∞\} .

From these observations and the fact that 𝒟̂_{𝒵}^{(m)} is a sheaf of p-adically complete algebras, we see that

Ê^{(m)} ⊂ Γ(𝒵, 𝒟̂_{𝒵}^{(m)}(∞)).

Consider the weak completion of the Weyl algebra in 2 variables

A_{2}(K)^{†} = \{ ∑_{\underline{k}∈N^{2},\underline{l}∈N^{2}} a_{\underline{l},\underline{k}}x^{l_{1}}y^{l_{2}}∂_{x}^{[k_{1}]}∂_{y}^{[k_{2}]}, a_{l,k} ∈ K, and ∃C > 0, η < 1 | |a_{\underline{k},\underline{l}}| < Cη^{|\underline{l}|+|\underline{k}|}\} ,

which is coherent from [Huy95b]. It is easy to see that

A_{2}(K)^{†} = \underrightarrow{lim}_{m} Ê_{Q}^{(m)}.

We endow A_{2}(K)^{†} with the inductive limit topology coming from this filtration.

Because the divisor ∞ over 𝒵 is ample, we can apply 4.5.1 of [Huy03] which tells us that

Γ(𝒵, 𝒟_{Z}^{†}(∞)) ≃ A_{2}(K)^{†}.

Moreover we have the following theorem (4.5.1 of [Huy03] and 5.3.4 of [Huy98]).

Theorem 1.5.2. The functor Γ(𝒵, .) (resp. RΓ(𝒵, .)) establishes an equivalence of categories between the category of left coherent 𝒟_{𝒵}^{†} (∞)-modules (resp. D_{coh}^{b}(𝒟_{𝒵}^{†} (∞))) and the category of left coherent A_{2}(K)^{†}-modules (resp. D_{coh}^{b}(A_{2}(K)^{†})).

Consider now the scheme 𝒵 = 𝒴 × 𝒴^{∨} endowed with the divisor ∞′ = p_{2}^{−1}(∞_{𝒴^{∨}} ) and 𝒟_{𝒵}^{†} (∞′) the ring of arithmetic differential operators with overconvergent coefficients along ∞′. In order to deal with coherent 𝒟_{𝒵}^{†} (∞′)-modules, denote

B_{2}(K)^{†} = \{ ∑_{l_{2}∈N,\underline{k}∈N^{2}} a_{l_{2},\underline{k}}y^{l_{2}}∂_{x}^{[k_{1}]}∂_{y}^{[k_{2}]}, a_{l_{2},\underline{k}} ∈ K, and ∃C > 0, η < 1 | |a_{l_{2},\underline{k}}|_{p} < Cη^{l_{2}+|\underline{k}|}\} ,

which we endow with the induced topology of A_{2}(K)^{†}. Consider F̂^{(m)} = Ê^{(m)} ⋂ B_{2}(K)^{†}. As before we observe that

F̂^{(m)} ⊂ Γ(𝒵, 𝒟̂_{𝒵}^{(m)}(∞′)),

which leads to the following inclusion

\underrightarrow{lim}_{m} F̂_{Q}^{(m)} = B_{2}(K)^{†} ⊂ Γ(𝒵, 𝒟_{𝒵}^{†} (∞′)).

We apply 2.3.3 of [NH04] to see that this is actually an equality.

Finally, we will also use the following division lemma (4.3.4.2 of [NH04]).

Theorem 1.6. i For any P ∈ A_{2}(K)^{†} there exists a unique (Q, R) ∈ A_{2}(K)^{†} × B_{2}(K)^{†} such that P = Q(−∂_{y} + πx) + R.

ii The maps P ↦ Q and P ↦ R are continuous. More precisely, if P ∈ Ê_{Q}^{(m)}, then Q ∈ Ê_{Q}^{(m+2)}, and R ∈ Ê_{Q}^{(m+2)} ⋂ B_{2}(K)^{†}.

2. Fourier transforms

2.1. Kernel of the Fourier transform.

In dimension 1, the duality pairing 𝒳×𝒳^{∨} → 𝔸̂_{𝒮}^{1} is given by t ↦ xy where t is the global coordinate on the right-hand side. It extends to δ : 𝒵 = 𝒴 × 𝒴^{∨} → ℙ̂_{𝒮}^{1}, by the formula t^{−1} ↦ x^{−1}y^{−1} on neighborhoods of ∞. Let L_{π} be the realization over ℙ̂_{𝒮}^{1} of the overconvergent Dwork F -isocristal. It is given by a connection ∇(1) = −πdt. We define K_{π} to be the 𝒟_{𝒵,Q}^{†} (∞)-module associated to the overconvergent F -isocristal 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).

2.2. Explicit descriptions of cohomological operations.

Let us consider the following diagram

\xymatrix{ & 𝒵 = 𝒴 × 𝒴^{∨} \ar[dr]^{p_{2}} \ar[dl]_{p_{1}} \\ 𝒴 &  & 𝒴^{∨}.}

For M, N two 𝒪_{𝒵} (^{†}∞)-modules, denote

M ˜⊗N = M ⊗_{𝒪_{𝒵} (^{†}∞)}^{L} N[−2].

Note that the sheaves p_{1}^{−1}𝒟_{𝒴}^{†}(∞_{𝒴}) (resp. p_{2}^{−1}𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )), are canonicaly subsheaves of rings of 𝒟_{𝒵}^{†} (∞). Cohomological operations involve sheaves 𝒟_{𝒵→𝒴}^{†} (∞), respectively 𝒟_{𝒴←𝒵}^{†} (∞), which are left (resp. right) coherent 𝒟_{𝒵}^{†} (∞)-modules and right (resp. left) p_{1}^{−1}𝒟_{𝒴}^{†}(∞_{𝒴})-modules, which can be explicitely described in our case. The module structures over 𝒟_{𝒴←𝒵}^{†} (∞) are obtained from these of 𝒟_{𝒵→𝒴}^{†} (∞) by twisting by the adjoint operator (1.3 of [Ber00]). In particuler underlying abelian groups of both sheaves are the same. Because the sheaves

𝒪_{𝒵} (^{†}∞) ⊗_{𝒪_{𝒵}} ω_{𝒵} and 𝒪_{𝒵} (^{†}∞) ⊗_{p_{1}^{−1}𝒪_{𝒴}} p_{1}^{−1}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 [NH04], we know that these sheaves admit a free resolution, as 𝒟_{𝒵}^{†} (∞)-modules

\xymatrix{0 \ar[r] & 𝒟_{𝒵}^{†} (∞) \ar[r] & 𝒟_{𝒵}^{†} (∞) \ar[r] & 0 \\  & P \ar@{|->}[r] & P ∂_{y}.}

For the sheaf 𝒟_{𝒴←𝒵}^{†} (∞), consider the map P ↦ ∂_{y}P. Actually, if we endow 𝒟_{𝒵}^{†} (∞) with the canonical structure of right (resp. left)-p^{−1}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 1.5.2 the sheaves 𝒟_{𝒵→𝒴}^{†} (∞) and 𝒟_{𝒴←𝒵}^{†} (∞) are acyclic for the global section functor.

Let us also remark that 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 the sections of A_{2}(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 itself by the adjoint operator.

For M ∈ D_{coh}^{b}(𝒟_{𝒴}^{†}(∞_{𝒴})) we state as usual

p_{1}^{!}(M) = 𝒟_{𝒵→𝒴}^{†} (∞) ⊗_{p_{1}^{−1}𝒟_{𝒴}^{†} (∞_{𝒴} )}^{L} p_{1}^{−1}M [1] ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞))

and for N ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞))

p_{2+}(N) = Rp_{2∗} (𝒟_{𝒴^{∨}←𝒵}^{†} (∞) ⊗_{𝒟_{𝒵}^{†} (∞)}^{L} N) ,

whose cohomology sheaves are not coherent in general. Fourier transform can now be defined.

2.3. Fourier transform.

Definition 2.3.1. For M ∈ D_{coh}^{b}(𝒟_{𝒴}^{†}(∞_{𝒴}))

ℱ(M) = p_{2+}(p_{1}^{!}M ˜⊗K_{π}).

Let us recall the fundamental results of [NH04].

Theorem 2.3.2. i (4.3.4 of [NH04]) there is a canonical isomorphism

𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} ))[−1] → ℱ(𝒟_{𝒴}^{†}(∞_{𝒴})),

ii (5.3.1 of loc. cit.) If M ∈ D_{coh}^{b}(𝒟_{𝒴}^{†}(∞_{𝒴})), then ℱ(M) ∈ D_{coh}^{b}(𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )).

2.4. Fourier transform with compact support.

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′_{2+}: D_{coh}^{b}(𝒟_{𝒵}^{†} (∞′)) → D_{coh}^{b}(𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )), which preserves coherence, since p_{2} is proper and thanks to the fact that ∞′ = p_{2}^{−1}(∞).

We will also use the scalar restriction functor ρ : D^{b}(𝒟_{𝒵}^{†} (∞)) → D^{b}(𝒟_{𝒵}^{†} (∞′)). For M in D_{coh}^{b}(𝒟_{𝒵}^{†} (∞)), we denote the dual

D_{𝒵} (M) = Rℋom_{𝒟_{𝒵}^{†} (∞)} (M, 𝒟_{𝒵}^{†} (∞) ⊗_{𝒪_{𝒵}} ω_{𝒵}^{−1}[2]) ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞)),

since the sheaf 𝒟_{𝒵}^{†} (∞) has finite cohomological dimension ([NH07]), and the corresponding dual functor D′_{𝒵} (M), for M ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞′)), (resp. D_{𝒴} and D_{𝒴^{∨}} for 𝒟_{𝒴}^{†}(∞_{𝒴}), resp. 𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )- modules).

The following division lemma will be crucial.

Lemma 2.4.1. Let 𝒰 = D_{+}(u_{1}) × D_{+}(v_{0}) or 𝒰 = D_{+}(u_{1}) × D_{+}(v_{1}) 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.

Proof. 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_{1})×D_{+}(v_{0}) will be endowed with coordinates x′ = 1/x and t = y, and D_{+}(u_{1})× D_{+}(v_{1}) with coordinates x′ and t = 1/y. Denote D^{†} = 𝒟_{𝒵}^{†} (∞)(𝒰), and D′^{†} = 𝒟_{𝒵}^{†} (∞′)(𝒰), which contains the algebras F̂_{Q}^{(m)} described in 1.5.

By considering the right 𝒪_{𝒵} -module structure on 𝒟_{𝒵}^{†} (∞), we see that an element Q ∈ D^{†} can be written

Q = ∑_{\underline{k}∈N^{2},\underline{l}∈Z^{2}} a_{\underline{l},\underline{k}}∂_{x′}^{[k_{1}]}∂_{t}^{[k_{2}]}x′^{l_{1}}t^{l_{2}},

such that

a_{\underline{l},\underline{k}} ∈ K and ∃C > 0, M > 0, | v_{p}(a_{\underline{l},\underline{k}}) ≥ \frac{max\{−l_{1}, 0\} + max\{−l_{2}, 0\} + |\underline{k}|}{C} − M,  (1)

(note that on D_{+}(u_{1}) × D_{+}(v_{0}) coefficients a_{(l_{1},l_{2}),\underline{k}} are equal to 0 if l_{2} < 0). With this notation Q ∈ D′^{†} if and only if ∀\underline{k} ∈ N^{2}, ∀\underline{l} | l_{1} < 0, a_{\underline{l},\underline{k}} = 0. We can also write Q this way

Q = ∑_{l_{1}∈Z} Q_{l_{1}}x′^{l_{1}},

with

Q_{l_{1}} = ∑_{{\underline{k}=(k_{1},k_{2})∈N×N \atop l_{2}∈Z}} a_{(l_{1},l_{2}),\underline{k}}∂_{x′}^{[k_{1}]}∂_{t}^{[k_{2}]}t^{l_{2}} ∈ Γ(𝒰, 𝒟_{𝒵}^{†} (∞′)).

Let us observe that the condition (1) is equivalent to the fact that there exist m > 0 such that

Q_{l_{1}} ∈ Γ(𝒰, 𝒟̂_{𝒵,Q}^{(m)}(∞′))

and elements T_{l_{1}} of Γ(𝒰, 𝒟̂_{𝒵}^{(m)}(∞′)) such that

Q_{l_{1}} = u_{l_{1}}p^{ν_{m}(−l_{1})}T_{l_{1}}, for some u_{l_{1}} ∈ K satisfying v_{p}(u_{l_{1}}) → +∞ if |l_{1}| → +∞.  (2)

We define

R′′ = ∑_{l_{1}∈N} Q_{l_{1}}x′^{l_{1}} ∈ Γ(𝒰, 𝒟̂_{𝒵,Q}^{(m)}(∞′)),

so that we can write down

Q = ∑_{l_{1}<0} u_{l_{1}}p^{ν_{m}(|l_{1}|)}T_{l_{1}}x^{|l_{1}|} + R′′,  (3)

with T_{l_{1}} ∈ Γ(𝒰, 𝒟̂_{𝒵,Q}^{(m)}(∞)) and v_{p}(u_{l_{1}}) → +∞ if |l_{1}| → +∞.

We will first prove (iii). Suppose that Q(−∂_{y} + πx) ∈ Γ(𝒰, 𝒟_{𝒵}^{†} (∞′)). Because of (3), we can suppose that

Q = ∑_{l_{1}<0} Q_{l_{1}}x′^{l_{1}}.

Now we are reduced to prove that Q = 0, that we can do in restriction to D_{+}(u_{1}) × D_{+}(v_{0}v_{1}), on which we choose x′ and y as coordinates.

Let us compute

Q(−∂_{y} + πx) = ∑_{l_{1}<0} (−Q_{l_{1}}∂_{y} + πQ_{l_{1}+1}) x′^{l_{1}}.

Therefore, Q(−∂_{y} + πx) is an element of Γ(𝒰, 𝒟_{𝒵}^{†} (∞′)) if and only if

Q_{−1}∂_{y} = 0, and  (4)

∀l_{1} ≤ −1, −Q_{l_{1}}∂_{y} + πQ_{l_{1}+1} = 0.  (5)

Let us decompose

Q_{−1} = ∑_{\underline{k}} β\underline{k}(y)∂_{x′}^{[k_{1}]}∂_{y}^{[k_{2}]},

with β\underline{k}(y) ∈ 𝒪_{𝒵,Q} (D_{+}(v_{0}u_{0}u_{1})), then we compute

Q_{−1}∂_{y} = ∑_{\underline{k}|k_{2}≥1} k_{2}β_{(k_{1},k_{2}−1)}(y)∂_{x′}^{[k_{1}]}∂_{y}^{[k_{2}]},

which is nul if and only if ∀\underline{k}, β\underline{k} = 0, i.e. Q_{−1} = 0. Thus by descending induction, one sees from (5) that ∀l_{1} ≤ −1, Q_{l_{1}} = 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 following equalities

p^{ν_{m}(|l_{1}|)}x^{|l_{1}|} = \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|}} ((−∂_{y} + πx) + ∂_{y})^{|l_{1}|}

= \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|}} |l_{1}|!∂_{y}^{[|l_{1}|]}

+ [∑_{r=1}^{|l_{1}|} ∑_{s=0}^{r−1}(−1)^{r−1+s} \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|−s}} \binom{|l_{1}|}{r}\binom{r − 1}{s}(|l_{1}| − 1 − s)!∂_{y}^{[|l_{1}|−1−s]}x^{s}] (−∂_{y} + πx)

= \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|}} \frac{|l_{1}|!}{q_{|l_{1}|}^{(m+2)}!}∂_{y}^{⟨|l_{1}|⟩_{(m+2)}}

+ [∑_{r=1}^{|l_{1}|} ∑_{s=0}^{r−1}(−1)^{r−1+s} \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|−s}} \binom{|l_{1}|}{r}\binom{r − 1}{s}\frac{(|l_{1}| − 1 − s)!}{q_{|l_{1}|−1−s}^{(m+2)}!}∂_{y}^{⟨|l_{1}|−1−s⟩_{(m+2)}}x^{s}] (−∂_{y} + πx).

Denote

R_{l_{1}} = \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|}} \frac{|l_{1}|!}{q_{|l_{1}|}^{(m+2)}!}∂_{y}^{⟨|l_{1}|⟩_{(m+2)}},

S_{l_{1}} = ∑_{r=1}^{|l_{1}|} ∑_{s=0}^{r−1}(−1)^{r−1+s} \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|−s}} \binom{|l_{1}|}{r}\binom{r − 1}{s}\frac{(|l_{1}| − 1 − s)!}{q_{|l_{1}|−1−s}^{(m+2)}!}∂_{y}^{⟨|l_{1}|−1−s⟩_{(m+2)}}x^{s},

c_{l_{1}}(r, s) = (−1)^{r−1+s} \frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|−s}} \binom{|l_{1}|}{r}\binom{r − 1}{s}\frac{(|l_{1}| − 1 − s)!}{q_{|l_{1}|−1−s}^{(m+2)}!}.

By definition, we have the following relation

p^{ν_{m}(|l_{1}|)}x^{|l_{1}|} = S_{l_{1}}(−∂_{y} + πx) + R_{l_{1}}.  (6)

Then, from estimations 1.4 we see that

v_{p} (\frac{p^{ν_{m}(|l_{1}|)}}{π^{|l_{1}|}} \frac{|l_{1}|!}{q_{|l_{1}|}^{(m+2)}!}) ≥ \frac{p^{2} − p − 1}{p^{m+2}(p − 1)}|l_{1}| − log_{p}(|l_{1}| + 1) − 1 → ∞ if |l_{1}| → +∞,

which proves that R_{l_{1}} ∈ Ê^{(m+2)} for |l_{1}| big enough. We also see that ∀r ≤ |l_{1}| − 1, s ≤ r,

v_{p} (c_{l_{1}}(r, s)) ≥ \frac{|l_{1}|}{p^{m+1}} − \frac{|l_{1}| − s}{p − 1} − log_{p}(|l_{1}| + 1) + \frac{|l_{1}| − 1 − s}{p − 1} − 1 − \frac{|l_{1}| − 1 − s}{p^{m+2}(p − 1)}

≥ (\frac{p^{2} − p − 1}{p^{m+2}(p − 1)}) |l_{1}| − log_{p}(|l_{1}| + 1) − 2 → ∞ if |l_{1}| → +∞,

which proves that S_{l_{1}} ∈ Ê^{(m+2)} for |l_{1}| big enough. As a consequence S_{l_{1}} and resp. R_{l_{1}} are elements of Γ(𝒰, 𝒟̂_{𝒵}^{(m+2)}(∞)) for |l_{1}| big enough (resp. Γ(𝒰, 𝒟̂_{𝒵}^{(m+2)}(∞′))).

Let Q ∈ D^{†}. We can use the description given in (3). Since |u_{l_{1}}| → 0 if |l_{1}| → +∞, we observe that

Q′ = ∑_{l_{1}<0} u_{l_{1}}T_{l_{1}}S_{l_{1}} ∈ Γ(𝒰, 𝒟̂_{𝒵,Q}^{(m+2)}(∞)),

and

R′ = ∑_{l_{1}<0} u_{l_{1}}T_{l_{1}}R_{l_{1}} ∈ Γ(𝒰, 𝒟̂_{𝒵,Q}^{(m+2)}(∞′)).

Moreover, we have the following equalities

Q = ∑_{l_{1}<0} u_{l_{1}}p^{ν_{m}(|l_{1}|)}T_{l_{1}}x^{|l_{1}|} + R′′

= ∑_{l_{1}<0} u_{l_{1}}T_{l_{1}} (S_{l_{1}}(−∂_{y} + πx) + R_{l_{1}}) + 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′^{†}. As in the previous case, we restrict ourselves to 𝒰 = D_{+}(u_{0}) × D_{+}(v_{0}v_{1}) and we decompose

Q = ∑_{l_{1}∈N} Q_{l_{1}}x′^{l_{1}},

The recursive formula (5) still holds and we get

πQ_{0} = 0, and
∀l_{1} ≥ 0, πQ_{l_{1}+1} = Q_{l_{1}}∂_{y}.

By induction, this proves that Q_{l_{1}} = 0 for all l_{1} ≥ 0 and thus that Q = 0.

The key lemma to define the Fourier transform with compact support is the following:

Lemma 2.4.2. Let M ∈ D_{coh}^{b}(𝒟_{𝒴}^{†}(∞_{𝒴}))), then

ρ_{∗}D_{𝒵} (p_{1}^{!}M ˜⊗K_{π}) ∈ D_{coh}^{b}(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 (1.5.1). It is thus enough to prove this lemma in the case of 𝒟_{𝒴}^{†}(∞_{𝒴}) itself.

If ℱ is an 𝒪_{𝒵} -coherent module, we denote

ℱ̃ = ℱ ⊗_{𝒪_{𝒵}} 𝒪_{𝒵} (^{†}∞).

Then by 2.1.1 of [NH04] 𝒯̃_{𝒵/S} (resp. 𝒯̃_{𝒵/𝒴} ) are free 𝒪_{𝒵} (^{†}∞)-modules of basis ∂_{x}, ∂_{y} (resp. ∂_{y}).

Let us reformulate lemma 4.2.4 of [NH04]. Let K_{•} the complex of induced 𝒟_{𝒵}^{†} (∞)-modules in degrees −1 and 0

0 → 𝒟_{𝒵}^{†} (∞) ⊗_{𝒪_{𝒵}^{†} (∞)} Λ^{1}𝒯̃_{𝒵/𝒴} →^{d} 𝒟_{𝒵}^{†} (∞) → 0,

where

d(P ⊗ ∂_{y}) = P · (∂_{y} + πx).

Lemma 2.4.3. In the derived category D_{coh}^{b}(𝒟_{𝒵}^{†} (∞)) this complex K_{•} is equal to the complex

p_{1}^{!}𝒟_{𝒴}^{†}(∞_{𝒴})˜⊗_{𝒪_{𝒵}^{†} (∞)} K_{π}[1],

which is nothing but the complex p_{1}^{∗}𝒟_{𝒴}^{†}(∞_{𝒴}) ⊗ K_{π} (in degree 0).

The augmentation map ε′ : 𝒟_{𝒵}^{†} (∞) → p_{1}^{∗}𝒟_{𝒴}^{†}(∞_{𝒴}) ⊗ K_{π} is given by

ε′(∂_{y}) = −πx ⊗ 1
ε′(∂_{x}) = (∂_{x} − πy) ⊗ 1.

In the rest of the proof, we identify the left induced 𝒟_{𝒵}^{†} (∞)-module 𝒟_{𝒵}^{†} (∞) ⊗_{𝒪_{𝒵}^{†} (∞)} Λ^{1}𝒯̃_{𝒵/𝒴} with 𝒟_{𝒵}^{†} (∞). Then Rℋom_{𝒟_{𝒵}^{†} (∞)} (K_{•}, 𝒟_{𝒵}^{†} (∞))[2] is represented by the following complex of right 𝒟_{𝒵}^{†} (∞)-modules, whose terms are in degrees −2 and −1

0 → 𝒟_{𝒵}^{†} (∞) →^{d′} 𝒟_{𝒵}^{†} (∞) → 0,  (7)

such that d′(P) = (∂_{y} + πx)P. Finally, we see that L_{•} = D(K_{•}) is represented in D_{coh}^{b}(𝒟_{𝒵}^{†} (∞)) by the following complex in degrees −2 et −1

0 → 𝒟_{𝒵}^{†} (∞) →^{d′′} 𝒟_{𝒵}^{†} (∞) → 0,  (8)

such that d′′(P) = P (−∂_{y} + πx).

Consider now the canonical map

𝒟_{𝒵}^{†} (∞′) → 𝒟_{𝒵}^{†} (∞) /𝒟_{𝒵}^{†} (∞)(−∂_{y} + πx) .

Over D_{+}(u_{0}) both sheaves 𝒟_{𝒵}^{†} (∞′) and 𝒟_{𝒵}^{†} (∞) coïncide. Let us study the situation over D_{+}(u_{1}). From the previous lemma 2.4.1, we see that this map is surjective over D_{+}(u_{1}) × D_{+}(v_{0}) and D_{+}(u_{1}) × D_{+}(v_{1}) and that over these open subsets the following complex is exact (using notations of 2.4.1)

0 → 𝒟_{𝒵}^{†} (∞′) →^{d′′} 𝒟_{𝒵}^{†} (∞′) → ρ_{∗} (𝒟_{𝒵}^{†} (∞) /𝒟_{𝒵}^{†} (∞)(−∂_{y} + πx)) → 0,

where d′′(P) = P (−x′∂_{y} +π), showing that ρ_{∗}D_{𝒵} (K_{•}) ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞′)), hence ρ_{∗}D_{𝒵} (p_{1}^{!}M ˜⊗K_{π}) ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞′)) for any M ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞)).

Finally, this leads us to the following definition.

Definition 2.4.4. For M ∈ D_{coh}^{b}(𝒟_{𝒴}^{†}(∞_{𝒴}))

ℱ_{!}(M) = p′_{2+}D′_{𝒵} ρ_{∗}D_{𝒵} (p_{1}^{!}M ˜⊗K_{π}) ∈ D_{coh}^{b}(𝒟_{𝒵}^{†} (∞′)).

Note that from the previous lemma 2.4.2, we know that ℱ_{!}(M) ∈ D_{coh}^{b}(𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )).

3. Comparison theorem

Proposition 3.1. Let M ∈ D_{coh}^{b}(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. There are canonical maps D′_{𝒵} (ρ_{∗}E) → D_{𝒵} (E) and also canonical maps

p′_{2+}D′_{𝒵} (E) → p_{2+}D_{𝒵} (E).

Applying this to E = ρ_{∗}D_{𝒵} (∞)(p_{1}^{!}M ˜⊗K_{π}) gives a canonical map

ℱ_{!}(M) → p_{2+}D_{𝒵} D_{𝒵} (p_{1}^{!}M ˜⊗K_{π}).

And we apply the biduality theorem 3.6 of [Vir00], to see that the RHS can be identified with ℱ(M).

Theorem 3.2. Let M ∈ D_{coh}^{b}(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 1.5.1, we can suppose that M = 𝒟_{𝒴}^{†}(∞_{𝒴}). This computation is the aim of the next subsection.

3.3. Computation of ℱ_{!}(𝒟_{𝒴}^{†}(∞_{𝒴})).

Let us consider 𝒰 one of the open sets D_{+}(u_{1}) × D_{+}(v_{0}) or D_{+}(u_{1}) × D_{+}(v_{1}) of 𝒵, denote by x′ and t coordinates over 𝒰. Then, over 𝒰, we have the following resolution of the sheaf 𝒟_{𝒴^{∨}←𝒵}^{†} (∞′) by right 𝒟_{𝒵}^{†} (∞′)-modules

\xymatrix{0 \ar[r] & 𝒟_{𝒵}^{†} (∞′) \ar[r] & 𝒟_{𝒵}^{†} (∞′) \ar[r] & 𝒟_{𝒴^{∨}←𝒵}^{†} (∞′) \ar[r] & 0 \\  & P \ar@{|->}[r] & ∂_{x′}P}

and exactly the same resolution for the sheaf 𝒟_{𝒵}^{†} (∞), replacing 𝒟_{𝒵}^{†} (∞′) by 𝒟_{𝒵}^{†} (∞). This proves that 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 the open set D_{+}(u_{0}) × D_{+}(v_{0}) where the previous isomorphism coincides with identity. Now we have the following lemma.

Lemma 3.3.1. 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, which is an isomorphism over D_{+}(u_{0}) × D_{+}(v_{0}) and thus an isomorphism (cf 4.3.7 of [Ber96]).

With the hypothesis of the lemma, we get

p_{2+}M = Rp_{2∗} (𝒟_{𝒴^{∨}←𝒵}^{†} (∞) ⊗_{𝒟_{𝒵}^{†} (∞)} M)
≃ Rp_{2∗} (𝒟_{𝒴^{∨}←𝒵}^{†} (∞′) ⊗_{𝒟_{𝒵}^{†} (∞′)} 𝒟_{𝒵}^{†} (∞) ⊗_{𝒟_{𝒵}^{†} (∞)} M)
≃ Rp_{2∗} (𝒟_{𝒴^{∨}←𝒵}^{†} (∞′) ⊗_{𝒟_{𝒵}^{†} (∞′)} ρ_{∗}(M))
≃ p′_{2+}M.

Applying this to M = K_{•}[−1] of 2.4.3, we see that

ℱ_{!}(𝒟_{𝒴}^{†}(∞_{𝒴})) = p′_{2+}D′_{𝒵} ρ_{∗}D_{𝒵} (K_{•}[−1])
≃ D_{𝒴} p′_{2+}ρ_{∗}D_{𝒵} (see [Vir04])(K_{•}[−1])
≃ D_{𝒴} p_{2+}D_{𝒵} (K_{•}[−1])

Now we need the following statement

Lemma 3.3.2. There is a canonical isomorphism

𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )[1] ≃ p_{2+}D_{𝒵} (K_{•}).

Proof. Using (8), we know that 𝒟_{𝒴^{∨}←𝒵}^{†} (∞)⊗_{𝒟_{𝒵}^{†} (∞)}^{L} D_{𝒵} (K_{•}) is represented by the following complex with terms in degrees −2 and −1

\xymatrix{0 \ar[r] & 𝒟_{𝒴^{∨}←𝒵}^{†} (∞) \ar[r] & 𝒟_{𝒴^{∨}←𝒵}^{†} (∞) \ar[r] & 0 \\  & P \ar@{|->}[r] & P (−∂_{y} + πx).}

Denote by ℰ the −1-cohomology group of this complex. Since the module 𝒟_{𝒴^{∨}←𝒵}^{†} (∞) is a right coherent 𝒟_{𝒵}^{†} (∞)-module, it is acyclic for the functor Γ(𝒵, .). It is also the case for ℰ by the long cohomology exact sequence and we can then compute Γ(𝒵, ℰ) as the cokernel of the map

\xymatrix{0 \ar[r] & Γ(𝒵, 𝒟_{𝒴^{∨}←𝒵}^{†} (∞)) \ar[r] & Γ(𝒵, 𝒟_{𝒴^{∨}←𝒵}^{†} (∞)) \ar[r] & 0 \\ & P \ar@{|->}[r] & P (−∂_{y} + πx)}.

In particular, we get an element 1 ∈ Γ(𝒵, ℰ), allowing us to consider a morphism φ : 𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} ) → Rp_{2∗}ℰ, that sends P to P · 1, where 1 ∈ R^{0}p_{2∗}ℰ.

On the other hand, from the previous lemma 3.3.1, we know that p_{2+}D_{𝒵} (K_{•}[−1]), thus ℰ, is a coherent 𝒟_{𝒴}^{†}(∞_{𝒴})-module. By 1.5.2, it is enough to prove that the morphism induced on 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 resolution given in 2.2, we identify Γ(𝒵, 𝒟_{𝒴^{∨}←𝒵}^{†} (∞)) with A_{2}(K)^{†}/∂_{x}A_{2}(K)^{†}. Finally, we get the following isomorphisms

Γ(𝒵, 𝒟_{𝒴^{∨}←𝒵}^{†} (∞)) /Γ(𝒵, 𝒟_{𝒴^{∨}←𝒵}^{†} (∞))(−∂_{y} + πx) ≃ A_{2}(K)^{†} /∂_{x}A_{2}(K)^{†} + A_{2}(K)^{†}(−∂_{y} + πx)
≃ B_{2}(K)^{†} /∂_{x}B_{2}(K)^{†}
≃ A_{1}(K)^{†}
≃ Γ(𝒴^{∨}, 𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )).

But RΓ(𝒵, ℰ) is isomorphic to Γ(𝒵, ℰ) placed in degree 0, and also to RΓ(𝒵, Rp_{2∗}ℰ). Because the cohomology sheaves of Rp_{2∗}ℰ are acyclic for Γ(𝒴^{∨}, .), the spectral sequence attached to composite functors Γ(𝒴^{∨}, .) and p_{2∗} degenerates, proving that R^{i}p_{2∗}ℰ = 0 for i ≠ 0. Finally the previous computation gives that p_{2∗}ℰ, and thus p_{2+}D_{𝒵} (K_{•})[−1] is isomorphic to 𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} ) (in degree 0) and the lemma.

We finally get

ℱ_{!}(𝒟_{𝒴}^{†}(∞_{𝒴})) ≃ D_{𝒴} p_{2+}D_{𝒵} (K_{•}[−1])
≃ D_{𝒴} (𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )[2])
≃ 𝒟_{𝒴^{∨}}^{†} (∞_{𝒴^{∨}} )[−1]
≃ ℱ(𝒟_{𝒴}^{†}(∞_{𝒴})) 2.3.2.

It remains to show that the canonical map ℱ_{!}(𝒟_{𝒴}^{†}(∞_{𝒴})) → ℱ(𝒟_{𝒴}^{†}(∞_{𝒴})) maps 1 to 1. For this we observe that the canonical map

p′_{2+}D′_{𝒵} ρ_{∗}D_{𝒵} (K_{•})[−1] → p_{2+}D_{𝒵} D_{𝒵} (K_{•})[−1]

maps 1 (considered as an element of the 0-cohomology group of these complexes) to 1, which is clear from the explicit computation of the complex K_{•} and the map of functors p′_{2+}D′_{𝒵} → p_{2+}D_{𝒵} . Then identification of D_{𝒵} D_{𝒵} (K_{•}) also maps 1 to 1, and this finally gives us the isomorphism

ℱ_{!}(𝒟_{𝒴}^{†}(∞_{𝒴})) ≃ ℱ(𝒟_{𝒴}^{†}(∞_{𝒴})).

References

[AM11] Tomoyuki Abe and Adriano Marmora. Product formula for p-adic epsilon factors. arxiv, pages 1–77, 2011.

[Ber96] P. Berthelot. 𝒟-modules arithmétiques I. Opérateurs différentiels de niveau fini. Ann. scient. Éc. Norm. Sup., 4^{e} série, t. 29, p.185–272, 1996.

[Ber00] P. Berthelot. D-modules arithmétiques II descente par Frobenius. Bull. Soc. Math. France, Mémoire 81, p. 1–135, 2000.

[Huy95a] C. Huyghe. Construction et étude de la Transformation de Fourier pour les D-modules arithmétiques. Thèse de Doctorat, Université de Rennes I, 1995.

[Huy95b] C. Huyghe. Interprétation géométrique sur l’espace projectif des A_{N}(K)^{†}-modules cohérents. C. R. Acad. Sci. Paris, t. 321, Série I, p. 587–590, 1995.

[Huy98] C. Huyghe. D^{†}-affinité des schémas projectifs. Ann. Inst. Fourier, t. 48, fascicule 4, p. 913–956, 1998.

[Huy03] C. Huyghe. Un théorème de comparaison entre les faisceaux d’opérateurs différentiels de Berthelot et de Mebkhout-Narvaez-Macarro. Journal of Algebraic Geometry, 12, No. 1, p. 147–199, 2003.

[KL85] N. Katz and G. Laumon. Transformation de Fourier et majoration de sommes exponentielles. Publ. Math. I.H.E.S., 62, p. 361–418, 1985.

[NH04] Christine Noot-Huyghe. Transformation de Fourier des 𝒟-modules arithmétiques. I. In Geometric aspects of Dwork theory. Vol. I, II, pages 857–907. Walter de Gruyter GmbH & Co. KG, Berlin, 2004.

[NH07] C. Noot-Huyghe. Finitude de la dimension homologique d’algèbres d’opérateurs différentiels faiblement complètes et à coefficients surconvergents. J. Algebra, 307(2), 2007.

[Vir00] A. Virrion. Dualité locale et holonomie pour les 𝒟-modules arithmétiques. Bull. Soc. Math. France, t. 321, p. 101–168, 2000.

[Vir04] Anne Virrion. Trace et dualité relative pour les 𝒟-modules arithmétiques. In Geometric aspects of Dwork theory. Vol. I, II, pages 1039–1112. Walter de Gruyter GmbH & Co. KG, Berlin, 2004.

Christine Huyghe IRMA, UFR de mathématiques Université Louis Pasteur 7, rue René Descartes 67084 STRASBOURG cedex FRANCE mél huyghe@math.unistra.fr, http://www-irma.u-strasbg.fr/˜ huyghe