For the reader’s convenience, we recall here the notions and results from the theory of algebraic D-modules that we need. Our references were [4, 3, 6, 7].

FRANCESCO BALDASSARRI AND ANDREA D’AGNOLO

Abstract. After works by Katz, Monsky, and Adolphson-Sperber, a compar- ison theorem between relative de Rham cohomology and Dwork cohomology is established in a paper by Dimca-Maaref-Sabbah-Saito in the framework of algebraic D-modules. We propose here an alternative proof of this result. The use of Fourier transform techniques makes our approach more functorial.

1. Review of algebraic D-modules

For the reader’s convenience, we recall here the notions and results from the theory of algebraic D-modules that we need. Our references were [4, 3, 6, 7].

1.1. Basic operations. Let X be a smooth algebraic variety over a field of charac- teristic zero, and let O _X and D _X be its structure sheaf and the sheaf of differential operators, respectively. Let Mod(D _X ) be the abelian category of left D _X -modules, D ^b (D X ) its bounded derived category, and D ^b _qc (D X ) the full triangulated subcate- gory of D ^b (D X ) whose objects have quasi-coherent cohomologies.

Let f : X − → Y be a morphism of smooth algebraic varieties, and denote by D X→Y and D Y ←X the transfer bimodules. We use the following notation for the operations of tensor product, inverse image, and direct image for D-modules ¹

⊗ : D ^b _qc (D X ) × D ^b _qc (D X ) − → D ^b _qc (D X ), (M, M ⁰ ) 7→ M ⊗ _O ^L

X

M ⁰ ,

f ^∗ : D ^b _qc (D _Y ) − → D ^b _qc (D _X ), N 7→ D X→Y ⊗ _f ^L

D

f ⁻¹ N , f + : D ^b _qc (D X ) − → D ^b _qc (D Y ), M 7→ Rf _∗ (D Y ←X ⊗ _D ^L

X

M).

If f : X − → Y and g : Y − → Z are morphisms of smooth algebraic varieties, then there are natural functorial isomorphisms

f ^∗ g ^∗ ' (g ◦ f ) ^∗ , (1.1)

g ₊ f ₊ ' (g ◦ f ) ₊ . (1.2)

For N , N ⁰ ∈ D ^b _qc (D Y ), there is a natural isomorphism (1.3) f ^∗ (N ⊗ N ⁰ ) ' f ^∗ N ⊗ f ^∗ N ⁰ .

For M ∈ D ^b _qc (D X ) and N ∈ D ^b _qc (D Y ), there is a projection formula ² (1.4) f ₊ (M ⊗ f ^∗ N ) ' f + M ⊗ N .

1 About the tensor product, note that M ⊗

X

M

' (M ⊗

X

D

) ⊗

X

M

' M ⊗

D_X

(D

⊗

X

M

), where M ⊗

X

D

(resp. D

⊗

X

M

) is given the natural stucture of left-right (resp. left-left) D

-bimodule, and ⊗

D_X

always uses up the “trivial” D

-module structure.

2 In the appendix we recall the proofs of base change and projection formulae.

1

Consider a Cartesian square of smooth algebraic varieties

(1.5) X ^{0 h}

0

//

f



X

f 



Y ^{0 h} // Y.

For M ∈ D ^b _qc (D X ), there is a base change formula ²

(1.6) f ₊ ⁰ h ^0∗ M[d X

− d X ] ' h ^∗ f + M[d Y

− d Y ], where d X denotes the dimension of X.

1.2. Relative cohomology. Let S be a closed reduced subscheme of X, and de- note by I _S ⊂ O X the corresponding ideal of O _X . For F ∈ Mod(O _X ) one sets ³

Γ _[S] (F ) = lim

−→ m

Hom _O

X

(O _X /I _S ^m , F ).

If M ∈ Mod(D _X ) one checks that Γ _[S] M has a natural left D X -module structure, and one considers the right derived functor

RΓ _[S] : D ^b _qc (D X ) − → D ^b _qc (D X ).

Let i : X \ S − → X be the open embedding, and M ∈ D ^b _qc (D X ). There is a distinguished triangle in D ^b _qc (D X )

(1.7) RΓ _[S] M − → M − → i + i ^∗ M −−→ . ⁺¹

For S, S ⁰ ⊂ X reduced closed subschemes, and M ∈ D ^b _qc (D X ), one has RΓ _[S] M ' M ⊗ RΓ _[S] O X ,

(1.8)

RΓ _[S] RΓ _[S

] M ' RΓ _[S∩S

] M.

(1.9)

Let f : X − → Y be a morphism of smooth varieties, Z ⊂ Y a reduced closed sub- scheme, and set S = f ⁻¹ (Z) ⊂ X. Then there is an isomorphism

(1.10) f + RΓ _[S] M ' RΓ _[Z] f + M.

Let Y be a closed smooth subvariety of X of codimension c Y , and denote by j : Y − → X the embedding. Recall that Kashiwara’s equivalence states that the functors M 7→ j ^∗ M[−c Y ] and N 7→ j + N establish an equivalence between the cat- egory Mod _qc (D _Y ) of quasi-coherent D _Y -modules, and the full abelian subcategory of Mod _qc (D _X ) whose objects M satisfy Γ _{[Y ]} M − ^∼ → M. This extends to derived categories. In particular, the functor

(1.11) j + : D ^b _qc (D Y ) − → D ^b _qc (D X ) is fully faithful, and for M ∈ D ^b _qc (D X ) one has

(1.12) RΓ _{[Y ]} M ' j + j ^∗ M[−c Y ].

3 In other words, for any open subset V ⊂ X, Γ

F (V ) = {s ∈ F (V ) : (I

|

)

s = 0, m  0}.

Note that if F is quasi-coherent, Hilbert’s Nullstellensatz implies that Γ

F ' Γ

F , the subsheaf

of F whose sections are supported in S.

1.3. Fourier-Laplace transform. To ϕ ∈ Γ (X; O X ) one associates the D X - module ⁴

D X e ^ϕ = D X /I ϕ , I ϕ (V ) = {P ∈ D X (V ) : P e ^ϕ = 0}, ∀V ⊂ X open.

For f : X − → Y a morphism of smooth algebraic varieties, and ψ ∈ Γ (Y ; O Y ), one has

(1.13) f ^∗ D _Y e ^ψ ' D _X e ^ψ◦f .

Let us denote by A ¹ X the trivial line bundle on X, and by t ∈ Γ (A ¹ X ; O _A

X

) its fiber coordinate. Let π : V − → X be a vector bundle of finite rank, ˇ π : ˇ V − → X be the dual bundle, γ V : V × X V − ˇ → A ¹ X be the natural pairing, and V ←− V × ^p

X V ˇ −→ ˇ ^p

V be the natural projections. The Fourier-Laplace transform for D-modules is the functor

F _V : D ^b _qc (D _V ) − → D ^b _qc (D V ˇ )

N 7→ p 2+ (p ^∗ ₁ N ⊗ γ _V ^∗ D _A

e ^t ).

The Fourier-Laplace transform is involutive, in the sense that (cf [10, Lemma 7.1 and Appendix 7.5])

(1.14) F V ˇ ◦ F V ' (− id V ) ^∗ .

Let f : V − → W be a morphism of vector bundles over X, and denote by ^t f : ˇ W − → ˇ V the transpose of f . Then for any N ∈ D ^b _qc (D V ) and P ∈ D ^b _qc (D W ) there are natural isomorphisms ⁵

F _W f ₊ N ' ( ^t f ) ^∗ F _V N , (1.15)

F V f ^∗ P ' ( ^t f ) + F W P.

(1.16)

If X is viewed as a zero-dimensional vector bundle over itself, the projection π : V − → X and the zero-section ˇ ι : X − → ˇ V are transpose to each other. Hence (1.16) gives for M ∈ D ^b _qc (D X ) and Q ∈ D ^b _qc (D V ˇ ) the isomorphisms ⁶

ˇ

ι + M ' F V π ^∗ M, (1.17)

ˇ ι ^∗ Q ' π + F V ˇ Q.

(1.18)

2. Dwork cohomology

Let π : V − → X be a vector bundle of rank r, let s : X − → ˇ V be a section of the dual bundle ˇ π : ˇ V − → X, and set ˜ s = id _V × _X s : V − → V × _X V . Recall that ˇ γ _V : V × _X V − ˇ → A ¹ X denotes the pairing, and let F ∈ Γ (V ; O _V ) be the function

F = t ◦ γ V ◦ ˜ s.

Let us denote by S the reduced zero locus of s in X, and by j : S − → X the embed- ding. The geometric framework is thus summarized in the following commutative

4 Equivalently, D

e

is the sheaf O

with the D

-module structure given by the flat connec- tion 1 7→ dϕ.

5 See the appendix for a proof.

6 Note that isomorphism (1.17) is the content of [5, Lemma 2.3], of which we have thus provided

a more natural proof.

diagram whose squares are Cartesian in the category of reduced schemes X ^{ˇ ι} //



V ˇ



V × _X V ˇ

p

oo ^γ

//

p

$$

A ¹ X

S

j

OO

j // X

s

OO

π V

oo

˜ s

OO

id

V.

oo

Generalizing previous results of [9, 13, 2], Theorem 0.2 of [5] gives the following link between relative cohomology and Dwork cohomology

Theorem 2.1. For M ∈ D ^b _qc (D _X ) there is an isomorphism RΓ _[S] M[r] ' π + (π ^∗ M ⊗ D V e ^F ).

Our aim here is to provide a more natural proof of this result.

Proof. For M = O X , the statement reads

(2.1) RΓ _[S] O X [r] ' π + D V e ^F . For a general M ∈ D ^b _qc (D _X ), there are isomorphisms

RΓ _[S] M ' M ⊗ RΓ _[S] O X by (1.8), and

π ₊ (π ^∗ M ⊗ D _V e ^F ) ' M ⊗ π ₊ D _V e ^F by (1.4).

It is thus sufficient to prove (2.1). Setting L = γ _V ^∗ D _A

X

e ^t , there is a chain of isomorphisms

D V e ^F ' ˜ s ^∗ L by (1.13)

' ˜ s ^∗ L ⊗ O V

' p 1+ ˜ s + (˜ s ^∗ L ⊗ O V ) by (1.2) ' p 1+ (L ⊗ ˜ s ₊ O V ) by (1.4) ' p 1+ (L ⊗ ˜ s + π ^∗ O X )

' p 1+ (L ⊗ p ^∗ ₂ s + O X ) by (1.6)

= F V ˇ s ₊ O X . Hence we have

π ₊ D V e ^F ' π + F V ˇ s ₊ O X

' ˇ ι ^∗ s + O X by (1.18), and to prove (2.1) we are left to establish an isomorphism

(2.2) ˇ ι ^∗ s ₊ O X ' RΓ [S] O X [r].

By (1.11), this follows from the chain of isomorphisms ˇ

ι ₊ ˇ ι ^∗ s ₊ O _X ' ˇ ι ₊ ˇ ι ^∗ s ₊ s ^∗ O V ˇ

' RΓ [ˇ ι(X)] RΓ [s(X)] O V ˇ [2r] by (1.12) ' RΓ _{[ˇ ι(S)]} RΓ _{[ˇ ι(X)]} O V ˇ [2r] by (1.9) ' RΓ _{[ˇ ι(S)]} ˇ ι ₊ O _X [r] by (1.12) ' ˇ ι + RΓ [S] O X [r] by (1.10).



Remark 2.2. Kashiwara’s equivalence allows one ⁷ to develop the theory of alge- braic D-modules on possibly singular varieties, so that the formulae stated in the previous section still hold. In this framework, (2.2) is obtained by

ˇ ι ^∗ s + O X ' j + j ^∗ O X [r − c S ] by (1.6) ' RΓ _[S] O X [r] by (1.12), where c _S denotes the codimension of S in X.

Appendix A. Appendix

A.1. Base change and projection formulae. The base change formula (1.6) is proved in [4, Theorem VI.8.4] for h a locally closed embedding ⁸ . Let us recall how to deal with the general case.

Proof of (1.6). The Cartesian square (1.5) splits into the two Cartesian squares X ^{0 (f}

0

,h

) //

f



Y ⁰ × X ^p

0

2

//

id

×f

 

X

f





Y ^{0 (id}

^,h) // Y ⁰ × Y ^p

// Y,

where p 2 and p ⁰ ₂ are the natural projections. Since (id Y

, h) is a closed embedding, by [4] the base change formula holds for the Cartesian square on the left hand side.

We are thus left to prove the base change formula for the Cartesian square on the right hand side. For M ∈ D ^b _qc (D _X ), one has the chain of isomorphisms

(id Y

×f ) + p ⁰ ₂ ^∗ M ' (id Y

×f ) + (O Y

 M) ' O _Y

 f + M

' p ^∗ ₂ f ₊ M,

where  denotes the exterior tensor product. 

Let us also recall, following [3], how projection formula is deduced from base change formula.

Proof of (1.4). Consider the diagram with commutative square X

δ

yy

f //

δ



Y

 δ



X × X ^f

00

// X × Y ^f

// Y × Y,

7 This is done for example in [3]. For S a reduced closed subschemes of a smooth variety X, the idea is to define Mod(D

) as the full abelian subcategory of Mod(D

) whose objects M satisfy Γ

M −→ M.

8 In the language of Gauss-Manin connections, the base change formula is stated in [1, § 3.2.6]

for h flat.

where δ X and δ Y are the diagonal embeddings, δ f is the graph embedding, f ⁰ = f × id Y , and f ⁰⁰ = id X ×f . Then there is a chain of isomorphisms

f + (M ⊗ f ^∗ N ) ' f + δ ^∗ _X (M  f ^∗ N ) ' f + δ ^∗ _X f ^00∗ (M  N )

' δ ^∗ _Y f ₊ ⁰ (M  N ) by (1.6) ' δ ^∗ _Y (f ₊ M  N )

' f + M ⊗ N .

 A.2. Fourier-Laplace transform. The formulae stated in section 1.3 for the Fourier-Laplace transform of algebraic D-modules have their analogues for the Fourier-Deligne transform of `-adic sheaves (see [12] or [11, §III.13]), and for the Fourier-Sato transform of conic abelian sheaves (see [8]). Apart from [10], we do not have specific references for the algebraic D-module case. We thus provide here some proofs.

Proof of (1.15) and (1.16). The following arguments are parallel to those in the proof of [12, Th´ eor` eme 1.2.2.4] or [8, Proposition 3.7.14]. Consider the diagram with Cartesian squares

V ˇ



W ˇ

t

f

oo

V × X V ˇ

p

OO

p

&&

V × X W ˇ oo α

r

OO

β //

r



W × X W ˇ

q

ff

q



V f // W,



where the morphisms p i , q i , r i , for i = 1, 2 are the natural projections. Note that γ V ◦ α = γ W ◦ β. The isomorphism (1.15) is obtained via the following chain of isomorphisms ⁹ , where we set L ₁ = D _A

X

e ^t . ( ^t f ) ^∗ F V N = ( ^t f ) ^∗ p 2+ (p ^∗ ₁ N ⊗ γ _V ^∗ L 1 )

' r 2+ α ^∗ (p ^∗ ₁ N ⊗ γ _V ^∗ L 1 ) by (1.6) ' r 2+ (α ^∗ p ^∗ ₁ N ⊗ α ^∗ γ _V ^∗ L 1 ) by (1.3) ' r 2+ (r ^∗ ₁ N ⊗ α ^∗ γ _V ^∗ L 1 ) by (1.1) ' r 2+ (r ^∗ ₁ N ⊗ β ^∗ γ ^∗ _W L 1 ) by (1.1) ' q 2+ β ₊ (r ^∗ ₁ N ⊗ β ^∗ γ _W ^∗ L 1 ) by (1.2) ' q 2+ (β + r ^∗ ₁ N ⊗ γ _W ^∗ L 1 ) by (1.4) ' q 2+ (q ^∗ ₁ f + N ⊗ γ _W ^∗ L 1 ) by (1.6)

= F _W f ₊ N .

9 Note that these arguments still apply if one replaces L

= D

A¹_X

e

with an arbitrary quasi- coherent D

A¹_X

-module. On the other hand, in order to prove (1.16) we will use the fact that the

Fourier transform is involutive.

Applying the functor F W ˇ to the isomorphism (1.15) with N = F V ˇ Q, Q ∈ D ^b _qc (D V ˇ ), and using (1.14), we get

f + F V ˇ Q ' F W ˇ ( ^t f ) ^∗ Q.

The isomorphism (1.16) is obtained from the one above by interchanging the roles

of f and ^t f . 

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