### 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}

−1### D

Y## 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 ⊗

_{O}

^{L}

X

### M

^{0}

### ' (M ⊗

_{O}

X

### D

_{X}

### ) ⊗

_{D}

^{L}

X

### M

^{0}

### ' M ⊗

^{L}

D_{X}

### (D

X### ⊗

_{O}

X

### M

^{0}

### ), where M ⊗

_{O}

X

### D

_{X}

### (resp. D

X### ⊗

_{O}

X

### M

^{0}

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

_{X}

### -bimodule, and ⊗

^{L}

D_{X}

### always uses up the “trivial” D

_{X}

### -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

^{0}

## 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

^{0}

## − d X ] ' h ^{∗} f + M[d Y

^{0}

## − 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}

0### ] M ' RΓ _{[S∩S}

0### ] 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

[S]### F (V ) = {s ∈ F (V ) : (I

_{S}

### |

_{V}

### )

^{m}

### s = 0, m 0}.

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

[S]### F ' Γ

_{S}

### 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}

1
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}

^{1}

### X V ˇ −→ ˇ ^{p}

^{2}

## 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}

1
X## 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

X### e

^{ϕ}

### is the sheaf O

X### with the D

X### -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

_{2}

## oo ^{γ}

^{V}

## //

### p

_{1}

## $$

## A ^{1} X

## S

### j

## OO

### j // X

### s

## OO

### π V

## oo

### ˜ s

## OO

### id

_{V}

## 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}

1
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

^{0}

### ) //

### f

^{0}

## Y ^{0} × X ^{p}

0

2

## //

### id

_{Y 0}

### ×f

## X

### f

## Y ^{0} ^{(id}

^{Y 0}

^{,h)} // Y ^{0} × Y ^{p}

^{2}

## // Y,

## where p 2 and p ^{0} _{2} are the natural projections. Since (id Y

^{0}

## , 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

^{0}

## ×f ) + p ^{0} _{2} ^{∗} M ' (id Y

^{0}

## ×f ) + (O Y

^{0}

## M) ' O _{Y}

^{0}

## 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

X## yy

### f //

### δ

f## Y

### δ

Y## X × X ^{f}

00

## // X × Y ^{f}

^{0}

## // 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

S### ) as the full abelian subcategory of Mod(D

X### ) whose objects M satisfy Γ

_{[S]}

### 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

2## OO

### p

1## &&

## V × X W ˇ oo α

### r

_{2}

## OO

### β //

### r

_{1}

## W × X W ˇ

### q

_{2}

## ff

### q

1## 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}

1
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

1### = D

A^{1}_{X}

### e

^{t}

### with an arbitrary quasi- coherent D

A^{1}_{X}