6 1.5 A first statement of the result

Radon and Fourier transforms for D-modules

Andrea D’Agnolo and Michael Eastwood

Version: July 17, 2002 Contents

1 Radon and Fourier transforms for D-modules 3

1.1 Review on algebraic D-modules . . . . 3

1.2 Review on the Fourier-Laplace transform . . . . 5

1.3 Review on the Radon transform(s) . . . . 6

1.4 Review on the blow-up transform(s) . . . . 6

1.5 A first statement of the result . . . . 7

1.6 Review on integral kernels . . . . 8

1.7 Basic regular holonomic kernels . . . . 9

1.8 Radon and Fourier transforms for D-modules . . . . 11

1.9 Twisted case . . . . 14

2 Radon and Fourier transforms for sheaves 14 2.1 Review on sheaves . . . . 14

2.2 Radon and Fourier transforms for sheaves . . . . 15

2.3 Link with the real blow-up . . . . 16

3 Applications 18 3.1 Radon transform of line bundles . . . . 18

3.2 Radon transform of closed forms . . . . 22

3.3 Euler complex . . . . 23

3.4 Radon transform of differential forms . . . . 28 3.5 Quantization of the Radon transform for differential forms . 29

INTRODUCTION

The Fourier and Radon hyperplane transforms are closely related, and one such relation was established by Brylinski [5] in the framework of holo- nomic D-modules. The integral kernel of the Radon hyperplane transform is associated with the hypersurface S ⊂ P × P^∗ of pairs (x, y), where x is a point in the n-dimensional complex projective space P belonging to the hyperplane y ∈ P^∗. As it turns out, a useful variant is obtained by

considering the integral transform associated with the open complement U of S in P × P^∗. In the first part of this paper we generalize Brylinski’s result in order to encompass this variant of the Radon transform, and also to treat arbitrary quasi-coherent D-modules, as well as (twisted) abelian sheaves. Our proof is entirely geometrical, and consists in a reduction to the one-dimensional case by the use of homogeneous coordinates.

The second part of this paper applies the above result to the quanti- zation of the Radon transform, in the sense of [8]. First we deal with line bundles. More precisely, let P = P(V) be the projective space of lines in the vector space V, denote by (•)◦ R the Radon transform associated with^D U⊂ P × P^∗, and for m ∈ Z set

m^∗= −m − n − 1, DP(m)= DP⊗_OOP(m),

where OP(m) is the −m-th tensor power of the tautological line bundle OP(−1). In [8] it was shown that the natural morphism

DP(−m^∗)

◦ R ⊗ det V −D → DP^∗(−m)

is an isomorphism for m < 0. Using the Fourier transform we give a different proof of this result in Theorem 6, as well as a description of the kernel and cokernel of the above morphism for m ≥ 0. Then we consider differential forms. More precisely, denote by Sp^P_• the Spencer complex.

Recall that the Spencer and de Rham complexes are interchanged by the solution functor, so that the shifted subcomplex Sp^P_{≥q [q]}describes the sheaf of closed q-forms. We establish in Theorem 7 the isomorphism

Sp^P_{≤q [q]}◦ R^D ←^∼− Sp^P_{≥n−q [n−q]}^∗ . (∗) Consider the maps P←− V \ {0}^π −→ V. Denoting by θ the Euler vector field,^j the sheaf π⁻¹Ω^q_P is identified with the subsheaf of j⁻¹Ω^q_Vwhose sections ω satisfy

Lθω = θ ω = 0,

where Lθdenotes the Lie derivative, and the interior product. We obtain (∗) by first relating in Theorem 8 the Radon transform of the sheaf of q- forms with the subsheaf of j⁻¹Ω^n+1−qV^∗ whose sections σ satisfy the Fourier transform of the above relations, namely

Lθσ = dσ = 0.

Acknowledgements

The authors wish to thank Masaki Kashiwara for useful insights and discussions.

1. RADON AND FOURIER TRANSFORMS FOR D-MODULES Let V and W be mutually dual (n + 1)-dimensional real vector spaces, P and P^∗ the associated projective spaces, and x = (x0, . . . , xn) and y = (y0, . . . , yn) dual systems of homogeneous coordinates. Consider the Leray form on P given by

ω(x) = Xn j=0

(−1)^jxjdx0∧ · · · ddxj· · · ∧ dxn,

and note that, setting ex = tx, one has d˜x = tⁿω(x)dt + tⁿ⁺¹dx. Let u(t) be one of the distributions 1, Y(t), 1/t, or δ(t) on the real line, so that bu(t) = δ(t), 1/t, Y(t), 1, respectively. Let ϕ(x) be a homogeneous function with homogeneity degree such that ϕ(x)u(hx, yi)ω(x) descends tob a relative density on P × P^∗ (e.g. if u = 1, thenbu = δ, and ϕ must satisfy the homogeneity relation ϕ(tx) = sgn(t)⁻ⁿt⁻ⁿϕ(x)). One then has the following formal relation between the Radon and Fourier transforms, the usual Radon hyperplane transform corresponding to the case u = 1,

Z

ϕ(x)u(hx, yi)ω(x) =b Z

ϕ(x)

Z

u(t)e^−thx,yidt

 ω(x)

= Z

ψ(ex)e^−hx,yie dxe for ψ(ex) = ϕ(x)t⁻ⁿu(t).

(It is quite delicate to make the above formula precise for functions, but [15]

provides a convenient framework.) The aim of this section is to estab- lish the corresponding relation for D-modules, thus generalizing a result of Brylinski [5].

1.1. Review on algebraic D-modules

For the reader’s convenience, we recall here the notions and results from the theory of D-modules that we need. Refer e.g. to [11, 18, 13] for the analytic case, and to [4, 1] for the algebraic case.

Let X be a smooth algebraic variety over a field k of characteristic zero, and let OX and DX be its structure sheaf and the ring of differen- tial operators, respectively. Let Mod(DX) be the abelian category of left DX-modules, D^b(DX) its bounded derived category, and D^b_q−coh(DX) (resp.

D^b

coh(DX)) the full triangulated subcategory of D^b(DX) whose objects have quasi-coherent (resp. coherent) cohomologies. To M ∈ D^b_coh(DX) one as- sociates its characteristic variety char(M), a closed involutive subvariety of the cotangent bundle T^∗X.

We use the following notations for the operations of external tensor

product, inverse image, and direct image for D-modules

D

 : D^b(DX) × D^b(DY) −→ D^b(DX×Y), Df^∗: D^b(DY) −→ D^b(DX),

Df∗: D^b(DX) −→ D^b(DY),

where f : X −→ Y is a map of smooth algebraic varieties. More precisely, denoting by DX−→Y and DY←−X the transfer bimodules, one has

Df^∗N = DX−→Y ⊗_f^L−1DY f⁻¹N , Df∗M = Rf_∗(DY←−X⊗_D^L

X M).

Recall that these operations preserve quasi-coherency, and if g : Y −→ Z is another map of smooth algebraic varieties, then there are natural isomor- phisms Dg∗Df∗M ' D(g ◦ f )_∗M, and Df^∗Dg^∗P ' D(g ◦ f )^∗P. Moreover, to any Cartesian square is attached a canonical isomorphism as follows

X^{0 f}

0

//

h⁰



Y⁰

h





X ^f // Y

Dh^∗Df∗M ' Df⁰_∗Dh^0∗M, M ∈ D^b_q−coh(DX).

The internal tensor product

⊗ : DD ^b(DX) × D^b(DX) −→ D^b(DX×X)

is defined by M1

D

⊗ M2 = Dδ^∗(M1

D

 M2), where δ : X ,→ X × X is the diagonal embedding. Recall that M1

⊗MD 2' M1⊗_O^L

XM2as OX-modules, and Df^∗(M1

⊗MD 2) ' Df^∗M1

⊗DfD ^∗M2. Moreover, one has the projection formula

Df∗(M⊗ Df^{D ∗}N ) ' Df∗M⊗ N ,^D M ∈ D^b_q−coh(DX), N ∈ D^b_q−coh(DY).

The duality functor

D_X : D^b(DX)^op−→ D^b(DX)

is defined by DXM = RHom_DX(M, DX⊗_OXΩ^⊗−1_X )^[dX], where dXdenotes the dimension of X and ΩX the sheaf of forms of maximal degree. Duality preserves coherency, but it does not preserve quasi-coherency, in general.

The functor

Df!: D^b(DX) −→ D^b(DY) is defined by Df!M = DYDf∗D_XM.

Consider the microlocal correspondence associated with f T^∗X←− X ×^fd Y T^∗Y −→ T^{fπ ∗}Y.

One says that N ∈ D^b_coh(DY) is non-characteristic for f if f_d⁻¹(T_X^∗X) ∩ f_π⁻¹(char(N )) ⊂ X ×Y T_Y^∗Y,

where T_X^∗X denotes the zero-section of T^∗X. Recall the following results Theorem 1. (i) The exterior tensor product

D

 preserves coherency and commutes with duality.

(ii) If f is proper, then Df∗ preserves coherency and commutes with du- ality. In particular, Df∗M ' Df!M for M ∈ D^b_coh(DX).

(iii) If N ∈ D^b_coh(DY) is non-characteristic for f , then Df^∗N is coherent and DXDf^∗N ' Df^∗D_YN . In particular, if f is smooth then Df^∗ preserves coherency and commutes with duality.

Let D^b_hol(DX) (resp. D^b_r-hol(DX)) be the full triangulated subcategory of D^b_coh(DX) consisting of holonomic (resp. regular holonomic) objects.

Holonomy is stable for all of the above operations, and regular holonomy is stable under tensor product, inverse image, and proper direct image.

1.2. Review on the Fourier-Laplace transform

Let V be the affine space associated with an (n+1)-dimensional vector space over k, and let V^∗ be the dual affine space. Denote by D(V) = Γ(V; DV) the Weyl algebra, and recall that since V is affine the two functors

D^b

q−coh(DV) ^RΓ(V;•) // D^b

q−coh(D(V))

DV⊗^L

D(V)(•)

oo

are quasi-inverse to each other. The formal relation P (x, ∂x)e^−hx,yi= Q(y, ∂y)e^−hx,yi

associates to each Q ∈ D(V^∗) a unique P ∈ D(V), called its Fourier trans- form. Since P1P2e^−hx,yi= P1Q2e^−hx,yi= Q2P1e^−hx,yi= Q2Q1e^−hx,yi, this gives a k-algebra isomorphism

D(V^∗)−^∼→ D(V)^op.

(Note that, choosing dual systems of coordinates V = Spec(k[x0, . . . , xn]) and V^∗= Spec(k[y0, . . . , yn]), the above isomorphism is described by yi7→

∂xi, ∂yi 7→ −xi.) Moreover, one has algebra isomorphisms D(V)^op' Γ(V; ΩV⊗_ODV⊗_OΩ^⊗−1_V )

' det V^∗⊗ D(V) ⊗ det V,

the identification ΩV' OV⊗ det V^∗being induced by T^∗V= V × V^∗. It is then possible to consider the functor associating to a quasi-coherent D(V)- module M the quasi-coherent D(V^∗)-module M^∧= det V^∗⊗M . Since this functor is exact, it induces a functor

∧ : D^b_q−coh(DV) −→ D^b_q−coh(DV^∗) (1.1) called the Fourier-Laplace transform. The Fourier-Laplace transform is an equivalence, it preserves coherency and holonomy, but it does not preserve regular holonomy, in general. (For references see e.g. [17, 5, 15].)

1.3. Review on the Radon transform(s)

Let P = P(V) be the n-dimensional projective space associated with V, and P^∗= P(V^∗) the dual projective space. Let us denote by S the smooth hypersurface of P × P^∗ defined by the homogeneous equation hx, yi = 0, and set U = (P × P^∗) \ S. Identifying P^∗with the family of hyperplanes in P, the set S describes the incidence relation “the point x ∈ P belongs to the hyperplane y ∈ P^∗”. Consider the smooth maps

Poo ^pS S ^qS // P^∗, Poo ^pU U ^qU // P^∗,

defined by restriction of the natural projections p and q from P × P^∗. To these maps are attached the pull-back–push-forward functors

Dq_{S ∗}DpS∗, Dq_U∗DpU∗: D^b_q−coh(DP) −→ D^b_q−coh(DP^∗). (1.2) The first functor is the D-module analogue of the usual Radon transform, consisting in “integrating along hyperplanes”. The second functor (cf [18, 2, 16]) is a small variation¹on the first one which has, amongst others, the advantage of giving an equivalence of categories.

Note that since pS and qS are smooth and proper, the first functor preserves coherency. Even though qU is not proper, it follows e.g. from Lemma 1 below that also Dq_U∗DpU∗ preserves coherency, as does the func- tor

Dq_U!DpU∗: D^b(DP) −→ D^b(DP^∗). (1.3) (For references see e.g. [8].)

1.4. Review on the blow-up transform(s)

Let ˙V= V \ {0} and consider the natural projection and embedding Poo ^π ˙V  ^j // V .

1in the sense that there is a distinguished triangle O^P∗ ⊗RΓ(P; Ω^P ⊗^L

DP

M) −→ Dq_{U ∗}DpU∗M −→ Dq_S∗DpS∗M−⁺¹−→, as follows e.g. from (1.9) and Lemma 1 below.

They induce an embedding (π, j) of ˙V as a locally closed subvariety of P×V.

Let fV₀be the closure of ˙V in P × V, a smooth subvariety, and consider the maps

Poo ^eπ Vf₀ ^e // V

obtained by restriction of the natural projections from P × V. Note thate

is the blow-up of the origin 0 in V, e is proper, and eπ is smooth. To these maps are attached the functors

Dj∗Dπ^∗, De^∗Deπ^∗: D^b_q−coh(DP) −→ D^b_q−coh(DV). (1.4) Using similar remarks as for the Radon transform one checks that these functors preserve coherency, as does the functor

Dj!Dπ^∗: D^b(DP) −→ D^b(DV). (1.5) 1.5. A first statement of the result

As a last piece of notation, let ˙V^∗= V^∗\ {0} and consider the natural projection and embedding

P^∗oo ^π ˙V^∗  ^j // V^∗.

The next theorem generalizes a result of Brylinski [5, Th´eor`eme 7.27], who obtained the isomorphism (1.7) assuming M regular holonomic. In or- der to help the reader in following the pull-back–push-forward procedures, let us summarize in the next diagram the maps that we will use. The starting point is P, and the target is ˙V^∗.

V ^∧ V^∗

f V₀

e

vvvv::

vv

e πHHHH$$H HH ˙V ?

jOO

π

oo ? _ S

qS

H$$H HH HH H

pS

{{wwwwwww ˙V ?^∗ OOj

π

P U

qU //

pU

oo P^∗

(1.6)

Theorem 2. For M ∈ D^b_coh(DP) there are natural isomorphisms in D^b(DV˙^∗)

Dπ^∗ Dq_U∗DpU∗M

' Dj^∗ Dj!Dπ^∗M
∧ , Dπ^∗ Dq_U!DpU∗M

' Dj^∗ Dj∗Dπ^∗M
∧ . For M ∈ D^b_q−coh(DP) there is a natural isomorphism in D^b(DV˙^∗)

Dπ^∗ Dq_S∗DpS∗M

' Dj^∗

De^∗Deπ^∗M
∧

. (1.7)

The statement may be visualized by the commutative diagram D^b(DV) ^Fourier // D^b(DV^∗)

pull-back

**

VV VV VV VV VV V

D^b(DV˙^∗) D^b(DP)

Blow-up

OO

Radon // D^b(DP^∗) ^pull-back 44h hh hh hh hh hh

In order to prove this theorem we will first restate it, using the language of integral kernels, as Theorem 3. This has the advantage of applying to quasi-coherent modules, and gives a reason for the strange looking pattern of ∗’s and !’s in the above formulae.

1.6. Review on integral kernels

Let X and Y be smooth algebraic varieties, and consider the projections Xoo ^p X × Y ^q // Y.

For K ∈ D^b(DX×Y) the functor

(•)◦ K : D^{D b}(DX) −→ D^b(DY)

M 7→ M◦ K = Dq^D ∗(Dp^∗M⊗ K)^D

is called integral transform with kernel K. More generally, if Z is another smooth algebraic variety and L ∈ D^b(DY ×Z), one sets

K◦ L = Dq^D _13∗(Dq12∗K⊗ Dq^D 23∗L) ∈ D^b(DX×Z),

where qij denotes the projections from X × Y × Z to the corresponding factors, so that for example q13(x, y, z) = (x, z). The bifunctor◦ preserves^D quasi-coherency, is associative in the sense that (M◦ K)^D ◦ L ' M^{D D}◦ (K◦^D L), and the identity functor corresponds to the regular holonomic kernel BX|X×X = Dδ∗OX, where δ : X ,→ X × X is the diagonal embedding.

One says that K ∈ D^b_coh(DX×Y) and L ∈ D^b_coh(DY ×Z) are transversal if (char(K) × T_Z^∗Z) ∩ (T_X^∗X × char(L)) ⊂ T_{X×Y ×Z}^∗ (X × Y × Z).

In particular, M ∈ D^b_coh(DX) is transversal to K if

(char(M) × T_Y^∗Y ) ∩ char(K) ⊂ T_X×Y^∗ (X × Y ).

In this case, assuming moreover that supp(K) is proper over Y , it follows from Theorem 1 that M◦ K is coherent, and^D

D_Y(M◦ K) ' D^D XM◦ D^D X×YK. (1.8)

1.7. Basic regular holonomic kernels

Let S be a smooth variety, let Z be a closed smooth subvariety of S of codimension d, set U = S \ Z, and consider the embeddings

jZ: Z ,→ S, jU: U ,→ S.

The simplest regular holonomic DS-modules attached to the stratification S = Z t U are

OS, BZ|S= Dj_{Z ∗}OZ, BU |S = Dj_{U ∗}OU, D_SBU |S = Dj_{U !}OU. As an alternative description, one has

BZ|S= RΓ[Z]OS [d], BU |S= RΓ[U ]OS,

where RΓ[Z]M ' Dj_{Z ∗}DjZ∗M[−d], and RΓ[U ]M ' Dj_{U ∗}DjU∗M. Recall that one has a distinguished triangle

RΓ[Z]M −→ M −→ RΓ_{[U ]}M−−→ .⁺¹ (1.9) The basic model is the stratification A¹_k = {0} t ˙A¹_k of the affine line A¹

k= Spec(k[t]), where one has the regular holonomic modules OA¹

k = DA¹

k/h∂ti = DA¹

k· 1, B_0|A¹

k = DA¹

k/hti = DA¹

k· δ, BA˙¹

k|A¹

k = DA¹

k/h∂tti = DA¹

k· 1/t, D_A1

kBA˙¹

k|A¹_k = DA¹

k/ht ∂ti = DA¹

k· Y .

(1.10)

Here we used the pattern

M = DA¹

k/hP i = DA¹

k· u to indicate that M is a cyclic DA¹

k-module with generator u and relation P u = 0.

Let now S be a closed smooth subvariety of X × Y , and consider the embedding

i : S ,→ X × Y, and the maps

Xoo ^pS S ^qS // Y , Xoo ^pU U ^qU // Y ,

obtained by restriction of the natural projections p and q from X ×Y . Note that Di∗OS' BS|X×Y, Di∗BZ|S' BZ|X×Y.

Lemma 1. For M ∈ D^b_q−coh(DX) there are natural isomorphisms in D^b(DY)

M◦ B^D _S|X×Y ' Dq_{S ∗}DpS∗M, M^D◦ Di∗BU |S ' Dq_{U ∗}DpU∗M,

M^D◦ OX×Y ' OY ⊗ DR(M), where DR(M) = RΓ(X; ΩX ⊗^L

DX M). If moreover M is coherent and transversal to Di∗B_{U |S}, and S is proper over Y , then there is an isomor- phism of functors from D^b_coh(DX) to D^b_coh(DY)

M^D◦ Di∗D_SB_{U |S} ' Dq_{U !}DpU∗M.

In order to check the transversality condition, note that char(Di∗B_{U |S}) ⊂ T_Z^∗(X × Y ) ∪ T_S^∗(X × Y ).

Proof. The first isomorphism is a particular case of the second one for Z = ∅, S = U . To prove the second isomorphism, note that for M ∈ D^b_q−coh(DX) there is the chain of isomorphisms

M^D◦ Di∗BU |S' Dq∗(Dp^∗M⊗ Di^D ∗Dj_{U ∗}OU)

' Dq∗Di∗Dj_{U ∗}(DjU∗Di^∗Dp^∗M⊗ O^D U) ' Dq_{U ∗}DpU∗M.

As for the third isomorphism, using the first one with S = X × Y we get M◦ O^D X×Y ' Dq∗Dp^∗M

' DaY∗DaX ∗M,

where aX: X −→ {pt} denotes the map to the variety reduced to a point.

Finally, for M ∈ D^b_coh(DX), the last isomorphism follows from the second one by (1.8), as follows

M◦ Di^D ∗D_SB_{U |S} ' M^D◦ DX×YDi∗B_{U |S} ' DY

(DXM)◦ Di^D ∗BU |S



' DYDq_{U ∗}DpU∗D_XM ' DYDq_{U ∗}D_UDpU∗M

= Dq_{U !}DpU∗M.

1.8. Radon and Fourier transforms for D-modules Consider the holonomic kernel (irregular at infinity)

L = DV×V^∗/I = DV×V^∗e^−hx,yi, (1.11) where I is the left ideal of differential operators P ∈ DV×V^∗ such that, formally, P e^−hx,yi = 0. Then L is the kernel attached to the Fourier- Laplace transform, since one has (see [17, §7.5])

M^∧' M◦ L,^D M ∈ D^b_q−coh(DV).

Concerning the Radon transform, it follows from Lemma 1 that the functors in (1.2) and (1.3) are given by composition with the regular holo- nomic kernels attached to the stratification P × P^∗ = S t U. According to (1.10), let us give these kernels the following names

R1= OP×P^∗, RY = DP×P^∗BU|P×P^∗, R_1/t= BU|P×P^∗, Rδ = BS|P×P^∗. (1.12) As for the blow-up, let E = fV₀\ ˙V be its exceptional divisor, a smooth hypersurface of fV₀. It follows from Lemma 1 that the functors in (1.4) and (1.5) are given by composition with the regular holonomic kernels attached to the stratification fV₀ = E t ˙V. According to (1.10), let us give these kernels the following names

S1= O_Vf0, SY= D_Vf0B_{V|f˙ V}

0, S_1/t= BV|f˙ V₀, Sδ= B_E|fV0. (1.13) Summarizing, one has

u = 1 Y 1/t δ

Ru= OP×P^∗ DP×P^∗BU|P×P^∗ BU|P×P^∗ BS|P×P^∗

(•)^D◦ Ru' OP∗⊗DR(•) Dq_{U !}Dp^U∗ Dq_{U ∗}Dp^U∗ Dq_{S ∗}Dp^S∗ Su= O_Vf

0

D_f

V₀B_V˙

| fV₀ B_V˙

| fV₀ B_E

| fV₀

(•)^D◦ Deı∗Su' De∗Deπ^∗ Dj!Dπ^∗ Dj∗Dπ^∗ B_0|V⊗DR(•) Consider the maps

f

V₀  ^eı // P× V, P^∗oo ^π V˙^{∗ j} // V^∗.

Theorem 3. Let M ∈ D^b_q−coh(DP), and let u be one of the four gener- ators in (1.10), so that

u = 1, Y, 1/t, δ; u = δ, 1/t, Y, 1;b respectively. Then there is a natural isomorphism in D^b(DV˙^∗)

Dπ^∗(M^D◦ Ru_b) ' Dj^∗(M^D◦ Deı^∗Su

◦ L).D

As we already pointed out, this statement implies Theorem 2.

Proof. Consider the maps

P× P^{∗ π} P× ˙V^∗

00

oo ^j

00

// P× V^∗

induced by P^∗ ←^π− ˙V^∗ −→ V^{j ∗}. Denote by S⁰⁰ the hypersurface of P × V^∗ defined by the equation hx, yi = 0, let U⁰⁰= (P × V^∗) \ S⁰⁰, and set

R⁰⁰₁= OP×V^∗, R⁰⁰_Y= DP×V^∗BU⁰⁰|P×V^∗, R⁰⁰_1/t= BU⁰⁰|P×V^∗, R⁰⁰_δ = BS⁰⁰|P×V^∗. One has

Dπ^∗(M^D◦ Ru_b) ' M◦ Dπ^{D 00∗}R_bu' M^D◦ Dj^00∗R⁰⁰_ub' Dj^∗(M◦ R^{D 00}_ub).

Then the statement is a corollary of the following proposition.

Proposition 1. There is an isomorphism in D^b(DP×V^∗) R⁰⁰_bu' Deı^∗Su

◦ L.D

Proof. Let us start by observing that fV₀ is the quotient of A¹_k× ˙V '

˙

V×PVf₀ by the action of the multiplicative group Gm given by c(t, x) = (c⁻¹t, cx). Let us denote by [[t, x]] the equivalence class of (t, x). Consider the commutative diagram

A¹

k A¹

k× ˙V



oo t ^τ //Vf₀



  ^eı // P × V

A¹

k× A¹_k

p1

OO

p2



A¹

k× ˙V × V^∗

 

˙γ⁰⁰

oo

p12

OO

p23



τ⁰

// fV₀× V^∗

e π⁰



q1

OO

e

M⁰MM&&M MM MM MM M

  ^eı0 // P × V × V^∗

q23

q13

qqqqq xxqqqqq

q12

OO

A¹

k oo ^˙γ ˙V × V^{∗ π0} // P × V^∗ V× V^∗ where pi, qi, pij, and qij are the natural projections,

t(t, x) = t, τ (t, x) = [[t, x]], ˙γ(x, y) = hx, yi, eı([[t, x]]) = ([x], tx), e([[t, x]]) = tx, eπ([[t, x]]) = [x],

˙γ⁰⁰ = idA¹

k× ˙γ, and f⁰ = f × idV^∗ for f = τ,eı, eπ, e, π. There are natural isomorphisms in D^b(DP×V^∗)

Deı^∗Su

◦ L ' DqD _13∗(Dq12∗Deı^∗Su

⊗ DqD 23∗L) ' Dq_13∗(Deı^0∗Dq1∗Su

⊗ DqD 23∗L) ' Dq_13∗Deı^0∗(Dq1∗Su

⊗ DeıD ^0∗Dq23∗L) ' Deπ∗⁰(Dq1∗Su

⊗ DeD ^0∗L).

There are natural isomorphisms in D^b(DV×V˙ ^∗) Dπ^0∗Deπ∗⁰(Dq1∗Su

⊗ DeD ^0∗L) ' Dp_23∗Dτ^0∗(Dq1∗Su

⊗ DeD ^0∗L) ' Dp_23∗(Dτ^0∗Dq1∗Su

⊗ DτD ^0∗De^0∗L) ' Dp_23∗(Dp12∗Dτ^∗Su

⊗ DτD ^0∗De^0∗L) ' Dp_23∗(Dp12∗Dt^∗(DA¹

k· u)⊗ D ˙γ^{D 00∗}L1) ' Dp_23∗(D ˙γ^00∗Dp1∗(DA¹

k· u)⊗ D ˙γ^{D 00∗}L1) ' Dp_23∗D˙γ^00∗(Dp1∗(DA¹

k· u)⊗ L^D 1) ' D ˙γ^∗((DA¹

k· u)◦ L^D 1) ' D ˙γ^∗(DA¹

k· bu) ' Dπ^0∗R⁰⁰_bu, where L1 = DA¹

k×A¹_ke^−tu is the one-dimensional Fourier-Laplace kernel, and DA¹

k· u is the cyclic module defined in (1.10). Summarizing, we have an isomorphism

Dπ^0∗(Deı^∗Su

◦ L) ' DπD ^0∗R⁰⁰_ub. One concludes by the following lemma.

Lemma 2. Let f : X −→ Y be a fibration with fiber ˙A¹_k = A¹_k\ {0}.

Then the functor Df^∗: Modq−coh(DY) −→ Modq−coh(DX) is exact and fully faithful.

Proof. Since f is smooth, Df^∗ is exact. Moreover, one has an isomor- phism

RHom_DX(Df^∗N1, Df^∗N2) ' RHom_DY(N1, Df∗Df^∗N2)^[−1]. (1.14) By the projection formula, Df∗Df^∗N2' Df∗OX

⊗ ND 2, and one has

Df∗OX ' Rf_∗

 OX

dX/Y

−−−→ Ω¹_X/Y

 ,

where Ω¹_X/Y, the sheaf of relative one-forms, sits in degree zero. Hence, locally on Y one has Df∗OX ' OY ⊕ OY [1]. Taking zero-th cohomology, (1.14) gives

Hom_DX(Df^∗N1, Df^∗N2) ' Hom_DY(N1, N2).