CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS AND FAITHFUL FLATNESS OF D

CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS AND FAITHFUL FLATNESS OF D

OVER D

by

Luis Narv´ aez Macarro & Antonio Rojas Le´ on

Abstract. — In these notes we prove the faithful flatness of the sheaf of infinite order linear differential operators over the sheaf of finite order linear differential operators on a complex analytic manifold. We give the Mebkhout-Narv´aez’s proof based on the continuity of the division of finite order differential operators with respect to a natural topology. We reproduce the proof of the continuity theorem given by Hauser-Narv´aez, which is simpler than the original proof.

Résumé (Continuité de la division des opérateurs différentiels et fidèle platitude deDX^∞sur DX)

Dans ce cours on d´emontre la fid`ele platitude du faisceau d’op´erateurs diff´erentiels lin´eaires d’ordre infini sur le faisceau d’op´erateurs diff´erentiels lin´eaires d’ordre fini d’une vari´ete analytique complexe lisse. La preuve que nous donnons est celle de Mebkhout-Narv´aez, qui utilise la continuit´e de la division d’op´erateurs diff´erentiels d’ordre fini par rapport `a une topologie naturelle. Nous r´eproduisons la preuve de Hauser-Narv´aez du th´eor`eme de continuit´e, qui est plus simple que la preuve originale.

Introduction

The sheaf OX of holomorphic functions on a complex analytic manifold X is the first natural example of left module over the sheaf of linear differential operatorsDX

onX. Here, as usual, differential operators have (locally) finite order. In fact, there is another natural sheaf of noncommutative rings extending DX, called the sheaf of linear differential operators of infinite order, D^∞_X, introduced by Sato. The leftDX- module structure onOX extends to a left D^∞X-module structure in such a way that D^∞X ⊗_DXOX=OX.

For any holonomic leftDX-moduleM, we know by the constructibility theorem of Kashiwara [9] (see also [12], [13]) that the complex of holomorphic solutions of M, R Hom_DX(M,OX), is constructible. The canonicalDX-linear biduality morphism

M−→R Hom_CX(R Hom_DX(M,OX),OX)

2000 Mathematics Subject Classification. — 32C38, 32S60.

Key words and phrases. — Infinite order differential operator, division theorem.

Both authors were partially supported by BFM2001-3207 and FEDER.

induces aD^∞X-linear morphism

(*) D^∞X ⊗_DXM−→R Hom_CX(R Hom_DX(M,OX),OX).

The local biduality theoremof Mebkhout asserts that (*) is an isomorphism for any holonomic module M (see [11, 11.3] in this volume). This theorem is an essential ingredient for the “full” Riemann-Hilbert correspondence, which establishes an equiv- alence between three categories: the bounded derived category of regular holonomic complexes of DX-modules, the bounded derived category of holonomic complexes of D^∞X-modules and the bounded derived category of analytic constructible complexes (see 11.4 in loc. cit.). The sheafD^∞_X does not have any known finiteness properties like DX, but to prove the full Riemann-Hilbert correspondence one needs to know that the extensionDX ⊂D^∞X is faithfully flat. This result has been stated and proved for the first time in [17] (see also [1]), and its proof depended on the microlocal machinery.

The aim of these notes is to give an elementary self-contained proof of the faithful flatness of the sheaf of differential operators of infinite order over the sheaf of dif- ferential operators of finite order. The method we follow is that of [14], whose first step consists in considering the ring of differential operators of infinite order as the completion of the corresponding ring of finite order for a natural topology, and then mimic Serre’s proof of the faithful flatness of the completion of a noetherian local ring over the ring itself [18]. The essential technical tool is the continuity of the Weierstrass-Grauert-Hironaka division of differential operators [2, 3]. We reproduce with detail the proof given in [8], which simplifies the original proof in [14]. As a complement we sketch the results of [15] for the case of differential operators with polynomial coefficients (Weyl algebra).

We would like to thank Herwig Hauser for a careful reading of these notes and for helpful suggestions.

1. Topological structure on rings of linear differential operators with analytic coefficients

LetX be a complex analytic manifold of pure dimensionn, countable at infinity.

Let us denote byOXthe sheaf of holomorphic functions and byDX the sheaf of linear differential operators (cf.[6]). For each open setU ⊂X, the spaceOX(U) endowed with the topology of uniform convergence on compact sets is a Fr´echet space, i.e.a complete metrizable locally convex space (it is also anuclearspace,cf.[5] for details).

The Banach open mapping theorem shows that the property of being continuous for a C-linear endomorphism P : OX → OX is a local property. For that, let {Ui} be an open covering ofX, that we can take as countable, such that each restriction P|U_i :OX|U_i →OX|U_i is continuous. For any open setU ⊂X, the canonical injection OX(U) ,→ QOX(U ∩Ui) is a closed inmersion by the open mapping theorem (its image is the kernel of the ˇCech map QOX(U ∩Ui) → QOX(U ∩Ui∩Uj) by the

sheaf condition). Hence, the continuity ofP(U) :OX(U)→OX(U) comes from the continuity ofQP(U∩U_i) :QOX(U ∩U_i)→QOX(U ∩U_i). As a consequence, the pre-sheaf ofC-linear continuous endomorphisms ofOX,Homtop(OX,OX), is actually a sheaf.

The following proposition is well-known (cf.[14], prop. 2.1.4):

Proposition 1.1. — For any continuous C-linear endomorphism P : OX → OX and for any system(U;x₁, . . . , x_n)of local coordinates ofX, there are unique holomorphic functionsaα∈OX(U),α∈Nⁿ, such that

P|_U = X

α∈Nⁿ

a_α 1 α!∂^α,

with ∂ = (∂/∂x1, . . . , ∂/∂xn) and lim_|α|→∞|aα|^1/|α| = 0 uniformly on any compact set of U. Equivalently, the function

(p, ξ)∈U×Cⁿ7−→ X

α∈Nⁿ

aα(p)ξ^α∈C is holomorphic.

From now on, we will denoteD^∞X =Homtop(OX,OX) and call itsheaf of infinite order linear differential operators. From the above proposition we deduce that it coincides with the sheaf of infinite order linear differential operators defined in [16, 17].

The following proposition is proved in [14], prop. 2.1.3.

Proposition 1.2. — Let P : OX → OX be a C-linear endomorphism. The following properties are equivalent:

a) P is continuous.

b) For any pair K, K⁰ ⊆ X of compact sets with K ⊂

◦

K⁰, there is a constant CK,K⁰ >0 such that|P(f)|K 6CK,K⁰|f|K⁰ for any holomorphic function f defined on a neighborhood of K⁰.

Corollary 1.3. — The sheaf DX of (finite order) linear differential operators is a sub- sheaf (of rings) of Homtop(OX,OX).

Proof. — LetP be a section ofDX over an open set U ⊂X. Since continuity is a local property, we can suppose thatU is a connected open set ofCⁿ. ThenP admits a unique expression

P = X

α∈Nⁿ,|α|6d

a_α 1 α!∂^α,

wheredis the order ofP and theaαare holomorphic functions onU. LetK, K⁰⊆U be a pair of compact sets as in proposition 1.2, b) and letf be a holomorphic function

on a neighborhood ofK⁰. From Cauchy inequalities we deduce that

|P(f)|K=

X

|α|6d

aα

1 α!∂^α(f)

_K 6 X

|α|6d

|aα|Kr^−|α||f|K⁰

whereris the distance betweenKandU−

◦

K⁰. By proposition 1.2, we conclude that P is continuous.

Definition 1.4 ([14], déf. 2.1.6). — For any open set U ⊆X, thecanonical topology of D^∞X(U) orDX(U) is defined as the locally convex topology given by the semi-norms p_(K,K⁰₎:P ∈D^∞X(U)7−→p_(K,K⁰₎(P) := sup{|P(f)|K/|f|K⁰ |f ∈OX(K⁰), f 6= 0}, indexed by pairs (K, K⁰) of compact sets inU withK⊂

◦

K⁰.

For any coordinate system (U;x1, . . . , xn) inX, we can use Cauchy inequalities as in corollary 1.3 and proposition 1.1 to prove that the map

X

α∈Nⁿ

aα(x)1

α!∂^α7−→ X

α∈Nⁿ

aα(x)y^α

is an isomorphism of locally convex vector spaces betweenD^∞X(U) endowed with the canonical topology and the space of holomorphic functions onU×Cⁿ endowed with the topology of uniform convergence on compact sets. This isomorphism depends on local coordinates and carries the spaceDX(U) into the space of holomorphic functions on U×Cⁿ which are polynomials with respect to the second factor. Consequently, D^∞X(U) is a Fr´echet (and nuclear) space andDX(U) is dense inD^∞X(U). We can write thenD^∞X(U) =D\X(U).

In fact, in [14,§2] it is proved thatD^∞X endowed with the canonical topology is a sheaf with values in the category of Fr´echetC-algebras.

Let us denote byOn,Dn,D^∞n the stalk at the origin of the sheavesO_Cⁿ,D_Cⁿ,D^∞_Cn

respectively. For ρ = (ρ1, . . . , ρn), L = (L1, . . . , Ln) in (R^∗+)ⁿ let us consider the pseudo-norm| − |^L_ρ :D^∞n →R+∪ {+∞}whose value atP=P

aβ∂^β=P

αβaαβx^α∂^β is

(1) |P|^L_ρ =X

β

|aβ|ρ|β|!L^β =X

αβ

|aαβ| · |β|!ρ^αL^β ∈R+∪ {+∞}.

Since β! 6 |β|! 6 n^|β|β!, we could also use β! instead |β|! in (1) to obtain an equivalent system of pseudo-norms. Nevertheless, the choice of|β|! is forced by the proofs of the majorations needed to obtain the norm estimates of theorem 2.11 (see [14, 2.2.4] and [8]).

Let us denote byD^∞n (ρ) the subspace ofD^∞n where| − |^L_ρ takes finite values for any L∈(R^∗+)ⁿand let us writeDn(ρ) :=Dn∩D^∞n (ρ). The semi-norms| − |^L_ρ,L∈(R^∗+)ⁿ, define a Fr´echet topology onD^∞n (ρ).

Following [8], we consider weights λ, µ ∈ (N^∗)ⁿ and, for real numbers s, t > 0, ρ = s^λ = (s^λ1, . . . , s^λn), L = t^−µ = (t^−µ1, . . . , t^−µn). When λ is fixed, we denote

| − |^µ,t_s :=| − |^L_ρ,D^∞n(s) :=D^∞n(ρ) andDn(s) :=Dn(ρ).

In the case where U is an open polycylinder of Cⁿ centered at 0 of polyradius σ=s^λ₀, 0< s₀6+∞, we have

D^∞_Cⁿ(U) = \

0<s<s0

D^∞n(s), D_Cⁿ(U) = \

0<s<s0

Dn(s),

and the canonical topology ofD^∞_Cⁿ(U) (resp.DCⁿ(U)) is the (topological) inverse limit of theD^∞n(s) (resp.Dn(s)), for 0< s < s0. In other words, the canonical topologies of D^∞_Cⁿ(U) andDCⁿ(U) are given by the semi-norms | − |^µ,t_s , 0 < s < s0, t^−µ 0 [14], 2.2.3. The last condition can be obtained with µ fixed and t → 0, or taking t=t(s)<1 andµ0.

For vectorsP= (P1, . . . , Pq)∈(D^∞n )^q, following [7] we also define

|P|^µ,t_s :=

q

X

i=1

|P_i|^µ,t_s s⁻⁽ⁱ⁻¹⁾,

whereλ∈(N^∗)ⁿ is fixed.

In the above situation, the product topology on D^∞_Cⁿ(U)^q and DCⁿ(U)^q is also given by the semi-norms | − |^µ,t_s , 0< s < s₀,t^−µ0.

2. The continuity theorem

In this section, we fix M₁, . . . , M_r ∈ D^qn and a total well ordering < in N²ⁿ compatible with sums (cf.[4, 1.3]). Whenever we speak about the ordering < in N²ⁿ× {1, . . . , q}we mean the ordering induced by<in the following way:

(α, β, i)<(α⁰, β⁰, j)⇐⇒







(α, β)<(α⁰, β⁰) or

(α, β) = (α⁰, β⁰) andi > j Given

N = (N1, . . . , Nq) =

q

X

i=1

Niei=

q

X

i=1

X

α,β

aαβix^α∂^βei∈D^qn, aαβi∈C,

where{ei}i=1...q stands for the canonical basis ofD^qn as a freeDn-module, we denote byN (N), theNewton diagramofN, the set of (α, β, i) inN²ⁿ× {1, . . . , q}such that a_αβi 6= 0 and byσ(N) itssymbol, i.e.the homogeneous component ofN of maximal degree with respect to the grading given by the total degree in∂:

σ(N) =

q

X

i=1

X

|β|=d

X

α

aαβix^α∂^βei, d= deg_T(N) = max deg(Ni).

TheexponentofN is exp(N) := min{(α, β, i)|aαβi6= 0,|β|=d}, and the correspond- ing monomial of σ(N) (and therefore of N) is, by definition, the initial monomial ofN.

Let (αj, βj, ij) be the exponent of Mj with respect to the given ordering (we can assume without loss of generality that its coefficient is 1). Also let M_j⁰ = M_j − x^αj∂^βjeij. We will denote byF ther-tuple (M1, . . . , Mr).

The following notion is needed in the continuity theorem 2.6:

Definition 2.1. — We say that a given weightλ∈(N^∗)ⁿ isadapted toF if for every positive constant K there existsµ∈Nⁿ withµi > K, λi for everyi= 1, . . . , n such that

λα_j−µβ_j−i_j < λα−µβ−i

for every j = 1, . . . , r and every (α, β, i) ∈ N(M_j⁰). We say that such a µ is K- admissible, or simply admissible, for (F, λ).

Lemma 2.2. — For any F = (M₁, . . . , M_r) as above, there exists a weightλ adapted toF.

Proof. — Consider first the case q = 1. Let π₁, π₂ : N²ⁿ = Nⁿ×Nⁿ → Nⁿ be the canonical projections. For every j = 1, . . . , r and every β ∈π2(N (Mj)) let M_β^j be the set of (α, β) in N (Mj) such thatα is minimal in A = {α : (α, β)∈ N (Mj)}

with respect to the componentwise order. The set Mβ^j is finite, in fact it consists of the elements (α₁, β), . . . ,(α_s, β), where{α₁, . . . , α_s} is the minimal set of generators of the ideal A +Nⁿ of Nⁿ. Therefore, the set M =S

j

S

β(M_β^j∪ {(0, β)}) is also finite.

Let (σ, ρ)∈Nⁿ×Nⁿ =N²ⁿ be a vector defining the given ordering restricted to the finite set M. We claim that λ = σ is adapted to F. Fix a positive constant K, and letpbe an integer such that p >max{K+|ρ|, σα+ρβ : (α, β)∈M}. Set µ= (p, . . . , p)−ρ. We have then

λα−µβ=σα+ρβ−p|β|.

We will show that the minimum of λα −µβ for (α, β) ∈ N (Mj) is attained in the exponent of Mj. First, we see that if λα−µβ is minimal, then (α, β) ∈ M. Otherwise, there would be (α⁰, β)∈ N (M_j), γ ∈ Nⁿ\{0} such that α =α⁰+γ, so λα−µβ=σγ+ (λα⁰−µβ)> λα⁰−µβ.

Furthermore, (α, β) must be in N(σ(Mj)). Otherwise, there would be (α⁰, β⁰)∈ N (Mj)∩M with|β⁰|>|β|. Thenλα−µβ=σα+ρβ−p|β|>σα+ρβ−p(|β⁰| −1)>

p−p|β⁰|> σα⁰+ρβ⁰−p|β⁰|=λα⁰−µβ⁰. Therefore, min{λα−µβ : (α, β)∈N (M_j)}= min{σα+ρβ−p|β|: (α, β)∈N (σ(Mj))∩M}. In this set,|β| is constant and the ordering is defined by (σ, ρ), so the minimum is attained in the smallest element of N (σ(Mj)) with respect to the ordering, i.e the initial monomial of Mj.

Now assumeq6= 1. LetMj=P

iαβa^j_iαβx^α∂^βei, and defineMj=P

iαβ|a^j_iαβ|x^α∂^β ∈ Dn. Let (αj, β_j) be the exponent ofMj with respect to the given ordering inN²ⁿ. Let (α_j, β_j, i_j) be the exponent of M_j. We have α_j =α_j and β_j =β_j. Otherwise, we would have (αj, βj) >(αj, β_j). Let i be such that (αj, β_j, i) ∈ N (Mj). Then we have by definition of the exponent that (αj, β_j, i) > (αj, βj, ij), and therefore (αj, β_j)>(αj, βj), which is in contradiction with the last inequality.

By the first part of the proof, there existsλadapted to (M1, . . . , Mr). Now given a positive constant K there is µ ∈ Nⁿ with µ_j > K/q and λ_j < µ_j such that λαj−µβj< λα−µβfor every (α, β)∈N (M⁰_j). Let us see thatλ=qλis adapted toF andµ=qµis K-admissible for (F, λ). Let (α, β, i)∈N (M_j⁰), then (α, β)∈N (M_j), hence λαj−µβj6λα−µβ by construction. Now we distinguish two cases:

If (α, β)>(αj, βj), thenλαj−µβj < λα−µβ. Butλas well as µare multiples of q, and therefore λαj−µβj 6λα−µβ−q, and λαj−µβj −ij < λαj−µβj 6 λα−µβ−q6λα−µβ−i.

If (α, β) = (αj, βj), then we must havei < ij, henceλαj−µβj−ij< λαj−µβj−i= λα−µβ−i. In either case, we get the desired inequality.

This completes the proof of the lemma.

Lemma 2.3. — Let F1, . . . ,Fm be a finite number of vectors whose coordinates are in D^qn as above (they may have distinct lengths). Then there exists λ∈Nⁿ which is adapted to all of them.

Proof. — This is a direct consequence of the following lemma applied to the vector constructed by concatenation ofF1, . . . ,Fm.

The following lemma is clear:

Lemma 2.4. — LetF= (M1, . . . , Mr)be a vector in(D^qn)^rand letG= (Mi₁, . . . , Mi_k), with 16i₁<· · ·< i_k 6r. Then everyλ∈Nⁿ adapted toF is also adapted toG.

Before stating the main theorem of this section we make one further definition:

Definition 2.5. — Let λ∈ (N^∗)ⁿ be a weight. A basis B of open neighborhoods of 0∈Cⁿ is said to be aλ-basisif it consists of open polycylinders of polyradiuss^λ for 0< s < s₀, for somes₀>0. We will say thatBis adapted toF if it is aλ-basis for someλadapted toF.

From the lemmas above it follows that we can always find a basis of neighborhoods of 0 adapted toF, and even a basis adapted to a finite number of vectorsF1, . . . ,Fm. After these preliminaries we are ready to state the continuity theorem of the division of linear differential operators:

Theorem 2.6. — LetF = (M1, . . . , Mr), withMi∈D^qnand letQi(F;E),i= 1, . . . , r (resp.R(F;E)) be the quotients (resp. the remainder) of the division ofE∈D^qnbyF

(see [4]). Then, for any weightλ∈(N^∗)ⁿ adapted to F, there exists a λ-basis B of open neighborhoods of 0 ∈ Cⁿ, such that for every U ∈ B the C-linear morphisms Qi(F;−) (resp.R(F;−)) map DCⁿ(U)^q into DCⁿ(U) (resp. into DCⁿ(U)^q). Fur- thermore,

Qi(F;−) :DCⁿ(U)^q−→DCⁿ(U), R(F;−) :D_Cⁿ(U)^q−→D_Cⁿ(U)^q are continuous with respect to the canonical topology.

The proof of theorem 2.6 will be obtained after some majorations, as in [14], [8], and it will not be finished until the end of 2.13. Our task consists of adapting the proof in [8] to the vector case. Roughly speaking, as explained in loc. cit., the key point is to approximate the Dn-linear map D^rn → D^qn defined by the finite system of vectorsF = (M1, . . . , Mr)∈(D^qn)^r instead of approximating the system itself by their initial monomials. This idea has been introduced in [7] in the commutative case of vectors of convergent power series.

Let (αj, βj, ij) be the exponent ofMj. For everyi= 1, . . . , n, letTj:D^qn→D^qn be theC-linear map defined byT_jx^α∂^βe_i=x^α∂^β+eje_i. GivenA=Pc_γδx^γ∂^δ∈Dn, we will denote byA^othe mapPcγδx^γT^δ :Dn →Dn, andA⁰=A−A^o(Ais considered here to be acting by multiplication on the left). Let also {∆1, . . . ,∆r,∆} be the partition ofN²ⁿ× {1, . . . , q}defined byM₁, . . . , M_r (see [3] and [4] in this volume).

We define now the following setsLandJ:

L={A∈D^rn: exp(M_j) +N(A_j)⊂∆_j, ∀j= 1, . . . , r}

J={B∈D^qn :N (B)⊂∆}

and the linear map u:L⊕J →D^qn given byu(A, B) =Pr

j=1AjMj+B.

From the division theorem ([4], th. 2.4.1) we see that LandJ are the sets where quotients and the remainder of the division by M₁, . . . , M_r are “allowed” to lie, the A_j and B are just the quotients and the remainder of the division of u(A, B) byF and the mapuis bijective.

We start by splittinguas a sumv+w1+w2, with v(A, B) =X

A^o_jx^αj∂^βje_ij +B w₁(A, B) =X

A⁰_jx^αj∂^βje_ij w2(A, B) =X

AjM_j⁰.

TheC-linear mapv is easily seen to be an isomorphism ofC-vector spaces, by defini- tion ofLandJ.

We follow the notation in the previous section regarding the seminorms | − |^µ,t_s . Let E ∈ D^qn, and (A, B) = v⁻¹(E) ∈ L⊕J, with Aj = P

γδa^j_γδx^γ∂^δ. Then,

E = P

jγδa^j_γδx^αj+γ∂^βj+δei_j +B. If we take the | − |^µ,t_s norm on both sides, and

keep in mind that (αj, βj, ij) +N (Aj)⊂ ∆j, N(B) ⊂∆ and that the sets ∆j,∆ are pairwise disjoint, we get:

(2) |E|^µ,t_s =X

j

X

γδa^j_γδx^αj+γ∂^βj+δe_ij

µ,t s

+ B

µ,t s

>X

jγδ

a^j_γδ

βj+δ

!s^{λ(αj+γ)−(ij−1)}t^{−µ(βj+δ)}. Proposition 2.7. — There is a constant C₁ >0 such that |(w1v⁻¹)E|^µ,s_s 6C₁s|E|^µ,s_s for everyE∈D^qn and for every µadmissible for (F, λ).

Proof. — Let (A, B) =v⁻¹(E), withA_j=P

γδa^j_γδx^γ∂^δ. First, we have

|(w1v⁻¹)E|^µ,s_s =|w1(A, B)|^µ,s_s =

X

j

A⁰_jx^αj∂^βjei_j

µ,s s

=

X

jγδ

a^j_γδx^γ(∂^δ−T^δ)x^αj∂^βjei_j

µ,s s

.

Expanding the inner product, we get

|(w1v⁻¹)E|^µ,s_s =

X

jγδ

a^j_γδx^γ X

0<ε6αj,δ

αj

ε δ!

(δ−ε)!x^αj−ε∂^βj+δ−εei_j

µ,s s

6X

jγδ

X

0<ε6αj,δ

|a^j_γδ| αj

ε δ!

(δ−ε)!|βj+δ−ε|!s^{λ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε)} 6X

j

X

0<ε6αj

X

δ>ε

X

γ

|a^j_γδ|2^|αj| δ!

(δ−ε)!|βj+δ−ε|!s^{λ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε)}. Therefore

|(w1v⁻¹)E|^µ,s_s

|E|^µ,ss

6 P

j

P

0<ε6α_j

P

δ>ε

P

γ|a^j_γδ|2^|αj| δ!

(δ−ε)!|βj+δ−ε|!s^{λ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε)} P

jγδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−(ij−1)−µ(βj+δ)} 6X

jε

P

δ>ε

P

γ|a^j_γδ|2^|αj| δ!

(δ−ε)!|βj+δ−ε|!s^{λ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε)} P

γδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−(ij−1)−µ(βj+δ)}

=X

jε

P

δ>ε

P

γ|a^j_γδ|2^|αj| δ!

(δ−ε)!|βj+δ−ε|!s^{λ(αj+γ−ε)−µ(βj+δ−ε)} P

γδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−µ(βj+δ)} 6X

jε

P

γδ|a^j_γδ|2^|αj| δ!

(δ−ε)!|βj+δ−ε|!s^{λ(αj+γ−ε)−µ(βj+δ−ε)} P

γδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−µ(βj+δ)} .

Now, using that

δ!

(δ−ε)! 6 |βi+δ|!

|βi+δ−ε|! for βi>0, we get

|(w₁v⁻¹)E|^µ,s_s

|E|^µ,ss 6X

jε

P

γδ|a^j_γδ|2^|αj||βj+δ|!s^{λ(αj+γ−ε)−µ(βj+δ−ε)} P

γδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−µ(βj+δ)}

=X

jε

2^|αj|s^(µ−λ)ε.

For eachj, there is only a finite number ofε6αj, so this last sum is finite. Moreover, sinceε > 0 and µ−λ has positive components, the exponent of s in every term of the sum is greater that 1. Assumings <1, we can therefore bound this sum byC₁s, withC1=P

i2^|αi|#{ε∈Nⁿ : 0< ε6αi}. We end up with the following bound (3) |(w₁v⁻¹)E|^µ,s_s 6C₁s|E|^µ,s_s

fors <1.

Proposition 2.8. — There is a constant C₂ > 0, and an s₀ > 0 such that

|(w₂v⁻¹)E|^µ,s_s 6 C₂s|E|^µ,s_s for 0 < s < s₀ and for every E ∈ D^qn and every µ admissible for (F, λ).

Proof. — We have

|(w₂v⁻¹)E|^µ,s_s =|w₂(A, B)|^µ,s_s =

X

j

A_jM_j⁰

µ,s s

=

X

j

(X

γδ

a^j_γδx^γ∂^δ)(X

αβi

c^j_αβix^α∂^βe_i)

µ,s s

=

X

jαβγδi

a^j_γδc^j_αβix^γ∂^δx^α∂^βe_i

µ,s s

=

X

jαβγδi

a^j_γδc^j_αβix^γ(X

ε6α,δ

α ε

δ!

(δ−ε)!x^α−ε∂^δ−ε)∂^βei

µ,s s

=

X

jαβγδεi

a^j_γδc^j_αβi α

ε δ!

(δ−ε)!x^γ+α−ε∂^β+δ−εei

µ,s s

6 X

jαβγδεi

|a^j_γδ||c^j_αβi| α

ε δ!

(δ−ε)!|β+δ−ε|!sλ(γ+α−ε)−(i−1)−µ(β+δ−ε)

= X

jαβεi

|c^j_αβi| α

ε

X

γδ

|a^j_γδ| δ!

(δ−ε)!|β+δ−ε|!sλ(α+γ−ε)−(i−1)−µ(β+δ−ε)

withεandδsatisfying 06ε6αandε6δ. Now, using (2), we get

|(w2v⁻¹)E|^µ,s_s

|E|^µ,ss

6 P

jαβεi|c^j_αβi| ^α_ε P

γδ>ε|a^j_γδ| δ!

(δ−ε)!|β+δ−ε|!sλ(α+γ−ε)−(i−1)−µ(β+δ−ε)

P

jγδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−(ij−1)−µ(βj+δ)} 6 X

αβεi

P

j|c^j_αβi| ^α_ε P

γδ|a^j_γδ| δ!

(δ−ε)!|β+δ−ε|!sλ(α+γ−ε)−(i−1)−µ(β+δ−ε)

P

jγδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−(ij−1)−µ(βj+δ)}

6 X

jαβεi

|c^j_αβi| α

ε

P

γδ|a^j_γδ| δ!

(δ−ε)!|β+δ−ε|!sλ(α+γ−ε)−(i−1)−µ(β+δ−ε)

P

γδ|a^j_γδ||βj+δ|!s^{λ(αj+γ)−(ij−1)−µ(βj+δ)} . We use the fact that _(δ−ε)!^δ! 6 |δ−ε|!^|δ|! 6 |β+δ−ε|!^|βj+δ|! and therefore

|(w2v⁻¹)E|^µ,s_s

|E|^µ,ss 6 X

jαβεi

|c^j_αβi| α

ε P

γδ|a^j_γδ||βj+δ|!sλ(α+γ−ε)−(i−1)−µ(β+δ−ε)

P

γδ|a^j_γδ||β_j+δ|!s^{λ(αj+γ)−(ij−1)−µ(βj+δ)}

= X

jαβεi

|c^j_αβi| α

ε

s^{λ(α−αj−ε)−(i−ij)−µ(β−βj−ε)}

6 X

jαβεi

|c^j_αβi|2^|α|s^{λ(α−αj−ε)−(i−ij)−µ(β−βj−ε)}.

For everyβ appearing in this sum as part of the exponent of a monomial in M_j⁰, let (α_jβ, β, i_jβ)∈N (M_j⁰) be such thatλα_jβ = min{λα: (α, β, i)∈N (M_j⁰)}. Then

X

jαβεi

|c^j_αβi|2^|α|s^{λ(α−αj−ε)−(i−ij)−µ(β−βj−ε)}

= X

jαβεi

|c^j_αβi|2^|α|s^(µ−λ)εs^{λ(α−αjβ)}s^{(λαjβ−µβ−i)−(λαj−µβj−ij)}

and the exponents (λα_jβ−µβ−i)−(λα_j−µβ_j−i_j) are integers greater or equal than one. Therefore fors <1 the last sum is bounded by

s X

jαβεi

|c^j_αβi|2^|α|s^(µ−λ)εs^{λ(α−αjβ)}6s X

ε

s^(µ−λ)ε X

jβ

X

iα

|c^j_αβi|2^|α|s^{λ(α−αjβ)}

and this series converges forssmall enough.

Lemma 2.9. — Given j, β, there are sjβ > 0 and Cjβ > 0 such that the series P

iα|c^j_αβi|2^|α|s^{λ(α−αjβ)}converges with sum bounded byC_jβ for0< s < s_jβ. Proof. — The power seriesP

iα|c^j_αβi|x^αdefines an analytic functionF(x) in a neigh- borhood of 0∈Cⁿ. Therefore, the functionf(s) =F(2s^λ1, . . . ,2s^λn) is analytic in a neighborhood of 0∈C. Its expansion as a power series near 0 is P

iα|c^j_αβi|2^|α|s^λα.

Letrbe its radius of convergence, and letsjβ <min{1, r}. The functionf is analytic in the disc|s|6s_jβ.

Since λα > λαjβ for every α, this series does not have any term in which the exponent of s is less than λαjβ. Hence f(s) has a zero of order at least λαjβ in s= 0, and therefore the functiong(s) =f(s)/s^λαjβ in analytic in the disc |s|6s_jβ. In particular, it is bounded inside the interval [0, sjβ] of the real line. And this is what we want, since the power series expansion of g in a neighborhood of 0 is P

iα|c^j_αβi|2^|α|s^{λ(α−αjβ)}. Lemma 2.10. — The series P

εs^(µ−λ)ε converges and its sum is uniformly bounded fors <1/2.

Proof. — Sinceµ_j> λ_j for everyj= 1, . . . , n, we have:

X

ε

s^(µ−λ)ε=X

ε

Y

j

s^{(µj−λj)εj} =Y

j

X

ε_j

s^{(µj−λj)εj} 6Y

j

X

ε_j

s^εj = 1

1−s ⁿ

62ⁿ

Now takes0= min{¹₂,minjβsjβ}andC2= 2ⁿP

jβCjβ. From the last two lemmas it follows that

(4) |(w2v⁻¹)E|^µ,s_s

|E|^µ,ss 6C2s for 0< s < s0.

From these two propositions we can deduce the following

Theorem 2.11. — There are constants s₀ > 0 and C > 0 such that for every E = u(A, B)∈D^qn we have

X

j

|Aj|^µ,s_s |Mj|^µ,s_s +|B|^µ,s_s 6C|E|^µ,s_s for0< s < s0 and for every µadmissible for(F, λ).

Proof. — Let Dn(s)_d denote the subspace of Dn(s) whose elements are germs of differential operators of degree at most d. So far we know that there are s0>0 and C1, C2>0 such that

|((w1+w2)v⁻¹)E|^µ,s_s 6|(w1v⁻¹)E|^µ,s_s +|(w2v⁻¹)E|^µ,s_s 6(C1+C2)s|E|^µ,s_s for 0< s < s0. Choose ε >0. Taking s <min{s0,_C^1−ε

1+C₂}, the map (w1+w2)v⁻¹ has norm | − |^µ,s_s less than 1−ε. Hence the series P

(−(w1+w₂)v⁻¹)ⁿ converges to a continuous endomorphism on every Dn(s)^q_d with norm at most 1/ε, since the map does not increase the degree on∂ andDn(s)^q_d is a Banach space for the| − |^µ,s_s norm. Therefore, this series defines a continuous endomorphism of the wholeDn(s)^q

which is nothing but the inverse of (Id +(w1+w2)v⁻¹). In particular we see that u= (Id +(w₁+w₂)v⁻¹)vis an isomorphism. Furthermore, we have the inequality

X

j

|A^o_jx^αj∂^βje_ij|^µ,s_s +|B|^µ,s_s =|v(A, B)|^µ,s_s

=|(Id +(w1+w₂)v⁻¹)⁻¹u(A, B)|^µ,s_s 6ε⁻¹|E|^µ,s_s for everyE=u(A, B)∈Dn(s)^q. We conclude with two lemmas.

Lemma 2.12. — For everyj = 1, . . . , rwe have|A^ox^αj∂^βje_ij|^µ,t_s >|x^αj∂^βje_ij|^µ,t_s |A|^µ,t_s for everyA∈Dn and everys, t >0.

Proof. — LetA=P

αβa_αβx^α∂^β. ThenA^ox^αj∂^βje_ij =P

αβa_αβx^α+αj∂^β+βje_ij and therefore

|A^ox^αj∂^βjeij|^µ,t_s =X

αβ

|aαβ| · |β+βj|!·s^{λ(α+αj)−(ij−1)}t^{−µ(β+βj)}

=s^{λαj−(ij−1)}t^−µβjX

αβ

|aαβ| · |β+βj|!·s^λαt^−µβ

>s^{λαj−(ij−1)}t^−µβj|βj|!X

αβ

|aαβ| · |β|!·s^λαt^−µβ

=|x^αj∂^βjei_j|^µ,t_s |A|^µ,t_s .

Lemma 2.13. — For everyj= 1, . . . , r, there is a constantCj >0such that|Mj|^µ,s_s 6 Cj|x^αj∂^βjei_j|^µ,s_s for0< s < s0.

Proof. — We have

|Mj|^µ,s_s

|x^αj∂^βjei_j|^µ,ss

=X

αβi

|c^j_αβi||β|!

|βj|!s^{λα−λαj+(ij−i)+µβj−µβ} 6X

αβi

|c^j_αβi|s(λα−µβ−i)−(λαj−µβj−ij)

Now with an argument analogous to that in the proof of 2.9, we see that the series P

α|c^j_αβi|s^λαdefines an analytic function in a neighborhood of 0, with a zero of order at leastµβ+i+ (λα_j−µβ_j−i_j) at the origin, hence our series defines a continuous function in a neighborhood of the origin, from where the existence of the constants Cj is deduced.

This concludes the proof of the theorem 2.11, by takingC=ε⁻¹·maxj{Cj}.

We see that the fact thatuis bijective implies the division theorem, more precisely, it implies the existence and uniqueness of the quotients and the remainder of the division of EbyM1, . . . , Mr, assuming that their Newton diagrams are contained in LandJ respectively.

Proof of Theorem 2.6. — Using the same notation as in the previous theorem, we have A_j =Q_j(F;E) and B =R(F;E). Let B be the basis of open neighborhoods of 0 consisting of open polycylinders of polyradiuss^λ with 0< s < s0. The canonical topology on DCⁿ(U)^q for any open polycylinder U centered at 0 of polyradius s^λ, s >0, is given by the seminorms| − |^µ,s_s0 ⁰, 0< s⁰ < sandµadmissible for (F, λ) (see section 1). Hence the bound in Theorem 2.11 implies the continuity ofQ_j andR.

Let us suppose now the following additional hypothesis on the chosen ordering:

(?) For every (α, β)∈N²ⁿ,(α, β)<(α⁰, β⁰) =⇒(0, β)<(0, β⁰) =⇒ |β|6|β⁰|.

There are many monomial orderings satisfying this condition. For instance, if<⁰ is a well ordering inNⁿ compatible with sums, we can set

(α, β)<(α⁰, β⁰) if











|β|<|β⁰| or

|β|=|β⁰|andβ <⁰ β⁰ or β =β⁰ andα <⁰α⁰.

We will see that under hypothesis (?) we can obtain a sharper result, namely (see [8]):

Theorem 2.14. — There are constants s₀ > 0, C > 0 such that for every E = u(A, B)∈D^qn we have

X

j

|A_j|^µ,t_s |M_j|^µ,t_s +|B|^µ,t_s 6C|E|^µ,t_s for every0< t < s < s₀ and for everyµ admissible for(F, λ).

The proof of the theorem in this case follows exactly the same steps as in the former case, the required hypothesis allows us to find admissible µ’s with arbitrarily large components satisfying the following additional condition:

µβj>µβ for every (α, β)∈N (M_j⁰) and for everyj = 1, . . . , r.

Proof. — We keep the notation used in Lemma 1. We will first check that β ∈ π2(N (σ(Mj))) whenµβis maximal. Otherwise there would be (α⁰, β⁰)∈N (Mj)∩M with|β⁰|>|β|, hence

µβ=p|β| −ρβ < p|β|6p|β⁰| −p < p|β⁰| −σα⁰−ρβ⁰< p|β⁰| −ρβ⁰ =µβ⁰. Let (α, β)∈ N(σ(Mj)). Then, (αj, βj)6(α, β) implies (0, βj)6(0, β) because of the hypothesis (?). Since (0, βj) and (0, β) are inM and the ordering restricted to this set is defined by the weights (σ, ρ), we have ρβ_j 6ρβ , henceµβ_j >µβ as we wanted to show.

This permits to verify the inequalities in the proof of the theorem for everyt < s, and not only fort=s.

It is possible to obtain a similar result if we restrict ourselves to the Weyl algebra A_n(C) (cf.[6], I.6) and consider the division operator inside this ring (cf.[4], Thm.

2.4.1, withL2(β) =|β|). The proof is basically the same one (now we do not have to take care of the convergence problems, since every element ofAn(C) consists only of a finite number of monomials). We will just state the corresponding results without giving the proofs, which can be found in [15].

In this case, the canonical topology onAn(C) that we consider is the one induced by the canonical topology onDCⁿ(Cⁿ) (or onD^∞_Cⁿ(Cⁿ)). It is defined by the family of seminorms (which in fact are now norms) parameterized bys^−λandt^−µfor suitableλ andµ, such thatstends to zero andt^−µ0. Here we say that a weightλ∈(N^∗)ⁿ is adapted to anr-tuple (M1, . . . , Mr) of elements ofAn(C) if for every positive constant K there existsµ∈Nⁿ with components greater thanK such that

λαj+µβj−ij > λα+µβ−i

for every (α, β, i)∈N (M_j⁰) and everyj = 1, . . . , r, where (α_j, β_j, i_j) is theexponent ofMj, defined in this case as follows:

Letσ(M) =P

i

P

αβaαβix^α∂^βei be the symbol ofM (defined as above). Theex- ponent ofM is max{(α, β, i)|aαβi6= 0}(this is a finite set in this case, so its maximum with respect to the ordering considered is well defined). Theinitial monomial of M is the monomial corresponding to its exponent.

In the same way as above, one can show that there is always a weightλ adapted to anyr-tupleF, and even one adapted to severalr-tuples (for distinctr’s) simulta- neously.

Theorem 2.15. — Let F = (M1, . . . , Mr), with Mi ∈ An(C)^q and let Qi(F;E), i= 1, . . . , r (resp.R(F;E)) be the quotients (resp. the remainder) of the division of E∈A_n(C)^q byF. Then theC-linear maps

Qi(F;−) :An(C)^q−→An(C), R(F;−) :An(C)^q−→An(C)^q are continuous with respect to the canonical topology.

Proposition 2.16. — There is a constantC1>0such that|(w1v⁻¹)E|^s_s^−µ−λ 6C1s|E|^s_s^−µ−λ

for everyE∈An(C)^q and0< s <1.

Proposition 2.17. — There are constantsC2>0ands0>0such that|(w2v⁻¹)E|^s_s^−µ−λ 6 C2s|E|^s_s^−µ−λ for0< s < s0 and for everyE∈An(C)^q.

Theorem 2.18. — The map u : L⊕J → A_n(C)^q is a bi-continuous isomorphism.

There are constantss0>0, C >0such that for everyE=u(A, B)∈An(C)^q we have X

j

|Aj|^s_s^−µ−λ|Mj|^s_s^−µ−λ+|B|^s_s^−µ−λ 6C|E|^s_s^−µ−λ

for0< s < s₀.