Continuous division of linear differential operators and faithful flatness of D∞X over DX

S´eminaires & Congr`es 8, 2004, p. 129–148 CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS AND FAITHFUL FLATNESS OF D∞ XOVER DX 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 de D∞ Xsur 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 OXof holomorphic functions on a complex analytic manifold Xis the first natural example of left module over the sheaf of linear differential operators DX on X. 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 left DXmodule structure on OXextends to a left D∞ X-module structure in such a way that D∞ X⊗DXOX=OX. For any holonomic left DX-module M, we know by the constructibility theorem of Kashiwara [9] (see also [12], [13]) that the complex of holomorphic solutions of M, R HomDX(M,OX), is constructible. The canonical DX-linear biduality morphism M−→ R HomCX(R HomDX(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. c S´eminaires et Congr`es 8, SMF 2004 130 L. NARV´ AEZ MACARRO & A. ROJAS LE ´ ON induces a D∞ X-linear morphism (*) D∞ X⊗DXM−→ R HomCX(R HomDX(M,OX),OX). The local biduality theorem of 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 equivalence 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 sheaf D∞ Xdoes not have any known finiteness properties like DX, but to prove the full Riemann-Hilbert correspondence one needs to know that the extension DX⊂D∞ Xis 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 differential 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 Let Xbe a complex analytic manifold of pure dimension n, countable at infinity. Let us denote by OXthe sheaf of holomorphic functions and by DXthe sheaf of linear differential operators (cf. [6]). For each open set U⊂X, the space OX(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 a nuclear space, cf. [5] for details). The Banach open mapping theorem shows that the property of being continuous for aC-linear endomorphism P:OX→OXis a local property. For that, let {Ui} be an open covering of X, that we can take as countable, such that each restriction P|Ui:OX|Ui→OX|Uiis continuous. For any open set U⊂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 ˇ Cech map QOX(U∩Ui)→QOX(U∩Ui∩Uj) by the S´ EMINAIRES & CONGR` ES 8 CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 131 sheaf condition). Hence, the continuity of P(U) : OX(U)→OX(U) comes from the continuity of QP(U∩Ui) : QOX(U∩Ui)→QOX(U∩Ui). As a consequence, the pre-sheaf of C-linear continuous endomorphisms of OX,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→OXand for any system (U;x1, . . . , xn)of local coordinates of X, there are unique holomorphic functions aα∈OX(U),α∈Nn, such that P|U=X α∈Nn aα 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)ξα∈C is holomorphic. From now on, we will denote D∞ X=Homtop(OX,OX) and call it sheaf 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→OXbe a C-linear endomorphism. The following properties are equivalent: a) Pis continuous. b) For any pair K, K0⊆Xof compact sets with K⊂ ◦ K0, there is a constant CK,K0>0such that |P(f)|K⩽CK,K0|f|K0for any holomorphic function fdefined on a neighborhood of K0. Corollary 1.3. — The sheaf DXof (finite order) linear differential operators is a subsheaf (of rings) of Homtop(OX,OX). Proof. — Let Pbe a section of DXover an open set U⊂X. Since continuity is a local property, we can suppose that Uis a connected open set of Cn. Then Padmits a unique expression P=X α∈Nn,|α|⩽d aα 1 α!∂α, where dis the order of Pand the aαare holomorphic functions on U. Let K, K0⊆U be a pair of compact sets as in proposition 1.2, b) and let fbe a holomorphic function SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 132 L. NARV´ AEZ MACARRO & A. ROJAS LE ´ ON on a neighborhood of K0. From Cauchy inequalities we deduce that |P(f)|K=X |α|⩽d aα 1 α!∂α(f)K ⩽X |α|⩽d |aα|Kr−|α||f|K0 where ris the distance between Kand U− ◦ K0. By proposition 1.2, we conclude that Pis continuous. Definition 1.4 ([14], déf.2.1.6). — For any open set U⊆X, the canonical topology of D∞ X(U) or DX(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 in Uwith K⊂ ◦ K0. For any coordinate system (U;x1, . . . , xn) in X, 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)yα is an isomorphism of locally convex vector spaces between D∞ X(U) endowed with the canonical topology and the space of holomorphic functions on U×Cnendowed with the topology of uniform convergence on compact sets. This isomorphism depends on local coordinates and carries the space DX(U) into the space of holomorphic functions on U×Cnwhich are polynomials with respect to the second factor. Consequently, D∞ X(U) is a Fr´echet (and nuclear) space and DX(U) is dense in D∞ X(U). We can write then D∞ X(U) = \ DX(U). In fact, in [14,§2] it is proved that D∞ Xendowed with the canonical topology is a sheaf with values in the category of Fr´echet C-algebras. Let us denote by On,Dn,D∞ nthe stalk at the origin of the sheaves OCn,DCn,D∞ Cn respectively. For ρ= (ρ1, . . . , ρn), L= (L1, . . . , Ln) in (R∗ +)nlet us consider the pseudo-norm |−|L ρ:D∞ n→R+∪{+∞} whose value at P=Paβ∂β=Pαβ aαβxα∂β is (1) |P|L ρ=X β |aβ|ρ|β|!Lβ=X αβ |aαβ|·|β|!ραLβ∈R+∪ {+∞}. Since β!⩽|β|!⩽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 by D∞ n(ρ) the subspace of D∞ nwhere |−|L ρtakes finite values for any L∈(R∗ +)nand let us write Dn(ρ) := Dn∩D∞ n(ρ). The semi-norms |−|L ρ,L∈(R∗ +)n, define a Fr´echet topology on D∞ n(ρ). S´ EMINAIRES & CONGR` ES 8 CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 133 Following [8], we consider weights λ, µ ∈(N∗)nand, for real numbers s, t > 0, ρ=sλ= (sλ1, . . . , sλn), L=t−µ= (t−µ1, . . . , t−µn). When λis fixed, we denote |−|µ,t s:= |−|L ρ,D∞ n(s) := D∞ n(ρ) and Dn(s):=Dn(ρ). In the case where Uis an open polycylinder of Cncentered at 0 of polyradius σ=sλ 0, 0 < s0⩽+∞, we have D∞ Cn(U) = \ 0<s<s0 D∞ n(s),DCn(U) = \ 0<s<s0 Dn(s), and the canonical topology of D∞ Cn(U) (resp. DCn(U)) is the (topological) inverse limit of the D∞ n(s) (resp. Dn(s)), for 0 < s < s0. In other words, the canonical topologies of D∞ Cn(U) and DCn(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 µfixed and t→0, or taking t=t(s)<1 and µ0. For vectors P= (P1, . . . , Pq)∈(D∞ n)q, following [7] we also define |P|µ,t s:= q X i=1 |Pi|µ,t ss−(i−1), where λ∈(N∗)nis fixed. In the above situation, the product topology on D∞ Cn(U)qand DCn(U)qis also given by the semi-norms |−|µ,t s, 0 < s < s0,t−µ0. 2. The continuity theorem In this section, we fix M1, . . . , Mr∈Dq nand 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) and i > j Given N= (N1, . . . , Nq) = q X i=1 Niei= q X i=1 X α,β aαβixα∂βei∈Dq n, aαβi ∈C, where {ei}i=1...q stands for the canonical basis of Dq nas a free Dn-module, we denote by N(N), the Newton diagram of N, the set of (α, β, i) in N2n× {1, . . . , q}such that aαβi 6= 0 and by σ(N) its symbol,i.e. the homogeneous component of Nof maximal degree with respect to the grading given by the total degree in ∂: σ(N) = q X i=1 X |β|=dX α aαβixα∂βei, d = degT(N) = max deg(Ni). SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 134 L. NARV´ AEZ MACARRO & A. ROJAS LE ´ ON The exponent of Nis exp(N) := min{(α, β, i)|aαβi 6= 0,|β|=d}, and the corresponding monomial of σ(N) (and therefore of N) is, by definition, the initial monomial of N. Let (αj, βj, ij) be the exponent of Mjwith respect to the given ordering (we can assume without loss of generality that its coefficient is 1). Also let M0 j=Mj− xαj∂βjeij. We will denote by Fthe r-tuple (M1, . . . , Mr). The following notion is needed in the continuity theorem 2.6: Definition 2.1. — We say that a given weight λ∈(N∗)nis adapted to Fif for every positive constant Kthere exists µ∈Nnwith µi> K, λifor every i= 1, . . . , n such that λαj−µβj−ij< λα −µβ −i for every j= 1, . . . , r and every (α, β, i)∈N(M0 j). We say that such a µis Kadmissible, or simply admissible, for (F, λ). Lemma 2.2. — For any F= (M1, . . . , Mr)as above, there exists a weight λadapted to F. Proof. — Consider first the case q= 1. Let π1, π2:N2n=Nn×Nn→Nnbe the canonical projections. For every j= 1, . . . , r and every β∈π2(N(Mj)) let Mj βbe the set of (α, β) in N(Mj) such that αis minimal in A={α: (α, β)∈N(Mj)} with respect to the componentwise order. The set Mj βis finite, in fact it consists of the elements (α1, β), . . . , (αs, β), where {α1, . . . , αs}is the minimal set of generators of the ideal A+Nnof Nn. Therefore, the set M=SjSβ(Mj β∪ {(0, β)}) is also finite. Let (σ, ρ)∈Nn×Nn=N2nbe a vector defining the given ordering restricted to the finite set M. We claim that λ=σis adapted to F. Fix a positive constant K, and let pbe 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(Mj), 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)∩Mwith |β0|>|β|. Then λα−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(Mj)}= 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. S´ EMINAIRES & CONGR` ES 8 CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 135 Now assume q6= 1. Let Mj=Piαβ aj iαβxα∂βei, and define Mj=Piαβ |aj iαβ|xα∂β∈ Dn. Let (αj, βj) be the exponent of Mjwith respect to the given ordering in N2n. Let (αj, βj, ij) be the exponent of Mj. We have αj=αjand βj=βj. Otherwise, we would have (αj, βj)>(αj, βj). Let ibe such that (αj, βj, i)∈N(Mj). Then we have by definition of the exponent that (αj, βj, i)⩾(αj, βj, ij), and therefore (αj, βj)⩾(αj, βj), which is in contradiction with the last inequality. By the first part of the proof, there exists λadapted to (M1, . . . , Mr). Now given a positive constant Kthere is µ∈Nnwith µj> K/q and λj< µjsuch that λαj−µβj< λα−µβ for every (α, β)∈N(M0 j). Let us see that λ=qλ is adapted to F and µ=qµ is K-admissible for (F, λ). Let (α, β, i)∈N(M0 j), then (α, β)∈N(Mj), hence λαj−µβj⩽λα −µβ by construction. Now we distinguish two cases: If (α, β)>(αj, βj), then λαj−µβj< λα −µβ. But λas well as µare multiples of q, and therefore λαj−µβj⩽λα −µβ −q, and λαj−µβj−ij< λαj−µβj⩽ λα −µβ −q⩽λα −µβ −i. If (α, β) = (αj, βj), then we must have i < ij, hence λαj−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, . . . , Fmbe a finite number of vectors whose coordinates are in Dq nas above (they may have distinct lengths). Then there exists λ∈Nnwhich is adapted to all of them. Proof. — This is a direct consequence of the following lemma applied to the vector constructed by concatenation of F1, . . . , Fm. The following lemma is clear: Lemma 2.4. — Let F=(M1, . . . , Mr)be a vector in (Dq n)rand let G=(Mi1, . . . , Mik), with 1⩽i1<· · · < ik⩽r. Then every λ∈Nnadapted to Fis also adapted to G. Before stating the main theorem of this section we make one further definition: Definition 2.5. — Let λ∈(N∗)nbe a weight. A basis Bof open neighborhoods of 0∈Cnis said to be a λ-basis if it consists of open polycylinders of polyradius sλfor 0< s < s0, for some s0>0. We will say that Bis adapted to Fif it is a λ-basis for some λadapted to F. From the lemmas above it follows that we can always find a basis of neighborhoods of 0 adapted to F, and even a basis adapted to a finite number of vectors F1, . . . , Fm. After these preliminaries we are ready to state the continuity theorem of the division of linear differential operators: Theorem 2.6. — Let F= (M1, . . . , Mr), with Mi∈Dq nand let Qi(F;E),i= 1, . . . , r (resp. R(F;E)) be the quotients (resp. the remainder) of the division of E∈Dq nby F SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 136 L. NARV´ AEZ MACARRO & A. ROJAS LE ´ ON (see [4]). Then, for any weight λ∈(N∗)nadapted to F, there exists a λ-basis Bof open neighborhoods of 0∈Cn, such that for every U∈Bthe C-linear morphisms Qi(F;−)(resp. R(F;−)) map DCn(U)qinto DCn(U)(resp. into DCn(U)q). Furthermore, Qi(F;−): DCn(U)q−→ DCn(U), R(F;−): DCn(U)q−→ DCn(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 Dr n→Dq ndefined by the finite system of vectors F= (M1, . . . , Mr)∈(Dq n)rinstead 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 of Mj. For every i= 1, . . . , n, let Tj:Dq n→Dq nbe the C-linear map defined by Tjxα∂βei=xα∂β+ejei. Given A=Pcγδxγ∂δ∈Dn, we will denote by Aothe map PcγδxγTδ:Dn→Dn, and A0=A−Ao(Ais considered here to be acting by multiplication on the left). Let also {∆1, . . . , ∆r,∆}be the partition of N2n× {1, . . . , q}defined by M1, . . . , Mr(see [3] and [4] in this volume). We define now the following sets Land J: L={A∈Dr n: exp(Mj) + N(Aj)⊂∆j,∀j= 1, . . . , r} J={B∈Dq n:N(B)⊂∆} and the linear map u:L⊕J→Dq ngiven by u(A, B) = Pr j=1 AjMj+B. From the division theorem ([4], th. 2.4.1) we see that Land Jare the sets where quotients and the remainder of the division by M1, . . . , Mrare “allowed” to lie, the Ajand Bare just the quotients and the remainder of the division of u(A, B) by F and the map uis bijective. We start by splitting uas a sum v+w1+w2, with v(A, B) = XAo jxαj∂βjeij+B w1(A, B) = XA0 jxαj∂βjeij w2(A, B) = XAjM0 j. The C-linear map vis easily seen to be an isomorphism of C-vector spaces, by definition of Land J. We follow the notation in the previous section regarding the seminorms |−|µ,t s. Let E∈Dq n, and (A, B) = v−1(E)∈L⊕J, with Aj=Pγδ aj γδxγ∂δ. Then, E=Pjγδ aj γδxαj+γ∂βj+δeij+B. If we take the | − |µ,t snorm on both sides, and S´ EMINAIRES & CONGR` ES 8 CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 137 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=XjXγδ aj γδxαj+γ∂βj+δeij µ,t s+B µ,t s ⩾Xjγδ aj γδβj+δ!sλ(αj+γ)−(ij−1)t−µ(βj+δ). Proposition 2.7. — There is a constant C1>0such that |(w1v−1)E|µ,s s⩽C1s|E|µ,s s for every E∈Dq nand for every µadmissible for (F, λ). Proof. — Let (A, B) = v−1(E), with Aj=Pγδ aj γδxγ∂δ. First, we have |(w1v−1)E|µ,s s=|w1(A, B)|µ,s s=X j A0 jxαj∂βjeij µ,s s =X jγδ aj γδxγ(∂δ−Tδ)xαj∂βjeij µ,s s. Expanding the inner product, we get |(w1v−1)E|µ,s s=X jγδ aj γδxγX 0<ε⩽αj,δ αj εδ! (δ−ε)!xαj−ε∂βj+δ−εeij µ,s s ⩽X jγδ X 0<ε⩽αj,δ |aj γδ|αj εδ! (δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε) ⩽X jX 0<ε⩽αjX δ⩾εX γ |aj γδ|2|αj|δ! (δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε). Therefore |(w1v−1)E|µ,s s |E|µ,s s ⩽PjP0<ε⩽αjPδ⩾εPγ|aj γδ|2|αj|δ! (δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε) Pjγδ |aj γδ||βj+δ|!sλ(αj+γ)−(ij−1)−µ(βj+δ) ⩽X jε Pδ⩾εPγ|aj γδ|2|αj|δ! (δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−(ij−1)−µ(βj+δ−ε) Pγδ |aj γδ||βj+δ|!sλ(αj+γ)−(ij−1)−µ(βj+δ) =X jε Pδ⩾εPγ|aj γδ|2|αj|δ! (δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−µ(βj+δ−ε) Pγδ |aj γδ||βj+δ|!sλ(αj+γ)−µ(βj+δ) ⩽X jε Pγδ |aj γδ|2|αj|δ! (δ−ε)!|βj+δ−ε|!sλ(αj+γ−ε)−µ(βj+δ−ε) Pγδ |aj γδ||βj+δ|!sλ(αj+γ)−µ(βj+δ). SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 144 L. NARV´ AEZ MACARRO & A. ROJAS LE ´ ON As in the previous case, it is also possible to obtain a sharper result if we assume the additional hypothesis (?) given earlier on the ordering. The new stronger statement is the following: Theorem 2.19. — If the chosen ordering satisfies hypothesis (?), the map u:L⊕J→ An(C)qis a bi-continuous isomorphism. There are constants s0>0,C > 0such that for every E=u(A, B)∈An(C)qwe have, for 0< t < s < s0, X j |Aj|t−µ s−λ|Mj|t−µ s−λ+|B|t−µ s−λ⩽C|E|t−µ s−λ. 3. Continuous scissions Proposition 3.1. — Let V⊂Cnbe an open neighborhood of 0,ri, qi⩾1,1⩽i⩽m integers and Fi:Dri V→Dqi V,1⩽i⩽ma finite family of DV-linear maps. There exists a weight λ∈(N∗)n, a λ-basis Bof open neighborhoods of 0∈Cnand a family of continuous scissions {σi U:DV(U)qi→DV(U)ri}U∈Bof Fi, i.e. : Fi(U)◦σi U◦Fi(U) = Fi(U), U ∈B,1⩽i⩽m compatible with restrictions, i.e. σi U|W=σi Wfor W⊂U. Proof. — First, let us write fi= (Fi)0and let us take a Gr¨ obner (or standard) basis Gi={Ni 1, . . . , Ni si}of im(fi)⊂Dqi nfor each i= 1, . . . , m (see [4] in this volume) , and consider the corresponding linear map gi:Dsi n→Dqi n. By shrinking Vif necessary, we can suppose that giis the stalk at 0 of a linear map Gi:Dsi V→Dqi V. Let us consider τi= (Q1(Gi;−), . . . , Qsi(Gi;−)) : Dqi n−→ Dsi n. Let λ∈(N∗)nbe a weight adapted to Gi, for every i= 1, . . . , m (see lemma 2.3). By theorem 2.6, there exists a λ-basis Bof open neighborhoods of 0 ∈Cn, such that for every U∈Band every ithe C-linear map τi U:= τi|DV(U)qi:DV(U)qi→DV(U)siis continuous. Furthermore, the fact that Giis a Gr¨ obner basis implies that gi◦τi◦gi= gi. By analytic continuation we obtain Gi(U)◦τi U◦Gi(U) = Gi(U),∀U∈B,∀i= 1, . . . , m. Let hi:Dsi n→Dri nbe the linear map such that fi◦hi=gi. By shrinking Vagain if necessary, we can suppose that hiis the stalk at 0 of a linear map Hi:Dsi V→Dri V. Let σi:= hi◦τi:Dqi n→Dri nand σi U:= hi(U)◦τi U:DV(U)qi→DV(U)ri,U∈B, which are continuous. From im(fi) = im(gi) (Giis a Gr¨ obner basis of im(fi)) and gi◦τi◦gi=giwe deduce fi◦σi◦fi=fi. Analytic continuation gives again Fi(U)◦σi U◦Fi(U) = Fi(U) for all U∈B, 1 ⩽i⩽m. S´ EMINAIRES & CONGR` ES 8 CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 145 Corollary 3.2. — Let V⊂Cnbe an open neighborhood of 0,ri, si, qi⩾1,i= 1, . . . , m integers and Dri V Fi −−−→ Dsi V Gi −−−→ Dti V, i = 1, . . . , m a finite family of exact sequences of DV-linear maps. There exists a weight λ∈(N∗)n and a λ-basis Bof open neighborhoods of 0∈Cnsuch that for any U∈Band any i= 1, . . . , m the sequence DV(U)riFi(U) −−−−−−→ DV(U)siGi(U) −−−−−−→ DV(U)ti is exact and topologically split. Proof. — By proposition 3.1, there exists a weight λ∈(N∗)n, a λ-basis Bof open neighborhoods of 0 ∈Cnand a family of continuous scissions {σi U:DV(U)qi→ DV(U)ri}U∈Bof Fi,i= 1, . . . , m compatible with restrictions. Let us write fi= (Fi)0, gi= (Gi)0and σi= lim U∈Bσi Ufor each i= 1, . . . , m. We have fi=fi◦σi◦fi and ker gi= im fi= ker(1 −fi◦σi). Then, for any U∈Band any M∈ker Gi(U) we have 0 = M0−fi(σi(M0)) = M−Fi(U)(σi U(M))0, and by analytic continuation we deduce M∈ker(1 −Fi(U)◦σi U)⊂im Fi(U). Proposition 3.3. — Let V⊂Cnbe an open neighborhood of 0,Mia coherent DVmodule, i= 1, . . . , m, and Dri V Fi −−−→ Dsi V πi −−−→ Mi−→ 0 a finite presentation. Then, for any λ-basis Bof open neighborhoods of 0∈Cnsuch that the morphisms Fi(U)split for U∈Band i= 1, . . . , m, the sequence DV(U)riFi(U) −−−−−−→ DV(U)siπi(U) −−−−−−→ Mi(U)−→ 0 is exact for U∈Band i= 1, . . . , m. Proof. — By shrinking Vif needed, we can suppose that the kernel of Fihas a good filtration on V(cf. [6], prop. 10). Then, for any compact polycylinder K⊂Vand any i= 1, . . . , m the sequence (5) DV(K)riFi(K) −−−−−−→ DV(K)siπi(K) −−−−−−→ Mi(K)−→ 0 is exact by the Cartan-Oka theorem (cf. loc. cit., prop.11). By proposition 3.1, there exist a weight λ∈(N∗)nand a λ-basis Bof open neighborhoods of 0 ∈Cnsuch that for any U∈Band any i= 1, . . . , m the maps Fi(U) : DV(U)ri→DV(U)sisplit, with scissions compatible with restrictions. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 146 L. NARV´ AEZ MACARRO & A. ROJAS LE ´ ON If Kis the closure of a U0∈B, the sequence (5) is the inductive limit of the sequences DV(U00)riFi(U00) −−−−−−−→ DV(U00)si−→ coker Fi(U00)−→ 0 with U00 ∈B,K⊂U00, and hence it splits. Now, for any U∈Bthe sequence DV(U)riFi(U) −−−−−−→ DV(U)siπi(U) −−−−−−→ Mi(U)−→ 0 is the projective limit of sequences (5), with K=U0⊂U,U0∈B, and consequently it is exact. 4. Faithful flatness of D∞ Xover DX Faithful flatness of the ring of differential operators of infinite order over the ring of differential operators of finite order has been stated for the first time by Sato, Kashiwara and Kawai in [17]. Their proof used microlocal methods. In this section we reproduce the proof given in [14], based on the continuity of division of differential operators studied in the precedent sections. Theorem 4.1. — For any complex analytic manifold X, the extension DX→D∞ Xis faithfully flat. Proof. — It is enough to prove that the ring extension Dn→D∞ nis faithfully flat. Let 0 −→ M1−→ M2−→ M3−→ 0 be an exact sequence of Dn-modules. It is the stalk at 0 of an exact sequence 0 −→ M1−→ M2−→ M3−→ 0 of DV-modules, where Vis an open neighborhood of 0. We can find a commutative diagram 0 0 0 0//M1// OO M2// OO M3// OO 0 0//Dr1 V// OO Dr2 V// OO Dr3 V// OO 0 0//Ds1 V// OO Ds2 V// OO Ds3 V// OO 0 0//Dt1 V// OO Dt2 V// OO Dt3 V// OO 0 S´ EMINAIRES & CONGR` ES 8 CONTINUOUS DIVISION OF LINEAR DIFFERENTIAL OPERATORS 147 with exact rows and columns. By propositions 3.1, 3.3 and corollary 3.2, there exist a weight λand a λ-basis of neighborhoods of 0 such that for any U∈Bthe diagram 0 0 0 0//M1(U)// OO M2(U)// OO M3(U)// OO 0 0//DV(U)r1// OO DV(U)r2// OO DV(U)r3// OO 0 0//DV(U)s1// OO DV(U)s2// OO DV(U)s3// OO 0 0//DV(U)t1// OO DV(U)t2// OO DV(U)t3// OO 0 has exact and topologically splitting rows and columns. Then, the corresponding diagram of topological completions has also exact rows and columns. As for any open polycylinder Wwe have \ DV(W) = D∞ V(W), we deduce \ Mi(U) = D∞ V(U)⊗DV(U)Mi(U) and the exactness of the sequence 0−→ D∞ V(U)⊗DV(U)M1(U)−→ D∞ V(U)⊗DV(U)M2(U)−→ D∞ V(U)⊗DV(U)M3(U)−→ 0 for every U∈B. Taking direct limits we obtain the exactness of 0−→ D∞ n⊗DnM1−→ D∞ n⊗DnM2−→ D∞ n⊗DnM3−→ 0, and so the extension Dn→D∞ nis flat. To conclude, we observe that the quotient topology on Mi(U), U∈B, is separated. Then Mi(U),→\ Mi(U) = D∞ V(U)⊗DV(U)Mi(U) and Mi,→D∞ n⊗DnMi. In particular Mi6= 0 implies D∞ n⊗DnMi6= 0 and the extension Dn→D∞ nis faithfully flat. Using the corresponding results for the Weyl algebra (Theorems 2.15 and 2.18) it is possible to obtain the faithful flatness for the extension An(C)→D∞ Cn(Cn), as done in [15]. References [1] J.E. Bj¨ ork –Rings of differential operators, North Holland, Amsterdam, 1979. [2] J. Brian¸con &Ph. Maisonobe – Id´eaux de germes d’op´erateurs diff´erentiels `a une variable, Enseign. Math. 30 (1984), p. 7–38. [3] F.J. Castro-Jim´ enez – Calcul de la dimension et des multiplicit´es d’un D-module monog`ene, C. R. Acad. Sci. Paris S´er. I Math. 302 (1986), p. 487–490. [4] F.J. Castro-Jim´ enez &M. Granger – Explicit Calculations in Rings of Differential Operators, in this volume. [5] R. Douady – Produits tensoriels topologiques et espaces nucl´eaires, in S´eminaire de g´eom´etrie analytique (A. Douady & J.-L. Verdier, eds.), Ast´erisque, vol. 16, Soci´et´e Math´ematique de France, 1974, p. 7–32. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 148 L. NARV´ AEZ MACARRO & A. ROJAS LE ´ ON [6] M. Granger &Ph. Maisonobe – A basic course on differential modules, in ´ El´ements de la th´eorie des syst`emes diff´erentiels, I, II [10], vol. I, p. 103–168. [7] H. Hauser &M. M¨ uller – A Rank Theorem for analytic maps between power series spaces, Publ. Math. Inst. Hautes ´ Etudes Sci. 80 (1994), p. 95–115. [8] H. Hauser &L. Narv´ aez-Macarro – Continuous division of differential operators, Ann. Inst. Fourier (Grenoble) 51 (2001), p. 769–778. [9] M. Kashiwara – On the maximally overdetermined systems of differential equations, Publ. RIMS, Kyoto Univ. 10 (1975), p. 563–579. [10] Ph. Maisonobe &C. Sabbah (eds.) – ´ El´ements de la th´eorie des syst`emes diff´erentiels, I, II, Les cours du CIMPA, Travaux en cours, vol. 45, 46, Hermann, Paris, 1993, summer school at CIMPA, Nice, 1990. [11] Z. Mebkhout – Le th´eor`eme de positivit´e, le th´eor`eme de comparaison et le th´eor`eme d’existence de Riemann, in this volume. [12] Z. Mebkhout &L. Narv´ aez-Macarro – D´emonstration g´eom´etrique du th´eor`eme de constructibilit´e, in Le formalisme des six op´erations de Grothendieck pour les Dmodules coh´erents, par Z. Mebkhout, Travaux en cours, vol. 35, Hermann, Paris, 1989, p. 248–253. [13] , Le th´eor`eme de constructibilit´e de Kashiwara, in ´ El´ements de la th´eorie des syst`emes diff´erentiels, I, II [10], vol. II, p. 47–98. [14] , Le th´eor`eme de continuit´e de la division dans les anneaux d’op´erateurs diff´erentiels, J. reine angew. Math. 503 (1998), p. 193–236. [15] A. Rojas Le´ on – Sobre la continuidad de la divisi´on en anillos de operadores diferenciales, Dep. ´ Algebra, Univ. Sevilla, 2001. [16] M. Sato – Hyperfunctions and partial differential equations, in Proc. Int. Conf. on Functional Analysis, Tokyo (1969), Tokyo Univ. Press, Tokyo, 1970, p. 91–96. [17] M. Sato, T. Kawai &M. Kashiwara – Microfunctions and pseudo-differential equations, in Hyperfunctions and pseudo-differential equations, Proc. Conf. Katata, (1971), Lect. Notes in Math., vol. 287, Springer-Verlag, 1973, p. 265–529. [18] J.-P. Serre – G´eom´etrie alg´ebrique et g´eom´etrie analytique, Ann. Inst. Fourier (Grenoble) 6(1956), p. 1–42. L. Narv´ aez Macarro, Departamento de Algebra, Facultad de Matem´aticas, Universidad de Sevilla, E-41012 Sevilla, Spain •E-mail : [email protected] A. Rojas Le´ on, Department of Mathematics, Princeton University, U.S.A. E-mail : [email protected] S´ EMINAIRES & CONGR` ES 8