A flatness property for filtered D-modules

arXiv:math/0505465v1 [math.AG] 23 May 2005 A flatness property for filtered D-modules F.J. Castro-Jim´enez and M. Granger Abstract Let Mbe a coherent module over the ring DXof linear differential operators on an analytic manifold Xand let Z1,· · · , Zkbe kgerms of transverse hypersurfaces at a point x∈X. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of Mat x, give rise to a multifiltration U•(M) of Mxas in Sabbah’s paper [9] and to an analytic standard fan in a way similar to [3]. We prove here that this standard fan is adapted to the multifiltration, in the sense of C. Sabbah. This result completes the proof of the existence of an adapted fan in [9], for which the use of [8] is not possible. 1 Introduction Let us consider a coherent module Mover the ring DXof differential operators on an analytic manifold X. For any smooth hypersurface Zof X, Malgrange and Kashiwara defined a filtration along Zfor DXand the notion of a good filtration for M. Given a set of transverse smooth hypersurfaces Z1,...,Zk, Sabbah considered in [9] multifiltrations of Mindexed by k-uples of relative integers. To be precise he dealt with linear combinations over Q+of the filtrations V(j)along each hypersurface Zjand with refinements VΓof the original multifiltration associated with each rational polyedral simplicial cone in the positive quadrant of (Qk)⋆. The original multifiltration is the one which corresponds to the case Γ = Nk. The aim of this paper is to clarify the flatness properties which appear in [9], namely to prove the existence of a fan E, such that for any cone Γ in this fan, the Rees module of the filtration VΓis flat over the affine ring AΓof the toric blowing-up of Ckassociated with the fan E. Such a fan is called an adapted fan in [9]. The reason for this clarification is that the proof in [9] depends on the appendix [8] in which the main tool is a division theorem in Rees rings of differential operators which is not correct as stated. Indeed, the infinite process that its proof suggests, would require monomials of unbounded degrees in the differential variables of the ring of operators. One of the main consequences of the existence of an adapted fan as developed in [9] is that we thus complete the proof of the existence of non trivial functional equations of Bernstein-Sato type for a k–uple of functions, following the argument of Sabbah in [9]. It should be emphasized here that this problem involves a DX×Ckmodule naturally associated with a k-uple of functions on X, and that in this situation a multifiltration, along the transverse hypersurfaces tj= 0 due to the factor Ck, appears in a natural way. We must note here that the proof of the existence of Bernstein-Sato equations has already been completed in Bahloul’s paper [4], by a different method which avoids the reference to a flatness property involved in the notion of an adapted fan. Bahloul uses instead the analytic standard fan as defined in [3], and described also in [5]. It is therefore not completely surprising that the adapted fan wanted in [9] turns out in fact to be the analytic standard fan. This emphasizes the interest of Bahloul’s proof which also has the advantage of being constructive. This constructiveness is valuable in an algebraic setting also. In both proofs, Bahloul’s and Sabbah’s , the latter which is completed by this paper, the main step is the proof of the goodness of the so-called saturated filtration. The details are given in section 2.3. The starting point of the proof of our main theorem (Theorem 4.3) is a criterion of flatness by M. Herrmann and U. Orbanz (see [7]), for graded modules over graded rings, where the grading is indexed by an arbitrary commutative group. In the statement of this result there is no reference to any finiteness property. In our case the indexing group will be Zk, and the ring a conical subring of the ring of Laurent polynomials in k variables with the obvious multigrading by monomials. The main ingredient of the proof of Theorem 4.3 is then 1 the existence of a simultaneous L–standard basis of a submodule N ⊂ Drfor all Lin a cone of the analytic standard fan of N(see [3]). The plan of the paper is as follows. In section 2, we recall with more details what V-multifiltrations, and Rees modules are, and the refinements of these multifiltrations with respect to a rational polyhedral simplicial cone. We end this section by defining the fiber at zero of these Rees modules seen as modules over the ring of an affine toric blowing up. We aim to prove that this is the fiber of a flat deformation if the cone is taken in the analytic standard fan. In section 3 we recall the notion of an analytic standard fan as developed in [3], and we sketch the easy generalisation which we need from the cyclic case treated in [3] to the general case. In section 4 we finally prove that the analytic standard fan is an adapted fan in the sense of [9]. Acknowledgements.- During the preparation of this paper both authors have been partially supported by Acci´on Integrada-Picasso HF2004-0117. First author has been also partially supported by MTM2004-01165. Both authors are grateful to the D´epartement de Math´ematiques et Applications of the ´ Ecole Normale Sup´erieure (Paris) where the first was invited during the preparation of the final part of this work. They also thank Jenifer Granger for her proofreading. 2 Multifiltrations, Rees rings and Rees modules Let us denote by O=C{x1,...,xn}the complex convergent power series ring and by Dthe ring of germs of linear differential operators with holomorphic coefficients (i.e. D=O[∂] = O[∂1, . . . , ∂n] where ∂iis the partial derivative with respect to xiand the product in Dis defined by the Leibnitz’s rule: ∂ia=a∂i+∂i(a), for each a∈ O). An operator Pin Dcan be written as P=X β∈Nn pβ(x)∂β=X α,β∈Nn pαβxα∂β where the first sum is finite, α= (α1,...,αn), β = (β1,...,βn), xα=xα1 1···xαn n,∂β=∂β1 1···∂βn n,pβ(x)∈ O and pαβ ∈C. For each i= 1,...,n, let us remember that the V-filtration on D, with respect to the hypersurface xi= 0, was defined by Malgrange and Kashiwara as: V(i) ℓ(D) = V(i) ℓ={P=X α,β∈Nn pαβxα∂β∈ D | βi−αi≤ℓ} for each ℓ∈Z. The family (V(i) ℓ)ℓ∈Zis an increasing exhaustive filtration on the ring D. For i= 1, the associated graded ring grV(1) (D) = M ℓ V(1) ℓ V(1) ℓ−1 is isomorphic to the ring C{x2, . . . , xn}[x1, ∂1, ∂2,...,∂n], graded by the so-called V(1)–graduation, where the homogeneous elements of V(1)–degree ℓare X α,β∈Nn;β1−α1=ℓ pαβxα∂β. We have similar descriptions for i= 2,...,k. We can also consider a V(i)-filtration on the free module Drjust by defining V(i) ℓ(Dr) = (V(i) ℓD)rand more generally, for any vector m= (m1,...,mr)∈Zr, one can define the shifted filtration V(i)[m]ℓ(Dr) = ⊕r j=1V(i) ℓ−mjD. Such a filtered module is called a V(i)-filtered free module. All the D-modules considered will be left modules. 2 Definition 2.1 Let Mbe a D-module. We say that a filtration U(i)(M)indexed by Zis a good V(i)-filtration if there exists a presentation M=Dr Nas a quotient by a left submodule Nof Dr, and a weight vector msuch that U(i) ℓ(M) = π(V(i)[m]ℓ(Dr)) where πis the projection Dr→ M. 2.1 V-multifiltrations and VΓ-multifiltrations Let us fix an integer ksuch that 1 ≤k≤n. For each s= (s1,...,sk)∈Zk, we shall denote Vs(D) = Tk i=1 V(i) si(D). The family {Vs(D)}s∈Zkdefines a multi-filtration on the ring D. To simplify we shall say that V•(D) is a k–filtration on Dor even, if no confusion is possible, a filtration on D. Let us consider a rational simplicial cone Γ in the first quadrant of the dual space (Qk)∗=HomQ(Qk,Q). We denote by ˇ Γ the dual cone of Γ, i.e. ˇ Γ = {a∈Qk|γ(a)≥0,∀γ∈Γ}. We associate with such a cone Γ the affine variety, denoted by SΓ, with coordinate ring equal to C[ˇ Γ∩Zk]. We will denote AΓ=C[ˇ Γ∩Zk] and A=C[Nk]. We denote by L(Γ) the set of primitive elements in the 1-skeleton of Γ. The multifiltration VΓon Dis defined as follows: For each s∈Zkwe define VΓ s(D) = X σ∈Zk|L(σ)≤L(s);∀L∈L(Γ) Vσ(D). Notice that the sum is indexed by σ∈s−ˇ Γ and that we have the inclusion Vs(D)⊂VΓ s(D). The family VΓ •(D) is a multifiltration of the ring D, indexed by s∈Zk. This means that : VΓ s(D).V Γ s′(D)⊂VΓ s+s′(D) and [VΓ s(D) = D. We may define, in a similar way to the case of one filtration, the notion of a free multi-filtered module and that of a good multifiltration of a finitely generated D-module M. For that purpose we chose a shift multivector n= (n(1),··· , n(r))∈(Zk)rcalled also a shift matrix, with columns n(i)∈Zk, and a presentation M=Dr Nof M. Definition 2.2 The multifiltered free module associated with nis the module Drendowed with the multifiltration indexed by s∈Zk, and defined as: V[n]s(Dr) = r M i=1 Vs−n(i)(D) Definition 2.3 A good multifiltration of M, is a filtration indexed by s∈Zk, of the type Us(M) = π(V[n]s(Dr)) = V[n]s(Dr) + N N for some presentation π:Dr→ M of M. In the obvious sense these multifiltrations are compatible with the multifiltered structure on the ring D. Remark that the i-th generator eiof Dris then of multidegree n(i)∈Zk. We may also endow Nwith the induced filtration Us(N) = V[n]s(Dr)∩ N, so that we also have Us(M) = V[n]s(Dr) Us(N). We may observe that the multifiltration on the free module Dris defined as the intersection of the V(i)– filtration with respect to the row vectors of n,ni= (n(1) i,··· , n(r) i)∈Zr, that is V[n]s(Dr) = V(1)[n1]s1(Dr)∩ · · · ∩ V(k)[nk]sk(Dr), 3 but the analogue for M, with respect to the good V(i)-filtrations as defined in definition 2.1 is not true since the inclusion: Us(M) = V(1)[n1]s1(Dr)∩ · · · ∩ V(k)[nk]sk(Dr) + N N⊂ k \ i=1 V(i)[ni]si(Dr) + N N= k \ i=1 U(i) si(M) may be strict. For each good filtration there is an associated Γ-filtration compatible with the multifiltration VΓ •(D) on D: For each s∈Zklet us consider UΓ s(M) = X σ∈Zk|L(σ)≤L(s);∀L∈L(Γ) Uσ(M). The multifiltration UΓ •(M) is a good filtration with respect to VΓ •(D). The goodness means here that we can verify, with the presentation of Mas above, that UΓ s(M) = π(V[n]Γ s(Dr)), is still the quotient of a filtration on the free module Drwhich is a direct sum of convenient shifts of the filtration VΓ •on the ring D. More precisely the involved filtration V[n]Γ •is defined by V[n]Γ s(Dr) = X σ∈Zk|L(σ)≤L(s);∀L∈L(Γ) V[n]σ(Dr)( = r M i=1 (VΓ s−n(i)(D)) and it remains to be remarked that: UΓ s(M) = X σ∈Zk|L(σ)≤L(s);∀L∈L(Γ) Uσ(M) = X σ∈Zk|L(σ)≤L(s);∀L∈L(Γ) V[n]σ(Dr) + N N=V[n]Γ s(Dr) + N N so that the goodness of the filtration U•(M) implies the goodness of UΓ •(M). 2.2 Rees rings and Rees modules. 2.2.1 Definition of Rees rings RV(D),RΓ(D)and of related Rees modules The Rees ring associated with the V-multifiltration on Dis defined by: RV(D) = M s∈Zk Vs(D)us where u= (u1,...,uk) are variables and the product in the Rees ring is induced by the natural product in the Laurent polynomial ring D[u±1] = D[u1, u−1 1,...,uk, u−1 k]. By definition the Rees ring RV(D) is a graded C-algebra with values group Zk, whose homogeneous elements are P usfor P∈Vs(D) and s∈Zk. Similarly given a D-module M=Dr/Nand a shift matrix n, we define from the good filtrations V[n]s(Dr), and Us(M), the Rees modules RV[n](Dr) = ⊕s∈ZkV[n]s(Dr)us RU(M) = M s∈Zk Us(M)us. Both have a natural structure of mutigraded left RV(D)-module and we shall consider RU(M) as a sub-group of M[u±1], the Laurent polynomials with coefficients in M. As Nis a left submodule of Dr, we can also consider on Nthe induced V-filtration: Vs(N) := N ∩ Vs[n](Dr) 4 for each s∈Zkas defined before (see 2.1). The abelian group RV(N) := M s∈Zk Vs(N)us is in fact an homogeneous submodule of RV[n](Dr). The RV(D)–module RU(M) is naturally isomorphic to the quotient RV[n](Dr)/RV(N). Definition 2.4 We call the module RU(M)the Rees module associated with the filtration U•(M). Let us now consider the filtration VΓ •(D) as in 2.1. Then the Rees ring associated with this filtration is defined in a similar way as RΓ(D) = M s∈Zk VΓ s(D)us⊂ D[u±1] the product being induced by the one of D[u±1]. If M=Dr Nis a finitely generated D-module and U•(M) is a good filtration on Mwith respect to V•(D) then, for any cone Γ as in 2.1, the abelian group RΓ(M) = M s∈Zk UΓ s(M)us⊂ M[u±1] is a left graded RΓ(D)-module. Definition 2.5 We call the module RΓ(M)the Rees module associated with the filtration U•(M)and the cone Γ. The fact that a filtration is good in the sense of previous section 2.1 is then equivalent to the fact that the Rees module of Mis finitely generated over RV(D). Similarly the goodness of UΓ •(M) is equivalent to the fact that the Rees module RΓ(M) as defined above, is finitely generated over RΓ(D). Since RΓDis a Noetherian ring, this implies in particular an Artin type property: If M′⊂ M is a submodule of a D-module Mendowed with a good filtration U•(M) then the induced filtration on M′, namely UΓ •(M)∩ M′is good. The ring RΓ(D) contains as a subring the C–algebra AΓ=C[ˇ Γ∩Zk]⊂C[u, u−1], all these rings being included in D[u, u−1]. Theorem 2.6 There is a fan Ein (Qk)∗ +such that for each cone Γ∈ E the Rees module RΓ(M)is flat over AΓ. We will prove a more precise form of this result we in section 4, see theorem 4.3, after having recalled in section 3 the notion of an analytic standard fan. 2.2.2 The fiber at 0 Let us denote by mthe ideal of RV(D) generated by (u1,...,uk). This is a two sided ideal since it is generated by central elements. The fiber at zero of RV(D) (resp. of RV[n](Dr)) is by definition the quotient ring (resp. the quotient module), RV(D) m(resp. RV[n](Dr) m.RV[n](Dr)). More generally the fiber at zero of RU(M) = RV[n](Dr) RV(N)is the quotient RU(M) mRU(M) which is naturally isomorphic to the quotient RV[n](Dr) RV(N) + m.RV[n](Dr). 5 Notice that the fiber at 0 of the module RU(M) can be zero for a non-zero RU(M), as shown in the following example. Let us denote by Ithe principal ideal of Dgenerated by the differential operator P= 1 + x2 1∂1and let us suppose k≥1. Then RV(I) is the principal ideal of RV(D) generated by P u0 1= 1 + (x2 1∂1u−1 1)u1and the fiber at zero of RV(D/I) is then zero since RV(I) + m.RV(D) contains P−(x2 1∂1u−1 1)u1= 1, so that it is equal to RV(D). Similarly for any k-dimensional cone Γ the C–algebra AΓ=C[ˇ Γ∩Zk] has a maximal ideal mΓ=C[ˇ Γ∩Zk\ {0}] because ˇ Γ is strictly convex. Then we define the fiber at zero of the Rees module RΓ(M) as RΓ(M) mΓRΓ(M).As is explained in [9], this fiber is a module over the ring RΓ(D) mΓRΓ(D)≃grΓD. 2.2.3 Description of Rees rings RV(D)and RΓ(D) It is useful to describe the Rees ring RV=RV(D) (resp. RΓ(D)) as subrings of the ring of relative differential operators DCn×Ck/Ck(resp DCn×SΓ/SΓ). This is nothing but an explicit version of the interpretation of a Rees ring as a ring of relative differential operators on the deformation of Y×0Γto its normal cone in Cn×SΓ,see [9]. We denote by SΓthe algebraic variety associated to the ring AΓ. Let us define first A=C{X′U, X′′}[X′, U, ∆] = C{X1U1,...,XkUk, Xk+1,...,Xn}[X′, U, ∆] where U= (U1,...,Uk), X′= (X1,...,Xk), X′′ = (Xk+1,...,Xn), X= (X1, . . . , Xn), ∆ = (∆1,...,∆n) are new variables satisfying the following relations, for i= 1,...,n: ∆iXi=Xi∆i+ 1 the other relations being trivial. The ring Ais graded with Zkas a values group (we will say that Ais a Zk–graded ring) , namely we have A=M s∈Zk As where Asis the set As={X α,β∈Nn;σ∈Nk fαβσXα∆βUσ∈ A | fαβσ ∈C{X′U, X′′}, σi+βi−αi=si;i= 1,...,k}. Proposition 2.7 There is an isomorphism of Zk–graded rings i=iV:RV(D)→ A defined by: •i(uj) = Ujfor j= 1,...,k. •i(xju−1 j) = Xjfor j= 1, . . . , k and i(xj) = Xjfor j=k+ 1,...,n. •i(∂juj) = ∆jfor j= 1,...,k and i(∂j) = ∆jfor j=k+ 1,...,n. 6 Proof.It is clear that iis injective and that, by the formula i(xα∂βus) = Xα∆β(Πk i=1Usi+αi−βi i), i(Vs(D)us) = Asfor all s∈Zk. Let us give now the same description for the ring RΓ(D). We shall do it in the only case of interest for us, when the cone Γ is basic, which means that if it is defined by kindependent linear forms {L1,...,Lk}generating the lattice Zk. For the sake of simplicity let us suppose k=n. In this situation we are going to make a change of variables in order to write AΓas a polynomial ring. For any iwe can write Li= (ℓi1,...,ℓik) and we can suppose by a suitable ordering of the linear forms Li that the determinant of the matrix L= (ℓij) equals 1. Let us write L′for the inverse matrix of L, then the columns {C1,...,Ck}of L′form a basis of the dual cone ˇ Γ. The family uC1,...,uCkgenerates the C–algebra AΓ=C[ˇ Γ∩Zk]. Here we write uCj=uc1j 1···uckj kwhere the cij’s are the entries of the column Cj. We will write Wi=UCifor i= 1,...,k. Similarly, we have Uj=WCj(L)for j= 1,...,k where Cj(L) is the j−th column of the matrix L. In section 4 we will only use the column Cj=Cj(L′). Let us remark here that if k < n then we can still define W, the relationship between Wand Ubeing exactly the same. Proposition 2.8 The ring RΓ(D)is isomorphic to the subring of A′=C{X, W}[∆], which contains all polynomials and in which only convergent power series with respect to the family of monomials XiUi=XiWCi(L) for i= 1,···k, and Xk+1,··· , Xn, in the general case k≤nare allowed. Proof.A monomial Xα∆βUs−β+αcan be written in terms of X, ∆, W as Xα∆βWL(s−β+α).All that remains to be done is to enumerate the monomials derived from the power series variables x1,··· , xn The ring AΓis identified to the subring C[W](= C[W1,...,Wk]) ⊂ RΓ(D) and this inclusion is flat. We shall prove this statement in 4.2. The fiber of RΓ(D) at the origin is by definition RΓ(D)⊗C[W] C[W] (W)which is isomorphic to a Weyl algebra, namely the Weyl algebra C[X, ∆]. This Weyl algebra is endowed with a Zk–graduation by weight(Xi) = −ǫi and weight(∆i) = ǫifor i= 1, . . . , k where ǫiis the vector in Zkwhose j−th coordinate is δij. 2.3 Multifiltrations and Bernstein-Sato functional equations Let us detail the functional equation problem raised in the introduction. It has already been remarked that it gives rise to a situation where a multifiltration along transverse hypersurfaces comes out in a natural way. These equations are of the type P(λ)fλ=b(λ)fλ1+1 1···fλk+1 k where λ= (λ1,··· , λk) is a k-uple of indeterminates, and f= (f1,··· , fk) is a k-uple of analytic functions on X. They are naturally written in the module OXλ1,··· , λk,1 f1···fkfλ, endowed with a structure of a DX[λ]- module, which can be extended in a way discovered by B. Malgrange to a DX×Ck-module structure with an action of the 2kvariables tj, ∂tjsuch that λj=−∂tjtj.The module Mthat we have then to consider is the module generated over the ring DX×Ckby fλ, with its naturally defined multifiltration V•(DX×Ck)·fλalong the hypersurfaces t1= 0,··· , tk= 0. The proof of Sabbah in [9] can be sketched as follows: let us define for any linear form L∈(Qk)⋆with positive rational coefficients ℓj≥0, a filtration of UL •(M), associated with the linear combination PℓjV(j)of the basic V- filtrations (see notations at 2). Precisely we set (see also the notation 3.1 below): VL •(DX×Ck) = {P=X µ,ν∈Nk Pµν(x, ∂x)tµ∂ν t,Xℓi(νi−µi)≤ •} UL •(M) = VL •(DX×Ck)·fλ Then the existence of a functional equation, in which b(λ) is a product of affine forms L(λ) + ccomes out from the following three steps: 7 1) There are Bernstein-Sato polynomials bL(λ), relative to each L, with functional equations bL(λ)fλ∈VL <0(DX×Ck)·fλ. 2) The saturated filtration Us(M) = \ L∈(Qk)⋆ UL L(s)(M) can be defined by using only a finite number of fixed linear forms. 3) The saturated filtration is good which is equivalent to the existence of a k-uple of integers κ, such that ∀s∈Nk, Us(M)⊂Us(M)⊂Us+κ(M). It is step 2) which, in the proof in [9], makes an essential use of the notion of an adapted fan, whose existence is proved in this paper, see theorem 2.6. In Bahloul’s paper [4], this step is made by constructive methods which do not use the flatness property. 3 The analytic standard fan In this section we will summarize the main results of [3]. Since these results are only given for a module D/I over Dwhich is a quotient by an ideal we will then show briefly how to adapt them to a module of the type Dr N. Let us remark that the fan used in this paper is obtained by a restriction of the fan in [3] to a subset of linear forms of the set Udefined below, namely the linear combinations of the filtrations V(i). Let Ube the set of linear forms Λ : R2n→R, Λ(α, β) = Pn i=1 eiαi+Pn i=1 fiβiwith ei+fi≥0 and ei≤0 for i= 1,...,n. If P=X αβ pαβxα∂β is an element in Dwe define ordΛ(P) –the Λ-order of P– to be the maximal value of Λ(α, β) for α, β such that pαβ 6= 0. The Λ-filtration FΛ,•(D) is defined by FΛ,ℓ =FΛ,ℓ(D) = {P∈ D | ordΛ(P)≤ℓ} for any ℓ∈R. We will write FΛ,<ℓ(D) := {P∈ D | ordΛ(P)< ℓ}. If ei= 0, fi= 1 for i= 1,...,n the corresponding Λ–filtration is nothing but the usual filtration by the order of differential operators. We shall denote it simply F•(D). Let us recall that we have fixed k≤n. For each linear form L∈(Qk)∗ +(i.e. the coefficients of Lare non-negative) we denote by e Lthe linear form on R2ndefined by e L(α, β) = L(β)−L(α) where L(α1,...,αn) = L(α1,...,αk). Notation 3.1 We define the filtration VL •on D, indexed by the set of values L(Zk), as: VL ℓ(D) = Fe L,ℓ(D) = {P∈ D | orde L(P)≤ℓ} Let us fix L∈(Qk)∗ +. The graded ring associated to the filtration VL=Fe Lon D, is by definition gre L(D) = M ℓ∈e L(Z2n) Fe L,ℓ Fe L,<ℓ . If no confusion is possible we shall write simply ordLand grL(D) instead of orde Land gre L(D). The graded ring grL(D) is a ring of differential operators and its structure is the following: suppose the coefficients of the form Lare (e1,...,ek) and suppose we also have ordered the variables to have ei>0 for 1≤i≤ℓfor some ℓ≤k. 8 Then the graded ring grL(D) is isomorphic to the ring C{xℓ+1,...,xn}[x1,...,xℓ, ∂1,...,∂n]. In this ring the graduation is induced by the weights weight(xi) = −ei,weight(∂i) = eifor 1 ≤i≤ℓand weight(xj) = weight(∂j) = 0 otherwise. There is only a finite number of types of these rings, one for each partition of {1,...,k}into two sets. Notation 3.2 For each P∈ D and for each d∈L(Zk)with ordL(P)≤dwe denote by σL d(P)–the L–symbol of Pof order d– the class of Pin Fe L,d/Fe L,<d. The principal symbol of Pis by definition σL(P) = σL d(P)if d= ordL(P). For P, Q ∈ D we have σL(PQ) = σL(P)σL(Q). For any left ideal Iin Dwe denote by grL(I) the graded ideal of grL(D)generated by the set {σL(P)|P∈I}. We set σL(0) = 0. We denote by D[t] the C-algebra O[∂, t] = C{x1,...,xn}[∂1,...,∂n, t] with relations (tbeing a new variable) •[t, a] = [t, ∂i] = [a, b] = [∂i, ∂j] = 0, •[∂i, a] = ∂a ∂xit, for a, b ∈ O and i= 1,...,n. The ring D[t] is isomorphic to the Rees ring associated with the order filtration F•on D. Since this Rees ring is by definition RF(D) = M ℓ∈Z Fℓ(D)vℓ⊂ D[v] = D ⊗CC[v] for a new variable v, we can define an isomorphism of graded rings ι:D[t]→ RF(D) by ι(a) = a=av0, ι(∂j) = ∂jv, and ι(t) = 1v=v. The natural graded structure of RF(D) can be translated on D[t]. An homogeneous element of degree d∈Z in D[t] is nothing but an expression X ℓ+|β|=d aℓ β∂βtℓ for some aℓ β ∈ O. For P=Pβpβ(x)∂β∈ D, the element Pvord(P)∈ RF(D) is called the homogenization of P, where ord(P) is the usual order of P. It is useful to see P vord(P)as an element of D[t]. The homogenization of Pis then denoted by h(P) and we have h(P) = X β pβ(x)∂βtd−|β| for d= ord(P) the usual order of P. For each linear form L∈(Qk)∗ +we can define in a natural way a filtration VL •(D[t]) on D[t], the L–order –denoted ordL(R)– of an element R=X ℓαβ rℓαβxα∂βtℓ being the maximal value of L(β)−L(α) for rℓαβ 6= 0. The associated graded ring is denoted by grL(D[t]). We have a natural ring isomorphism from grL(D[t]) onto grL(D)[t], where with the notations as before we have [∂i, xj] = δijt,δij being the Kronecker symbol. For each left ideal Iin Dwe denote by h(I) the left homogeneous ideal of D[t] generated by {h(P)|P∈I}. As in the case of D, we denote by grL(h(I)) the homogeneous ideal –in grL(D[t])– generated by the set of principal symbols σL(G) of the elements Gin h(I). The main result of [3] is the following Theorem 3.3 Let Ibe a non-zero left ideal of Dand let h(I)be the associated homogenized ideal in D[t]. Then there exists a partition Eof Uinto convex rational polyhedral cones such that for any Γ∈ E the ideals grΛ(h(I)) and grΛ(I)do not depend on Λ∈Γ. 9