Explicit calculations in rings of differential operators

S´eminaires & Congr`es 8, 2004, p. 89–128 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS by Francisco J. Castro-Jim´enez & Michel Granger Abstract. — We use the notion of a standard basis to study algebras of linear differential operators and finite type modules over these algebras. We consider the polynomial and the holomorphic cases as well as the formal case. Our aim is to demonstrate how to calculate classical invariants of germs of coherent (left) modules over the sheaf Dof linear differential operators over Cn. The main invariants we deal with are: the characteristic variety, its dimension and the multiplicity of this variety at a point of the cotangent space. In the final chapter we shall study more refined invariants of D-modules linked to the question of irregularity: The slopes of a D-module along a smooth hypersurface of the base space. Résumé (Calculs explicites dans l’anneau des opérateurs différentiels). — Dans ce cours on d´eveloppe la notion de base standard, en vue d’´etudier les alg`ebres d’op´erateurs diff´erentiels lin´eaires et les modules de type fini sur ces alg`ebres. On consid`ere le cas des coefficients polynomiaux, des coefficients holomorphes ainsi que le cas des alg`ebres d’op´erateurs `a coefficients formels. Notre but est de montrer comment les bases standards permettent de calculer certains invariants classiques des germes de modules (`a gauche) coh´erents sur le faisceaux Ddes op´erateurs diff´erentiels lin´eaires sur Cn. Les principaux invariants que nous examinons sont : la vari´et´e caract´eristique, sa dimension et sa multiplicit´e en un point du fibr´e cotangent. Dans le dernier chapitre nous ´etudions des invariants plus fins des D-modules qui sont reli´es aux questions d’irr´egularit´e : les pentes d’un D-module, le long d’une hypersurface lisse. 2000 Mathematics Subject Classification. — 13N10, 13P10, 16S32. Key words and phrases. — D-modules, Gr¨ obner basis, slopes. F.C.: Partially supported by DGESIC-PB97-0723; BFM2001-3164 and FQM-218. Both authors partially supported by Picasso-HF2000-0044. c S´eminaires et Congr`es 8, SMF 2004 90 F.J. CASTRO-JIM´ ENEZ & M. GRANGER Introduction The purpose of these notes is to make an account of explicit methods, using the notion of a standard basis, which could be used in studying algebras of linear differential operators and finite type modules over these algebras. We consider in parallel each of the following cases: coefficients in a ring of polynomials k[x1, . . . , xn] for the Weyl algebra An(k), in the ring of germs of holomorphic functions at 0 ∈Cnfor Dn, or in the ring of formal power series for c Dn. We denote Rany of these rings of operators and Bthe corresponding commutative ring of coefficients. Our aim is to demonstrate how to calculate classical invariants of germs of coherent (left) modules over the sheaf Dof linear differential operators over Cn. In practice we shall look at finite type modules over Dnor c Dn. The main invariants we are dealing with are: the characteristic variety, and the multiplicity of this variety at a point of the cotangent space. See [25] and [19] for an introduction to the theory of D-modules and for the definition of the characteristic variety, of its dimension and and of its multiplicity. In the last chapter we shall study more refined invariants of R-modules linked to the question of irregularity: The slopes of a Dn-module or an An(k)-module along a smooth hypersurface of the base space. In these notes we deal mainly with the case of monogenic modules M=R/I with Ia (left) ideal of R. We provide an algorithm to build standard bases of Iand in the context of chapter II these bases yield a special kind of system of generators for which the module of relations is easy to describe. There is a straightforward generalisation for the case M=Rp/Ninvolving a submodule Nof Rp. Then continuing the process of building standard bases for submodules we can thus obtain a (locally) free resolution of M. The techniques used are the notion of privileged exponents with respect to an ordering and a theorem of division. They were introduced by H.Hironaka (cf. [26] or [1]). In the polynomial case the notion of a standard basis was developed by Buchberger under the name of a Gr¨ obner basis in [13] where he also gives an algorithm for its calculation. The commutative case is treated in chapter I, where we recall the notions of a privileged exponent of a polynomial or a power series with respect to a convenient ordering, the definition of a standard basis and the algorithm for calculating it, which is the Buchberger’s algorithm in the polynomial case. We also draw attention to the elegant proof in the convergent case taken from Hauser and Muller (cf. [20].) We finish by giving some applications in commutative algebra such as calculating multiplicities, syzygies, and the intersections of ideals. In chapter II, we consider division processes in algebras of operators which are compatible with a filtration which may either be the filtration by the order of operators or in the particular case of An(k), the Bernstein filtration by the total order. At the same time, for the sake of completeness we treat a weighted homogeneous version of these filtrations. Using a compatible ordering on monomials we again develop a division algorithm and an algorithm for the construction of a standard basis. These S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 91 algorithms are very similar to those developed in chapter I, since in fact a division by a family of operators {P1, . . . , Pr}, or by a standard basis of an ideal Iinduces the same object via the principal symbols in the commutative associated graded rings. The references for these results are [11] and [14]. Let us also notice that it is only in the case of k[x1, . . . , xr] or An(k) that the suitable orderings used in chapters I and II are well orderings and therefore that the algorithms are effective. In the power series case they depend on formal or convergent processes in the local rings of series. In chapter III we give an algorithm for the calculation of the slopes of a coherent R-module along a smooth hypersurface Yof knor Cnin the neighbourhood of a point of Y. The material is essentially taken from our work with A.Assi [2] where however only the case of An(k) is considered. The notion of a slope of a coherent D-module Mwas introduced by Y. Laurent under the name of a critical index. He considers, in the more general context of microdifferential operators a family of filtrations Lr=pF +qV (with ra rational number such that 0 ⩽r=p/q ⩽+∞), which is an interpolation between the filtration by the order Fand the V-filtration of Malgrange and Kashiwara (cf. [22]). The critical indices are those for which the Lr-characteristic variety of Mis not bihomogeneous with respect to Fand V. Laurent proved in loc. cit. the finiteness of the number of slopes and then C.Sabbah and F. Castro proved the same result in [30] by using a local flattener. In [28] Z. Mebkhout introduced the notion of a transcendental slope of a holonomic D-module M, as being a jump in the Gevrey filtration Irr(r) Y(M) of the irregularity sheaf IrrY(M). The irregularity sheaf is the complex of solutions of M with values in the quotient of the formal completion along Yof the structural sheaf O, by Oitself. By the main result of [28], it is a perverse sheaf, and Irr(r) Y(M) is the subperverse sheaf of solutions in formal series of Gevrey type ralong Y. In [23] Laurent and Z.Mebkhout proved that the transcendental slopes of an holonomic D-module are equal to the slopes in the sense of Laurent called algebraic slopes. The analogue in dimension one is Malgrange’s paper [27] for the perversity of the irregularity sheaf and Ramis’s paper [29] for the theorem of the comparison of slopes. In chapter III, we recall the principle of the algorithm of calculation of the algebraic slopes of an R-module that we developed in [2] and we give some supplementary information. Here the additional difficulty is that the linear form Lrwhich yields the similarly called filtration now possesses a negative coefficient in the variable x1. Although we can still speak of privileged exponents and standard bases, the standard bases are no longer systems of generators of the ideal Iwhich we consider but only induce a standard basis of the graded associated ideal. A more serious consequence of non-positivity, is that the straightforward division algorithm does not work inside finite order operators. The way to solve this problem is to homogenize the operators in R[t] with respect to the order filtration or, in the case of An(k), with respect to the Bernstein filtration. We notice in chapter III, following a remark made by L.Narv´aez SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 92 F.J. CASTRO-JIM´ ENEZ & M. GRANGER [16] that we can simplify the original proof in [2] by considering on An[t] a different structure as a Rees ring. Another improvement to [2] lies in the distinction between the slopes in the sense of Laurent and the values of rfor which the ideal Igives a non-bihomogeneous graded ideal grLr(I). We call those r, the idealistic slopes of I. In [2] we considered only this set of slopes and proved its finiteness; this paper however already contains the hard part of the algorithm of the calculation of algebraic slopes. Let us end this introduction by pointing out two other extensions of the original material of our paper [2]. First we make the same algorithm work for the rings of operators Dn, or c Dn. Secondly we give some significant examples of the calculations of slopes: the slopes of the direct image of DCe1/xkby an immersion in C2, with respect to a smooth curve Ytangent to the support. This example contains idealistic slopes which end up not being algebraic slopes. Finally, we calculate the slopes of DC2e1/(yp−xq)along any line through the origin. Added on March 21, 2003. — This paper was written in September 1996, as material for a six hour course given in the CIMPA summer school “Differential Systems” (Sevilla, September 1996). Consequently, the bibliography is outdated. Since then, many papers have been published about the computational aspects in D-modules theory. We have therefore decided to add, after the references, a complementary list of recent publications on the subject. 1. Division theorems in polynomial rings and in power series rings 1.1. Let kbe a field, with an arbitrary characteristic unless otherwise stated. Let n be a positive integer . We denote by: •k[X] = k[X1, . . . , Xn] the ring of polynomials with coefficients in kand variables X1, . . . , Xn. •k[[X]] = k[[X1, . . . , Xn]] the ring of formal power series with coefficients in k and variables X1, . . . , Xn. •k{X}=k{X1, . . . , Xn}the ring of convergent power series with coefficients in kand variables X1, . . . , Xn, if k=Ror C.(1) If f∈k[[X]], f6= 0, we write f=Pα∈NnfαXαwhere fα∈k. If f∈k[X]f6= 0, then this sum is finite. The set N(f) = {α∈Nn|fα6= 0}is called the Newton diagram of the power series or of the polynomial f. 1.2. L-degree and L-valuation. — Let L:Qn→Qbe a linear form with non negative coefficients. Definition 1.2.1. — Let 0 6=f∈k[X]. We define the L-degree of f(and we denote it by degL(f)) as being max{L(α)|fα6= 0}. We set degL(0) = −∞. (1)Or, more generally, a complete valued field. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 93 Definition 1.2.2. — Let 0 6=f∈k[[X]]. We define the L-valuation of f(which we denote by valL(f)) as being min{L(α)|fα6= 0}. We set valL(0) = +∞. We have degL(fg) = degL(f) + degL(g) if f, g ∈k[X] and valL(fg) = valL(f) + valL(g) if f, g ∈k[[X]]. Definition 1.2.3. — Let 0 6=f∈k[[X]]. We call the sum inL(f) = PL(α)=valL(f)fαXα the Linitial form of the power series f(2). Let Ibe an ideal of k[[X]]. We call the ideal of k[[X]] generated by {inL(f)|f∈I}, the initial ideal of I. We denote it by InL(I) (or simply In(I)) Notation. — The following notation will be useful. If f=PαfαXαis a power series, we set inL,ν(f) = PL(α)=νfαXα. When no confusion can occur, we write inν(f) instead of inL,ν(f). We have: f=Pνinν(f). Definition 1.2.4. — Let 0 6=f∈k[X]. We call the sum finL(f) = PL(α)=degL(f)fαXα the L-final form of the polynomial f. Let Ibe an ideal of k[X]. We call the ideal of k[X] generated by {finL(f)|f∈I}the final ideal of I. We denote it by FinL(I) (or simply by Fin(I)). 1.3. Orderings in Nn. — Let <be a total well ordering on Nncompatible with sums (i.e. if α, β ∈Nnand α < β then we have α+γ < β +γfor any γ∈Nn). Let L:Qn→Qbe a linear form with non negative coefficients . The relation <L, defined by: α <Lβif and only if L(α)< L(β) or L(α) = L(β) and α < β is a total well ordering on Nncompatible with sums. 1.4. The privileged exponent of a polynomial or of a power series. — The notion of the privileged exponent of a power series is due to H.Hironaka. It was introduced in [26] (see also [1], [10]). We fix, once and for all, a total well ordering <, compatible with sums, in Nn. Let L:Qn→Qbe a linear form as above. Definition 1.4.1. — Let f=PαfαXα∈k[X], f 6= 0. We call: •The n-uple expL(f) = max<L{α|fα6= 0}, the L-privileged exponent of f •The monomial mpL=fexpL(f)XexpL(f), the L-privileged monomial of f Let f=PαfαXα∈k[[X]], f 6= 0. We call: •The n-uple expL(f) = min<L{α|fα6= 0}, the L-privileged exponent of f. •The monomial mpL=fexpL(f)XexpL(f), the L-privileged monomial of f. (2)If all the coefficients of Lare positive, then the initial form of a power series is a polynomial. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 94 F.J. CASTRO-JIM´ ENEZ & M. GRANGER When it becomes necessary, we shall use the more precise notation, exp<L(f) = expL(f) and mp<L(f) = mpL(f). In all the cases, when no confusion can result, we shall write exp(f) instead of expL(f) and mp(f) instead of mpL(f). Note 1.4.2. — When f∈k[X], f 6= 0, we shall take care not to confuse the privileged exponent of the polynomial fwith the privileged exponent of the power series f, in spite of the notation. If necessary, we shall use the notation expp(f) for the privileged exponent of the polynomial fand exps(f) for the privileged exponent of the power series f. Proposition 1.4.3. — Let f, g ∈k[X](resp. f, g ∈k[[X]]) be non zero elements. We have: (1) exp(fg) = exp(f) + exp(g). (2) mp(fg) = mp(f) mp(g). (3) If exp(f)6= exp(g)then exp(f+g) = max <L{exp(f),exp(g)}(resp. exp(f+g) = min <L{exp(f),exp(g)}). Let Ibe a non zero ideal of k[X] (resp. k[[X]]). We denote E<L(I) = {expL(f)|f∈Ir{0}}. When no confusion can result, we write E(I) instead of E<L(I). Because of 1.4.3, we have E(I) + Nn= E(I). We denote by mp(I), the ideal of k[X] generated by the family of monomials {mp(f)|f∈I}(3). Proposition 1.4.4. — Let Ibe a non zero ideal of k[X](resp. k[[X]]). Then we have: E(I) = E(mp(I)) = E(Fin(I)) (resp. E(I) = E(mp(I)) = E(In(I))). Proof. — By definition, for every non zero polynomial f, we have exp(f) = exp(fin(f)) and exp(f) = exp(mp(f)) (see 1.4.1). If fis a non zero power series, then we have: exp(f) = exp(in(f)) and exp(f) = exp(mp(f)) (see 1.4.1). Note 1.4.5. — With the notations of 1.4.2, if fis a power series such that in(f) is a polynomial, (this condition is verified if every coefficient in the linear form Lis positive) then we have, in general, exp(f)6= expp(in(f)). Assume that every coefficient in the linear form Lis positive (we then just say that Lis a positive linear form). Consider the ordering CLdefined on Nnby the formula: αCLβif and only if L(α)< L(β) or L(α) = L(β) and β < α (3)This is a monomial ideal, which means that a polynomial fis an element of the ideal if and only if every monomial of fis in the ideal. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 95 This is a total well ordering (4) on Nncompatible with the sum. If fis a power series, then we have: exp<L(f) = exps<L(inL(f)) = exppCL(inL(f)). Proposition 1.4.6. — Let E⊂Nnsuch that E+Nn=E. Then Econtains a finite family of generators; In other words, there exists a finite family F⊂Esuch that E=∪α∈F(α+Nn). Proof. — This is a version of Dickson’s lemma. The proof is by induction on n. For n= 1 a (finite) family of generators is given by the smallest element of E(for the usual ordering in N). Assume that n > 1 and that the result is true for n−1. Let E⊂Nnbe such that E+Nn=Nn. We can assume that Eis non empty. Let α∈E. For any i= 1, . . . , n and j= 0, . . . , αiwe consider the bijective mapping φi,j :Ni−1×{j}×Nn−i−→ Nn−1 (β1, . . . , βi−1, j, γi+1, . . . , γn)7−→ (β1, . . . , βi−1, γi+1, . . . , γn) and we denote Ei,j =φi,j(E∩(Ni−1×{j}×Nn−i)). It is clear that Ei,j +Nn−1=Ei,j and by the induction hypothesis there is a finite subset Fi,j ⊂Ei,j generating Ei,j. The family F={α}∪∪i,j(φi,j)−1(Fi,j)generates E. The proof above is taken from [18]. Remark. — The previous proposition can be rephrased as follows: Any monomial ideal in k[X]is finitely generated. This is a particular case of the Hilbert basis theorem. In the same way we can see that any increasing sequence Ekof subsets of Nn, stable under the action of Nn, is stationary. We shall often use this property called the Noetherian property for Nn. We can adapt the proof above to show that, given E⊂Nnas in the proposition, we can find in any set of generators, a finite subset of generators of E. This proves in particular that in any system of generators made of monomials of a monomial ideal of k[X], we can find a finite subset of generators. This is Dickson’s lemma. Definition 1.4.7. — Let Ibe a non zero ideal of k[X] (resp. k[[X]]). A standard basis(5) of I, relative to L(or L-standard basis of I) is any family f1, . . . , fmof elements in Isuch that E(I) = ∪m i=1(expL(fi) + Nn). Remark. — There always exist a standard basis for I, because of the definition of E(I) and 1.4.6. (4)If the form Lhas at least one non positive coefficient the previous formula defines a total ordering over Nn, but not a well ordering. (5)The notion of a standard basis, introduced by H. Hironaka in [21], is similar to the notion of a Gr¨ obner basis, introduced by Buchberger in [13]. We shall come back to this analogy later. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 96 F.J. CASTRO-JIM´ ENEZ & M. GRANGER 1.5. Here are the divisions. — We shall prove here that a standard basis of an ideal Iis a system of generators of this ideal. With any m-uple (α1, . . . , αm) of elements of Nnwe shall associate a partition(6) ∆1, . . . , ∆m,∆ of Nnin the following way. We set: ∆1=α1+Nn,∆i+1 = (αi+1 +Nn)r(∆1∪···∪∆i) if i⩾1, ∆ = Nnr(∪m i=1∆i) Theorem 1.5.1. — Let (f1, . . . , fm)be an m-uple of non zero elements of k[[X]] (resp. of k[X]). We denote by ∆1, . . . , ∆m,∆the partition of Nnassociated with (exp(f1), . . . , exp(fm)). Then, for any fin k[[X]] (resp. in k[X]) there exists a unique (m+ 1)-uple (q1, . . . , qm, r)of elements of k[[X]] (resp. of k[X]) such that: 1) f=q1f1+···+qmfm+r, 2) exp(fi) + N(qi)⊂∆i, i = 1, . . . , m, 3) N(r)⊂∆. If kis either Ror Cand if the fiare convergent power series, then for any convergent power series fthe series qiand rare convergent. Remark. — The element qiin the theorem is called the i-th quotient and ris called the remainder of the division of fby (f1, . . . , fm). We shall denote the remainder by r(f;f1, . . . , fm). Of course, the quotients as well as the remainder depend on the well ordering <L. Proof of theorem 1.5.1. — Assume that two (m+ 1)-uples, (q1, . . . , qm, r) and (q0 1, . . . , q0 m, r0), satisfy the conditions of the theorem. We have: (1) m X i=1 (qi−q0 i)fi+r−r0= 0 If qi6=q0 ithen exp((qi−q0 i)fi)∈∆i. If r6=r0then exp(r−r0)∈∆. Since ∆1, . . . , ∆m,∆ is a partition of Nn, the equality (1) is only possible if qi=q0 ifor any iand if r=r0. This proves the uniqueness in the theorem. We shall now prove the existence. Let us first consider the polynomial case. Since the set Nnis well ordered with respect to <L, we use an induction on unitary monomials of k[X]. If Xα= 1 (i.e. if α= (0, . . . , 0)), then either exp(fi)6= (0, . . . , 0) for any iand in this case it is enough to write 1 = Pm i=1 0fi+ 1, or there exists an integer jsuch that exp(fj) = (0, . . . , 0). In this case fjis a non zero constant.(7) Assume that jis minimal. We write 1 = Pi6=j0·fi+ (1/fj)fj+ 0. This proves the result at the first step of the induction. Assume that the result is proved for any βsuch that β <Lα. Let jbe such that α∈∆j. If there is no such jwe write Xα=Pm i=1 0fi+Xα. If (6)We use the word partition in a broad sense, which means that an element of the family may be empty. (7)We use here the fact that for the well ordering <L, (0,...,0) is the first element of Nn. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 97 jexists, let γ∈Nnbe such that α= exp(fj) + γ. We can write, Xα=1 cjXγfj+gj where cjis the coefficient of the privileged monomial of fjand all the monomials in gjare smaller (with respect to <L) than α. By the induction hypothesis there exists (q0 1, . . . , q0 m, r0) satisfying the conditions of the theorem for f=gj. In particular we have: Xα=X i6=j q0 ifi+1 cj Xγ+q0 jfj+r0. This proves the result for α. Thus, existence is proved for the polynomials. We say that a polynomial gis L-homogeneous if all its monomials have the same L-degree. It is clear in the proof above that if fis L-homogeneous of L-degree d∈Qand if fiis L-homogeneous of L-degree di∈Q(for any i) then the quotient qi, if it is non zero is L-homogeneous of L-degree d−di, and the remainder r, if it is non zero is L-homogeneous of L-degree d. Assume now that fis a power series. Let us now see the existence in that case, first assuming that Lis a positive linear form (see 1.4.5). Any non zero power series f=PαfαXαcan be represented, in a unique way, as a sum f=Pν∈L(N2)fνwhere fν=PL(α)=νfαXαis a L-homogeneous polynomial. By definition (see 1.2.2) we have: valL(f) = min{ν|fν6= 0}. Because of 1.4.5 we have, for any i: exp(fi) = exppCL(in(fi)) and we can apply the division, in the polynomial case, of in(f) by (in(f1), . . . , in(fm)). There exists a (unique) (m+ 1)-uple (σ1, . . . , σm, ρ) such that in(f) = m X i=1 σiin(fi) + ρ and satisfying the conditions similar to 2) and 3) in the theorem. The following notations will be useful: σi(f) = σi,ρ(f) = ρand for any power series g,bg=g−in(g). We have: f= in(f) + b f= m X i=1 σi(f)fi+ρ(f) + b f− m X i=1 σi(f)b fi We introduce the following notation: s0(f) = f, s(f) = s1(f) = b f− m X i=1 σi(f)b fi, sj(f) = s(sj−1(f)). We have: •valL(sj+1(f)) >valL(sj(f)) for any j. •degL(σi(sj+1(f))) >degL(σi(sj(f))) for any iand any j. •degL(ρ(sj+1(f))) >degL(ρ(sj(f))) for any iand any j. •For any i, the series X j⩾0 σi(sj(f)) SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 104 F.J. CASTRO-JIM´ ENEZ & M. GRANGER we have: FHSA(k) = dimkA Amk+1 = #{α∈(NnrE(I)) | |α|⩽k}. Proof. — Let us consider the formal power series case, the convergent case being similar. We have a natural isomorphism of vector spaces A/Amk+1 ≃k[[X]]/(I+mk+1). For the ordering <Lwe have the equality E(I+mk+1) = E(I)∪E(mk+1). Indeed, it is enough to prove the inclusion E(I+mk+1)⊂E(I)∪E(mk+1), the other being obvious. Let f∈Iand g∈mk+1. If val(f)<val(g) then in(f+g) = in(f) and thus exp(f+g) = exp(f)∈E(I). If val(f)⩾val(g) then val(f+g)⩾min{val(f),val(g)}⩾ val(g)⩾k+ 1. Whence f+g∈mk+1. We end the proof of the proposition by applying 1.5.3. Let us denote by ℘the set of the subsets {1, . . . , n}. We introduce the following notations: •For each σ∈℘we write: –S(σ) = {α∈Nn|αi= 0 if i∈σ} –T(σ) = S({1, . . . , n}rσ) – #σ= cardinal of σ •For each non-empty subset E⊂Nnsuch that E+Nn=E: –cd(E) = min{#σ|S(σ)∩E=∅} –d(E) = n−cd(E) Proposition 1.9.4. — Let ∅6=E⊂Nnbe such that E+Nn=E. Let σ∈℘be such that #σ=cd(E). Then the set {α∈T(σ)|(α+S(σ)) ∩E=∅} is finite. Proof. — We remark that the set defined in the proposition is the complement of p(E) in T(σ), pbeing the natural projection of Nnonto T(σ). Since p(E) is stable by addition in T(σ), this complement could only be infinite if it contained a coordinate axis in T(σ), which would contradict the minimality of the cardinal of σ. Let us denote by eσ(E) the cardinal of the set defined in the previous proposition and by e(E) the sum e(E) = X #σ=cd(E) eσ(E) Theorem 1.9.5. — With the notations above we have: (1) d(E(I)) = dim(A) (2) e(E(I)) = e(A). Proof. — See [15], [7]. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 105 2. Division theorems in the rings of differential operators 2.1. The aim of this section is to adapt the division theorems proved in chapter I to the case of the rings of differential operators and to give some applications: The calculation of free resolutions, of characteristic varieties and of multiplicities. The references are [11] and [14]. Let kbe a field of characteristic zero. We denote: •An(k) = k[X,∂] = k[X1, . . . , Xn;∂1, . . . , ∂n] the Weyl algebra, i.e. the ring of linear differential operators with polynomial coefficients in nvariables. •b Dn(k) = k[[X]][∂] = k[[X1, . . . , Xn]][∂1, . . . , ∂n] the ring of linear differential operators with formal power series in nvariables as coefficients. •Dn(k) = k{X}[∂] = k{X1, . . . , Xn}[∂1, . . . , ∂n] the ring of linear differential operators with convergent power series in nvariables as coefficients, if k=Ror Cor, more generally, a complete valued field of characteristic zero. For the sake of brevity we shall write when no confusion is possible: An,c Dn,Dn. We denote by Rany of these three rings. If Pis an operator we develop it in the following way: P=X (α,β)∈N2n a(α,β)Xα∂β=X β∈Nn fβ∂β where a(α,β)∈k, fβ∈k[X],k[[X]] or k{X}. We call the following subset of N2n, denoted by N(P), the Newton’s diagram of P: N(P) = {(α, β)∈N2n|a(α,β)6= 0} 2.2. The order of an operator. — We fix a linear form Lon Q2nwith non negative coefficients, whose restriction L2to {0}×Qnhas strictly positive coefficients. This condition is only necessary in the case of power series coefficients. Definition 2.2.1. — Let 0 6=P∈R=An,c Dnor Dn. We define the L2-order of P(and we denote it by ordL2(P)) as being max{L2(β)|fβ6= 0}. We set ordL2(0) = −∞. We have ordL2(PQ) = ordL2(P) + ordL2(Q) for any operators Pand Q. For each k∈L2(Qn), we write FL2 k(R) = {P∈R|ordL2(P)⩽k}. The family FL2 •(R) is an increasing filtration of the ring R. Let grL2 k(R) (or, more briefly, grk(R)) denote the quotient FL2 k(R)/FL2 <k(R). We call the mapping σL2 k: Fk(R)→grk(R) the symbol function of order k. Definition 2.2.2. — Let P∈Fk(R)rF<k(R). We call σL2 k(P) the L2-principal symbol of P. We denote the L2-principal symbol of ∂iby ξi. Thus, σL2 k(P) = PL2(β)=kfβξβ. We shall write it simply σL2(P). SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 106 F.J. CASTRO-JIM´ ENEZ & M. GRANGER The ring grL2(R) = ⊕ kgrL2 k(R) is commutative and isomorphic to the ring B[ξ1, . . . , ξn] where as the case may be B=k[X],k[[X]],or k{X}. Definition 2.2.3. — Let Ibe an ideal (9) of R. We call the ideal of grL2(R), denoted by grL2(I), generated by {σL2(P)|P∈I}the L2-graded ideal associated with I. Definition 2.2.4. — Let Ibe an ideal of R. We call the set {(x,ξ)∈k2n|σL2(P)(x,ξ) = 0 for all P∈I}, denoted by CharL2(R/I), the L2-characteristic variety of the R-module R/I. When R=Anwe also have the possibility of mixing the variables Xand ∂: Definition 2.2.5 (The L-Bernstein filtration). — Let P∈An(k). We call the integer max{L(α, β)|a(α,β)6= 0} the L-order of P(and we denote it by ordL(P)). The L-principal symbol of Pis the sum σL(P) = PL((α,β))=ordL(P)a(α,β)Xαξβ. We have once again the notion of graded ideal associated with an ideal Iof An and the notion of L-characteristic variety of An/I, for the L-Bernstein filtration. On the other hand when L2(β) = β1+···+βn, the filtration induced by L2is the usual filtration by the order of operators with respect to derivation variables. 2.3. Orderings in N2nand the privileged exponent of an operator. — Let < be a total well ordering on N2ncompatible with sums. We define an ordering denoted by <L, on N2n, in a different way according to whether we are in Anor with power series coefficients. •In An: (α, β)<L(α0, β0) if and only if        L2(β)< L2(β0) or L2(β) = L2(β0) and L(α, β)< L(α0, β0) or L2(β) = L2(β0), L(α, β) = L(α0, β0) and (α, β)<(α0, β0) This is a total well ordering compatible with sums. •In c Dnor Dn: (α, β)<L(α0, β0) if and only if        L2(β)< L2(β0) or L2(β) = L2(β0) and L(α, β)> L(α0, β0) or L2(β) = L2(β0), L(α, β) = L(α0, β0) and (α, β)>(α0, β0) Definition 2.3.1. — Let P∈An,c Dnor Dn. We call the 2n-uple expL(P) = max<L{(α, β)|a(α,β)6= 0}, the L-privileged exponent of P. (9)All the ideals under consideration are left ideals. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 107 Remark. — We have in every case the formula expL(P) = expL(σL2(P)) with σL2(P)∈k[X,ξ], k[[X]][ξ] or k{X}[ξ] respectively, the two last rings being seen as subrings of k[[X,ξ]] or of k{X,ξ}and the privileged exponents being taken in the sense of the first chapter. Then we can state the following propositions which can be proved exactly as in the first chapter: Proposition 2.3.2. — Let P, Q ∈R. We have: 1) exp(PQ) = exp(P) + exp(Q). 2) If exp(P)6= exp(Q)then exp(P+Q) = max<L{exp(P),exp(Q)}. For each non zero ideal Iof Rlet E<L(I) denote the set {expL(P)|P∈Ir{0}}. If no confusion is possible we write E(I) instead of E<L(I). We have, by 2.3.2, E(I) + N2n= E(I) and as we prove in 1.4.6 we have: Proposition 2.3.3. — Let E⊂N2nbe such that E+N2n=E. Then there is a finite subset F⊂Esuch that E=∪(α,β)∈F((α, β) + N2n). Definition 2.3.4. — Let Ibe a non zero ideal of R. We call any family P1, . . . , Pmof elements in Isuch that E(I) = ∪m i=1(expL(Pi) + N2n), a standard basis of I, relative to L(or an L-standard basis of I) Remarks 1) There always exists a standard basis of Iby definition of E(I) and 2.3.3. 2) In the case of Anwe can also consider the L-Bernstein filtration, and the following ordering similar to the one given in the preceding chapter up to the change of ninto 2n: (α, β)<L(α0, β0) if and only if L(α, β)< L(α0, β0) or L(α, β) = L(α0, β0) and (α, β)<(α0, β0) 2.4. More divisions. — The statements below narrowly follow those in the preceding chapter and we shall only give the proofs of the points specific to the case of the operators. With each m-uple ((α1, β1), . . . , (αm, βm)) of elements of N2n, we associate a partition ∆1, . . . , ∆m,∆ of N2nin the same way as in chapter I. We set: ∆1= (α1, β1) + N2n,∆i+1 = ((αi+1, βi+1) + N2n)r(∆1∪···∪∆i) if i⩾1, ∆ = N2nr(∪m i=1∆i). Theorem 2.4.1. — Let (P1, . . . , Pm)be an m-uple of non zero elements of Rand let ∆1, . . . , ∆m,∆be the partition of N2nassociated with (exp(P1), . . . , exp(Pm)). Then, for any Pin R, there is a unique (m+ 1)-uple (Q1, . . . , Qm, R)of elements in R, such that: (1) P=Q1P1+···+QmPm+R. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 108 F.J. CASTRO-JIM´ ENEZ & M. GRANGER (2) exp(Pi) + N(Qi)⊂∆i, i = 1, . . . , m. (3) N(R)⊂∆. Proof. — Uniqueness can be proved as in the commutative case. For existence, we consider σL2(P) and σL2(Pi) as elements of k[X,ξ] (resp. k[[X,ξ]],or k{X,ξ}), which are L2-homogeneous with respect to the variable ξ. Let us write the division in the sense of chapter I, in any of the three cases: σL2(P) = m X i=1 qiσL2(Pi) + r, the qi(X,ξ) and r(X,ξ) being polynomials and L2-homogeneous with respect to variables ξ(since the coefficients of L2are strictly positive). Suppose that d= ordL2(P) and that di= ordL2(Pi). Then the degrees of the quotients and of the remainder are given by the relations: ordL2(qi) = d−di,or qi= 0,ordL2(r) = dor r= 0. Let then Qiand Rbe the obvious operators such that qi=σL2(Qi) and r=σL2(R) (for example if qi=PL2(β)=d−dia(α,β)Xαξβ, Qi=PL2(β)=d−dia(α,β)Xα∂β). Then the operator P0=P−Pm i=1 QiPi−Ris of L2-order strictly smaller than d. We remark that the Qiand Rhave the properties 2) and 3) above since qiand rhave the corresponding properties and exp(Pi) = exp(σL2(Pi)). We end the proof by an induction (finite since the coefficients of L2are >0) on the L2-order. Remark. — The element Qiin the theorem is called the i-th quotient and Ris called the remainder of the division of Pby (P1, . . . , Pm). The remainder will be denoted by R(P;P1, . . . , Pm). Remark. — It follows from the proof that for any division P=Q1P1+···+QmPm+R as in the theorem we have max{maxi{expL(QiPi)},expL(R)}= expL(P) and as a consequence max{maxi{ordL2(QiPi)},ordL2(R)}= ordL2(P). Remark. — We have a similar (and simpler to prove) division theorem in the ring grL2(R) = B[ξ]. We let the reader state (and prove) a division theorem in An, relative to the L-Bernstein filtration. See [14]. Corollary 2.4.2. — Let Ibe a non zero ideal of R(or grL2(R)) and let P1, . . . , Pmbe a family of elements of I. The following conditions are equivalents: 1) P1, . . . , Pmis a standard basis of I. 2) For any Pin R, we have: P∈Iif and only if R(P;P1, . . . , Pm) = 0. Corollary 2.4.3. — Let Ibe a non zero ideal of R(or grL2(R)) and let P1, . . . , Pmbe a standard basis of I. Then P1, . . . , Pmis a system of generators of I. These two statements can be proved exactly as in the commutative case. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 109 Remark. — Let Ibe an ideal of R. Then {P1, . . . , Pm}is a standard basis of Iif and only if {σ(P1), . . . , σ(Pm)}is a standard basis of grL2(I). 2.5. The calculation of a standard basis and its applications. — Let P1, P2 be two operators with privileged exponents (α1, β1),(α2, β2). As in chapter I, we call the semisyzygy of P1, P2the operator M1P1−M2P2=S(P1, P2) where M1, M2are two monomials whose exponents ν1, ν2are such that ν1+ (α1, β1) = ν2+ (α2, β2) and minimal for this property and furthermore such that the leading coefficients satisfy c(M1)c(P1) = c(M2)c(P2) so that we get expL(S(P1, P2)) <LexpL(M1P1) = expL(M2P2). We have again: Proposition 2.5.1. — Let P1, . . . , Prbe a system of generators of the ideal Iof Rsuch that for any (i, j)the remainder of the division of S(Pi, Pj)by (P1, . . . , Pr)is zero. Then, {P1, . . . , Pr}is a standard basis of the ideal I. Proof. — We deduce the proof from the result in the commutative case by considering the σ(Pi)∈B[ξ1, . . . , ξn], and by using the fact that Piand σ(Pi) have the same privileged exponent. If MiPi−MjPj=S(Pi, Pj) = A1P1+···+ArPris a division, we have ordL2(AkPk)⩽ordL2(MiPi) = ordL2(MjPj). We set mi=σ(Mi), ak=σνk(Ak) where νk= ordL2(MiPi)−ordL2(Ak), and then we get the relation: miσ(Pi)−mjσ(Pj) = a1σ(P1) + ···+arσ(Pr). This is a division in k[X,ξ], k[[X,ξ]] or k{X,ξ}as the case may be. Furthermore, it is L2-homogeneous, hence in B[ξ]. Thus, {σ(P1), . . . , σ(Pr)}gives a standard basis of the ideal which they generate in the above rings hence also in B[ξ]. It remains to prove that the σ(Pi)’s generate gr(I). We consider P∈Iand we write: P=A1P1+···+ArPr(∗) If ordL2(P)< δ = max(ordL2(AkPk)), we have a1σ(P1) + ··· +arσ(Pr) = 0, where ak=σδ−ordL2(Pk)(Ak). We deduce from 1.6.4 the fact that in B[ξ], L2-homogeneous relations between the σ(Pk) are generated by those which come from the divisions of semisyzygies. This allows us to change the relation (∗) in order to lower δ. We finally obtain a decomposition (∗) for which δ= ordL2(P) in which case we have σ(P) = a1P1+···+arPr∈gr(I). Let I⊂Rbe an ideal given by a system of generators P1, . . . , Ps. The process that we are going to describe enables us to build a standard basis (P1, . . . , Ps, Ps+1, . . . , Ps+t) by a finite sequence of divisions. This algorithm is the analogue for algebraic differential operators of Buchberger’s [13] (see 1.6.3). SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 110 F.J. CASTRO-JIM´ ENEZ & M. GRANGER •Assume that (P1, . . . , Ps, Ps+1, . . . , Ps+q) are already built and define Eq= Ss+q k=1(exp(Pk) + N2n). •If there is (i, j) such that the remainder of the division of S(Pi, Pj) by (P1, . . . , Ps+q) is non zero, let us choose the first of these (i, j) (for the lexicographic ordering) and denote by Ps+q+1 the remainder thus obtained. Thus we have Eq⊂Eq+1 and Eq6=Eq+1 ⊂E(I). •By a Noetherian argument, this process stops and there exists an integer tsuch that Es+t= E(I). This can be detected by the lack of a non zero remainder since then (P1, . . . , Ps, . . . , Ps+t) is a standard basis. •We can eliminate (one by one) the Pkwhose privileged exponents are contained in the N2n-subset generated by the remaining exponents. Application 1. The calculation of the characteristic variety of a R-module of type R/I Proposition 2.5.2. — Let (P1, . . . , Pr)be a L-standard basis of the ideal Iof R. Then the equations of the L2-characteristic variety of R/I are: σ(P1)(X,ξ) = ··· =σ(Pr)(X,ξ) = 0 Indeed the equations σ(P)(X,ξ) = 0 for all P∈Iare linear combinations of these equations. Application 2. Free resolutions of an R-module of type R/I. — Let (P1, . . . , Pr) be a standard basis of the ideal Iof R. Let Sbe the module of relations between the operators Pk. This module is the set of r-uples R= (A1, . . . , Ar)∈Rrsuch that A1P1+···+ArPr= 0. We say that Ris of order kif k= max(ordL2(AiPi)) and we set: σk(R) = (σk−d1(A1), . . . , σk−dr(Ar)). Let us denote the relations following from the division of semisyzygies by Ri,j and ri,j =σ(Ri,j). Proposition 2.5.3. — We have an exact sequence: Dr(r+1)/2ϕ −→ Drψ −→ D→D/I with: ψ(Q1, . . . , Qr) = Q1P1+···+QrPr, ϕ((Ai,j)) = XAi,jRi,j. Proof. — This is equivalent to stating that the relations between the P`are generated by the relations Ri,j. If Ris such a relation, σ(R) = ris a homogeneous relation between the σ(P`), of degree k= ordL2(R). By the commutative analogue (see 1.6.4), we can write r=Pλi,jri,j with ord(λi,j) + ki,j ⩽kwhere ki,j = ord(Ri,j). We choose Λi,j ∈Rsuch that σ(Λi,j) = λi,j. Then, R0=R−PΛi,jRi,j is a relation between the operators P`of L2-order < k. We conclude by an induction on the L2-order. Application 3. Elimination of variables in Anand intersection of ideals. — Rename the vector (x1, . . . , xn, ∂1, . . . , ∂n) as (y1, . . . , yn, yn+1, . . . , y2n) and consider new variables z1, . . . , zn, zn+1, . . . , z2n. Let τbe a permutation of 2nsymbols and denote S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 111 zi=yτ(i). Denote by ρthe inverse of τ. Then Anis isomorphic to the k-algebra generated by z1, . . . , zn, zn+1, . . . , z2nwith relations [zρ(i), zρ(j)] = 0 (i⩽j) except for j=i+nin which case [zρ(i), zρ(j)] = −1. Let Ibe a left ideal of Anand kbe an integer 0 ⩽k⩽2n−1. We denote by An,k the subalgebra of Angenerated by zk+1, . . . , z2n. We define Ik=I∩An,k. The (left) ideal Ikof An,k is the set of operators in Iwhich depend only on zk+1, . . . , z2n. We write I2n=k∩I. The ideal Ikis called the k-th elimination ideal of I. We shall return later to this definition. Using the lexicographic ordering on N2n(as in 1.7.1) we can prove the following results which are similar as well as the proofs to those in 1.7 and 1.8.1. Lemma 2.5.4. — Let Pbe an element of An. Then mp<lex (P)is in An,k if and only if Pis in An,k. Theorem 2.5.5. — Let Ibe a left ideal of Anand kan integer such that 0⩽k⩽2n. Let Gbe a standard basis of the ideal Irelative to the lexicographic ordering. Let Gk=G∩An,k. Then we have: (1) If Gk=∅then Ik= (0). (2) If Gk6=∅then Gkis a standard basis of the ideal Ikrelative to the lexicographic ordering. Let I, J be two left ideals of An. Let θbe a new indeterminate. We denote by Ie (resp. Je) the extension of the ideal I(resp. J) to the ring An[θ] (here θis a central element). If his an element of k[θ] we denote by hIe(resp. hJe) the product of the ideals(10) (h) and Ie(resp. (h) and Je). With these notations we have: Theorem 2.5.6. — Let I, J be two left ideals of An. Then I∩J= (θIe+(1−θ)Je)∩An. Remark. — The theory of standard bases can be easily generalized to the case of sub–modules of RN, see [14]. For that purpose we only have to adapt the notions of ordering and of privileged exponents to exponents in N2n×{1, . . . , N}. By applying this to the calculation of a standard basis of ker(ϕ) and then of the successive kernels, we build a free resolution of any R-module Mof finite presentation, whence for example a realization of the complex of solutions and of the De Rham complex RHomR(M,O) and ΩnL ⊗M. This is algorithmic in the algebraic case. 2.6. An example: The characteristic cycle of O[1/f]for a quasihomogeneous fin two variables.— In this example we are dealing with the form L2(i, j) = i+j. In this case and more generally in the case of the diagonal form L2on Qn, we refer to [25, 19] for the definition of the multiplicity at a point of the cotangent space. The characteristic cycle of a coherent D-module is the linear (10)these are ideals of the ring An[θ] SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 112 F.J. CASTRO-JIM´ ENEZ & M. GRANGER combination of the irreducible components of the characteristic variety, each counted with its multiplicity at a generic point. Let f∈C[x, y] be a quasi-homogeneous polynomial. We denote by w1and w2the weights of variables and by χthe Euler vector field: χ=w1x∂x+w2y∂y We have χ(f) = f. We verify that O[1/f] = D·1 f, because the Bernstein polynomial of fhas no ⩽−2 integer root (see [31]). It is easier to deal with the quotient O[1/f]/O and we find that the annihilator ideal of its generator c`(1 f) is the ideal generated by the following three operators: •P1=f0 y∂x−f0 x∂y •P2=w1x∂x+w2y∂y+ 1(= χ+ 1) •P3=f Let us first consider the case f=yp+c1xq1yp−p1+···+ckxkq1yp−kp1+··· with q1> p1⩾1, p= 0 or 1(mod p1) and w2= 1/p,q1w1=p1w2. In this situation we verify by computing the semisyzygies that {P1, P2, P3}is a standard basis for the ordering (of series type) associated with L(j, i, β, α) = j+i+ α+βthe monomial with the same L-order being further ordered by y > x > ∂y> ∂x. The privileged exponents are respectively: (p−1,0,0,1),(0,1,0,1),(p, 0,0,0). By applying 1.9 we can compute the multiplicity at the origin of O[1/f]/Owhich is therefore (p−1)+0+p+0+0+0 = 2p−1. The characteristic cycle has the following form: sT∗ 0(C2) + 1.T∗ f−1(0)(C2) for some integer s. The multiplicity of f−1(0) at the origin being pwe get from this 2p−1 = s+ 1.p, or: s=p−1. For the case f=x·g, where gis a polynomial as in the previous case we refer to [9]. 3. Generalized division theorems. The calculation of slopes The reference for this chapter is [2] for the case of the Weyl algebra. We denote by Rany of the rings An,Dnor c Dn. 3.1. Orders and filtrations with respect to a smooth hypersurface. Let Ybe a hypersurface of Cndefined by x1= 0. Given a linear form L(a, b) = pa +qb on Q2(with non negative and relatively prime integer coefficients p, q), we define the L-order along Yof P=P(x, ∂) in Rdenoted by ordL(P), as the maximum of L(|β|, β1−α1) for (α, β) in the Newton diagram of P. To shorten we write here x instead of Xand ∂instead of ∂. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 113 Notice that here Lis a linear form on Q2whereas in the previous chapters this letter was used to denote a linear form on Q2nwhose part is now taken by: e L(α, β) = L(|β|, β1−α1) = (p+q)|β|−q(β2+···+βn+α1). Let FL,•(R) be the filtration induced by the L-order on Ri.e. FL,k is the set of operators Psuch that ordL(P)⩽k. Let F(resp. V) denote the filtration associated with the linear form L(a, b) = a(resp. L(a, b) = b). By extension we also write F (resp. V) for the corresponding linear forms. If L6=F, V the graded ring associated with this filtration grL(R) = L k∈Z FL,k(R)/FL,k−1(R) is isomorphic to one of the graded commutative rings C[x, ξ] = C[x1, . . . , xn, ξ1, . . . , ξn] or C{x2, . . . , xn}[x1, ξ1, . . . , ξn] or C[[x2, . . . , xn]][x1, ξ1, . . . , ξn] where the degree of the monomial xαξβis L(|β|, β1−α1). If L=F, the filtration FL,•is the filtration by the order of operators. The graded ring grV(R) is isomorphic to one of the rings An, C{x2, . . . , xn}[x1, ∂1, . . . , ∂n] or C[[x2, . . . , xn]][x1, ∂1, . . . , ∂n] where the degree of the monomial xα∂βis β1−α1. Given an ideal Iof Rlet grL(I) be the graded ideal associated with the filtration induced by FL,•on I. The ideal grL(I) is generated by the set {σL(P)|P∈I}where σL(P) is the principal symbol of Pwith respect to L. By definition, if L6=V, σL(P) = X L(|β|,β1−α1)=ordL(P) pα,βxαξβ. If Lis the form V, the symbol of Pwith respect to Vis the differential operator σV(P) = X β1−α1=ordV(P) pα,βxα∂β. Notice that for L6=V, (α, β)→L(|β|, β1−α1) is a linear form whose coefficients on the βiare all strictly positive. What follows works in the same way for any family of linear forms of this type for which the variables αihaving non-positive coefficients are fixed and for which ordL([P, Q]) <ordL(P)+ordL(Q) whence grL(R) is commutative. We shall not write this generalization. In the case of an ideal of An, the following lemma shows how to deal with the ideal generated by Iin Dn(or in c Dn) and conversely: Lemma 3.1.1. — Let Ibe an ideal in An. Then grL(DnI) = grL(Dn) grL(I). More precisely, if F={P1, . . . , Pr}is a system of generators of Isuch that G={σL(Pi)}r i=1 generates grL(I), then Ggenerates grL(DnI)over grL(Dn). Remark. — We shall see later that such a family Fcan be calculated effectively starting from a system of generators of the ideal I. Proof. — See [2]. The same result is valid in c Dn. SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 120 F.J. CASTRO-JIM´ ENEZ & M. GRANGER Recall, to end this chapter, the process which allows us to determine these idealistic slopes. Of course this is an algorithm only in the case of An. •We determine an L-standard basis {P1, . . . , Pr}of Iwhere Lis For a previously determined slope. We make sure that it is also a V-standard basis of grL(I). •We determine the form L(1) with minimal slope >slope(L) such that one of the σL(1) (Pi) is not bihomogeneous. Precisely L(1) is the linear form with smallest slope greater than slope(L) appearing in the (F, V )-Newton diagram of the operators Pi. •By a finite division process we can decide whether one of the bihomogeneous components of one of the σL(1) (Pi) is not an element of grL(1) (R). In this case L(1) is a new idealistic slope. In the other case we can modify Piin order to eliminate L(1), and obtain a basis which is standard for Land for L(1). We prove in [2] that this type of cancellation can happen only a finite number of times before we come upon a new slope or upon V. 3.6. Examples of calculations of slopes Example 1. — In this example we consider the direct image of the DC-module DCe1/vk, by an immersion in C2and the slopes relative to a hypersurface tangent to the support. The advantage of this example is that we can carry out all the calculations in many cases and that it shows idealistic slopes which are not slopes. For k∈Nwe write: M=DCe1/vk≃DC DC(vk+1∂v+ 1),N=i+M≃DC2 DC2(vk+1∂v+ 1) + DC2u where iis the immersion C→C2given by i(v) = (0, v). We want to calculate the slopes of Nalong the curve vm+u= 0. We carry out the change of variables: u=x−ym, v =y. We have: ∂u=∂x, ∂v= ∂y+mym−1∂x. We then find that Nis the quotient of DC2by the ideal Igenerated by the following operators: •P0 1=yk+1∂y+myk+m∂x+ 1 •P2=ym−x We then have to look at the slopes along x= 0. In what follows we say that the slope is −p/q if L(a, b) = pa +qb. Subexample 1.1: m= 1. — We find the slope −k. We are in the same situation as for the calculation of the slope of Nalong v= 0. This is also a particular case of the following. Subexample 1.2: k=mp. — We then find the slope −p. We have: P0 1=ymp+1∂y+mym(p+1)∂x+ 1 = (y∂y−mp)ymp +m∂xxp+1 + 1 = (y∂y−mp)xp+mxp+1∂x+m(p+ 1)xp+ 1 (mod DP2) This gives the presentation of N: S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 121 •P1=mxp+1∂x+xpy∂y+mxp+ 1 •P2=ym−x We choose the ordering <Ffor which the variables are ordered as follows: ∂x> ∂y> y > x. We then find that (P1, P2) is an F-standard basis. Indeed the remainder of the division of: S(P1, P2) = ymP1−mxp+1∂x by {P1, P2}is zero. In this standard basis the privileged exponent of Piis also the privileged exponent of σV(σF(Pi)), so that by looking at P1we can say that the −p is an idealistic slope. For that we verify that σL(P1) = mxp+1ξ+xpyη +mxp+ 1 is not bihomogeneous (11). This last result comes for example from the fact that 1/∈grV(grF(I)) (N6= 0). There is no other slope because for any L0of slope >−p, grL0(I) = (1). Subexample 1.3: m= 2 and k= 2n−1. — By changing P0 1=y2n∂y+ 2y2n+1∂x+ 1 modulo P2=y2−x, as in the preceding subexample, we find I=DP1+DP2, with: P1= 2xny∂x+xn∂y+ 1. The first semisyzygy S(P1, P2) = yP1−2xn∂xP2divided by {P1, P2}gives a remainder P3whence the following generators for I: •P1= 2xny∂x+xn∂y+ 1 •P2=y2−x •P3= 2xn+1∂x+xny∂y+y+ 2xn We find that the remainders of the divisions of S(P1, P3) = xP1−yP3and S(P2, P3) = −2xn+1∂xP2+y2P3by {P1, P2, P3}are zero. Thus this is an F-standard basis which is also as in the preceding subexample a V-basis of grF(I). Let Lbe the linear form corresponding to the first eventual slope, the slope −n. We have σL(P3) = 2xn+1ξ+xnyη+y. It is impossible that y∈grV(grF(I) because Iwould contain the two elements of order 0, y2−xand y+xφ(x, y) and Nwould be supported by the origin. Thus we have pointed out an idealistic slope of the ideal I. It is not a slope of DC2/I because the equations of the L-characteristic variety are: •σL(P1) = 2xnyξ = 0 •σL(P2) = y2= 0 •σL(P3) = 2xn+1ξ+xnyη +y= 0 and the associated reduced variety is bihomogeneous with equations y=xξ = 0. Let us set Fi=σL(Pi) and look for a V-standard basis of grL(I). Let us remark that the privileged exponent of F3has changed and is now the monomial y. The semisyzygy S(F1, F3) = −F1+ 2xnξF3gives the remainder: F4= 4x2n+1ξ2+ 2x2nyξη, and we find that all the other remainders are zero, so that (11)Here Lis the linear form of slope −p SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 122 F.J. CASTRO-JIM´ ENEZ & M. GRANGER {F1, F2, F3, F4}is a V-standard basis of grL(I). We can lift the preceding semisyzygy in D, as −P1+ 2xn∂xP3, and this gives the operator: P4= 4x2n+1∂2 x+ 2x2ny∂x∂y+ 4(n+ 2)x2n∂x+ 2nx2n−1y∂y−xn∂y+ 4nx2n−1−1. The family {P1, P2, P3, P4}is therefore a system of generators of Iwhich gives a standard basis of grV(grL(I)) and by P4we point out the linear form L0of slope −(n−1/2). By the algorithm above, it is enough to verify that σL0(P4) = 4x2n+1ξ2+ 2x2nyξη −1 is not in grV(grL(I)), which amounts to find out that 1 /∈grV(grL(I)) = (2xnyξ, y2, y, 4x2n+1ξ2+ 2x2nyξη). Finally, there is no other slope because for any form L00 of slope >−(n−1/2) we have 1 = −σL00 (P4)∈grL00 (I). Example 2: De1/(yp−xq)(with G. Brevet). — To apply our algorithm to the determination of the slopes of the D-module generated by e1/(yp−xq), it is necessary to know the annihilator in Dof the function e1/(yp−xq). The answer to this last question was given by J. Brian¸con and Ph. Maisonobe in [12] in the more general case where fis quasi-homogeneous with an isolated singularity. The annihilator is: D(fχ + 1) + D(∂f ∂x ∂ ∂y −∂f ∂y ∂ ∂x) where χis a vector field such that χ(f) = f. Thus we have: I= AnnD(e1/(yp−xq)) = D(P1, P2) with P1=pyp−1∂x+qxq−1∂y, P2=qyp+1∂y−pxq+1∂x−2qxqy∂y+pq Furthermore, we take as an L-ordering (where Lis a linear form on Q2with rational positive coefficients) the ordering on N4defined as follows: (i, j, α, β)<L(i0, j0, α0, β0)⇐⇒                  L(α+β, α −i)< L(α0+β0, α0−i0) or L(α+β, α −i) = L(α0+β0, α0−i0) and i+j > i0+j0 or    L(α+β, α −i) = L(α0+β0, α0−i0), i+j=i0+j0 and (α, β, j, i)<lex(α0, β0, j0, i0) 1. The calculation of a standard basis of Ifor the form L=F. — If q > p, we have: mpF(P1) = pyp−1∂xand mpF(P2) = qyp+1∂y. Let us set then ∆1= (0, p−1,1,0)+N4 and ∆2= ((0, p + 1,0,1) + N4)r∆1. The syzygy relative to P1and P2is equal to: S(P1, P2) = qy2∂yP1−p∂xP2. The remainder of the division gives a third operator P3=p2xq+1∂x2+ 2pqxqy∂x∂y+q2xq−1y2∂y2 +p2(q+ 1)xq∂x+q2(p+ 1)xq−1y∂y−p2q∂x. Proposition 3.6.1. — For 2⩽p < q,{P1, P2, P3}is an F-standard basis of I. S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 123 Proof. — We have mpF(P3) = p2xq+1∂x2and ∆3= ((q+1,0,2,0)+N4)r(∆1∪∆2). We then have to prove that the remainders of the divisions by (P1, P2, P3) of the semisyzygies S(P1, P3) and S(P2, P3) are zero. We find first: S(P1, P3) = pxq+1∂xP1−yp−1P3 = (p2xq+1yp−1∂x2+pqx2q∂x∂y+pq(q−1)x2q−1∂y) −p2xq+1yp−1∂x2+ 2pqxqyp∂x∂y+q2xq−1yp+1∂y2+p2(q+ 1)xqyp−1∂x +q2(p+ 1)xq−1yp∂y−p2qyp−1∂x =··· = ((pq −2q−p)xq−2qxqy∂y+pq)P1−qxq−1∂yP2. Let us denote by Q1= (pq −2q−p)xq−2qxqy∂y+pq the quotient relative to P1in this division. We must now deal with the semisyzygy S(P2, P3) = p2xq+1∂x2P2−qyp+1∂yP3. Instead of directly applying the division algorithm we are going to use the above equalities: yp−1P3=Q1P1+qxq−1∂yP2 p∂xP2= (qy2∂y−(p−1)qy)P1−P3 and we denote by Q0 1=qy2∂y−(p−1)qy the quotient relative to P1. Thus we have on one hand: qy2(∂yyp−1−(p−1)yp−2)P3= (qy2∂y−(p−1)qy)yp−1P3 = (qy2∂y−(p−1)qy)(Q1P1+qxq−1∂yP2) and the obtained quotients for P1and P2are allowed for the division. We have, on the other hand: p2xq+1∂2 xP2=pxq+1∂x(Q0 1P1−P3) and the obtained quotients are allowed for the division. This shows that S(P2, P3) has by division a zero remainder. 2. The calculation of the slopes. — Let us draw first the Newton polygons associated with the operators P1, P2, P3: 6 V - F (1,0) • (1,−q+1) • N(P1) 6 V - F (1,0) •• (1,−q) • N(P2) 6 V - F (1,1) BBBB • (2,−q+1) • • N(P3) SOCI´ ET´ E MATH´ EMATIQUE DE FRANCE 2004 124 F.J. CASTRO-JIM´ ENEZ & M. GRANGER Since {P1, P2, P3}is an F-standard basis of Iand a system of generators of I, the ideal grF(I) is generated by the principal symbols σF(P1) = pyp−1ξ+qxq−1η σF(P2) = qyp+1η−pxq+1ξ−2qxqyη σF(P3) = p2xq+1ξ2+ 2pqxqyξη +q2xq−1y2η2 Proposition 3.6.2. — (σF(Pi))1⩽i⩽3is a V-standard basis of grF(I). Proof. — We have mpV(σF(P1))=pyp−1ξ, mpV(σF(P2))=qyp+1ηand mpV(σF(P3))= p2xq+1ξ2. The divisions by σF(P1), σF(P2), σF(P3) give: S(σF(P1), σF(P2)) = qy2ησF(P1)−pξσF(P2) = σF(P3)≡0, S(σF(P1), σF(P3)) = pxq+1ξσF(P1)−yp−1σF(P3) =−2q2xqyησF(P1)−qxq−1ησF(P2)≡0, S(σF(P2), σF(P3)) = p2xq+1ξ2σF(P2)−qyp+1ησF(P3)) =−2q2xqy3η2σF(P1)−q2xq−1y2η2σF(P2)−pxq+1ξσF(P3)≡0. This proves the proposition. We now have to consider the linear form L(L < F) with the greatest possible slope such that one of the principal symbols of one of the Piis not bihomogeneous. We have: L(a, b) = qa +b(slope equal to −q). Proposition 3.6.3. — The D-module De1/(yp−xq)has only the slope −qalong the hypersurface x= 0. Proof First step: −qis a slope. We know (see [2]) that if L < Λ< F , then grΛ(I) = grV(grF(I)). If grL(I) was bihomogeneous, then it would also be equal to grV(grF(I)). But by the previous proposition, we have: grV(grF(I)) = (σV(σF(P1)), σV(σF(P2)), σV(σF(P3)))C{y}[x, ξ, η] = (pyp−1ξ, qyp+1η, p2xq+1ξ2+ 2pqxqyξη +q2xq−1y2η2)C{y}[x, ξ, η] Since σL(P3) = σV(σF(P3)) −p2qξ and σV(σL(P3)) = −p2qξ it is therefore enough to prove that ξ6∈ grV(grF(I)): this can be seen by writing ξas a linear combination of σV(σF(Pi)) then by evaluating at x=y= 0 (we find then ξ≡0!). Second step: there is no other slope. Take V < L0< L. Let us show that L0is not a slope. We have: (σL0(P1), σL0(P2), σL0(P3))C{y, x}[ξ, η]⊂grL0(I) S´ EMINAIRES & CONGR` ES 8 EXPLICIT CALCULATIONS IN RINGS OF DIFFERENTIAL OPERATORS 125 that is (yp−1ξ, yp+1η, ξ)⊂grL0(I) and so CharL0 (De1/(yp−xq))⊂ {y=ξ= 0}∪{η= ξ= 0}. Thus this characteristic variety is of dimension 2 and: qgrL0(I) = (ξ, y) or (ξ, η) or (ξ, yη). pgrL0(I) is therefore bi-homogeneous and L0is not a slope. 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