A computational approach to the D-module of meromorphic functions

arXiv:math/0204162v1 [math.AG] 12 Apr 2002 A Computational Approach to the D-module of Meromorphic Functions F.J. Castro-Jim´enez and J.M. Ucha September, 2001 Abstract Let Dbe a divisor in Cn. We present methods to compare the D-module of the meromorphic functions O[∗D] to some natural approximations. We show how the analytic case can be treated with computations in the Weyl algebra. 1 Introduction Let us denote by O=OCnthe sheaf of holomorphic functions on X=Cn. Consider a point p∈Cn.Der(Op) is the Op-module of C-derivations of Op. The elements in Der(Op) are called vector fields. Let D⊂Xbe a divisor (i.e. a hypersurface) and p∈D. A vector field δ∈Der(Op) is said to be logarithmic with respect to Dif δ(f) = af for some a∈ Op, where fis a local (reduced) equation of the germ (D, p)⊂(Cn, p). The Op-module of logarithmic vector fields (or logarithmic derivations) is denoted by Der(log D)p. This yields an O-module sheaf denoted by Der(log D) (see [15]). Let us denote by D=DXthe sheaf (of rings) of linear differential operators with holomorphic coefficients on X=Cn. A local section Pof D(i.e. a linear differential operator) is a finite sum P=Pαaα∂αwhere α= (α1,... ,αn)∈Nn,aαis a local section of Oand ∂= (∂1,... ,∂n) with ∂i=∂ ∂xiin some local chart. For any divisor D⊂Cnwe denote by O[⋆D] the sheaf of meromorphic functions with poles along D. It follows from the results of Bernstein-Bj¨ork ([1], [2]) on the existence of the b-function for each local equation fof D, that O[⋆D] is a left coherent D-module. Kashiwara proved that the dimension of its characteristic variety is nand then that O[⋆D] is holonomic, [12]. We will consider some D-modules associated to any divisor D: •The (left) ideal Ilog D⊂ D generated by the logarithmic vector fields Der(log D). •The (left) ideal e Ilog D⊂ D generated by the set {δ+a|δ∈Ilog Dand δ(f) = af}. More generally, the ideals e I(k) log Dgenerated by the set {δ+ka |δ∈Ilog Dand δ(f) = af} 1 •The modules Mlog D=D/Ilog D,f Mlog D=D/e Ilog Dand more generally f M(k) log D= D/e I(k) log D. The inclusion e I(k) log D⊂AnnD(1/fk) yields to a natural morphism φk D:f M(k) log D→ O[⋆D] defined by φk D(P) = P(1/fk) where Pdenotes the class of the operator P∈ D modulo e I(k) log D. The image of φk Dis D1 fk, i.e. the D-submodule of O[⋆D] generated by 1/fk. Considering the general ideals e I(k) log Dis a suggestion of Prof. Tajima. The point is the well known chain of inclusions D · f−1⊂ D · f−2⊂ · · · ⊂ D · fk=D · fk−1=···=O[⋆D], where kis least integer root of the b-function. We are interested in the germ (D, p)⊂(Cn, p) for a fixed point p∈D. So we will work in the ring of germs of linear differential operators Dp. We suppose that p= 0 ∈Cnand from now on, we will denote D=D0. In this context we will use the Weyl algebra An(C) as a subring of D. Under a computational point of view the divisor Dwill be defined by a polynomial f∈C[x1,... ,xn]. The b-function of fis computable by [14] and we give a direct method to present O[∗D]. If the calculation is intractable –as sometimes happens in the examples– we also present an indirect method to deduce that O[∗D] and the modules f M(k) log Ddo not coincide. The method is strongly based in the following result of [13]: Theorem 1.1. The restriction to Dof the sheaf Exti D(O[∗D],O)is zero for i≥0. More precisely, we prove that, under certain algorithmic conditions, some cohomology groups are not zero. The interest of this second method has been tested in [18], [9] and [10]. We point out how the algorithms presented in [17] –that only calculate cohomology groups in the algebraic case– could also be useful in some analytic situations. It is important to underline that our methods manage the analytic case. As the inclusion An(C)⊂ D is flat, the computation of syzygies and free resolutions in the Weyl algebra yields to the analogous computations in D. 2 Comparison algorithms We propose in this section two methods to compare the logarithmic modules presented above. It is important to remark that the computation of the analytic Der(log D) can be made for a divisor Dif its local equation is a polynomial f∈C[x1,... ,xn] . Simply compute, using Gr¨obner basis, a system of generators of (f1, . . . , fn, f) where fi=∂f ∂xi because the inclusion of the Weyl algebra in Dis flat. 2 2.1 Direct comparison The first method is complete but needs the calculation of the b-function. Experimental evidences show that if the divisor is not locally Euler homogeneous (i.e. there is no δ∈Der(log D) such that δ(f) = f) the b-function is hard to compute. More precisely, the problem seems to be the calculation of AnnD[s](fs) and the use of certain elimination orders during the calculation of Gr¨obner basis (here D[s] stands for the polynomial ring, with the indeterminate scommuting with D). Nevertheless, the method is applicable to all the Euler homogeneous divisors we have considered (and to some not Euler homogeneous). Algorithm 2.1. INPUT: A local equation f= 0 of a divisor D; 1. Compute the b-function of f. Let −α0be the least integer root. 2. Compute the ideal AnnD(1/fα0). 3. Compute a set of generators {s1,... ,sr}of Syz(f1,... ,fn, f). The ideal e I(α0) log Dis generated by the elements sj      ∂1 . . . ∂n −α0     ∈ D. 4. Compare AnnD(1/fα0) and e I(α0) log D. OUTPUT: O[∗D]≃f M(α0) log D⇔AnnD(1/fα0) = e I(α0) log D. The correctness of the algorithm is obvious as O[∗D]≃ D 1 fα0≃ D/AnnD(1/fα0). 2.2 Indirect deduction: a sufficient condition This second method is an alternative way when you can not obtain the b-function. In the worst case, it only needs the computation of a free resolution of f M(α) log Dfor any integer α≥1. More precisely, the algorithm looks for a technical condition in some step of the free resolution. In many examples, it is enough to compute only the first syzygies1. Definition 2.2. If 0→ Drsϕs −→ · · · → Dr2ϕ2 −→ Dr1ϕ1 −→ D π −→ M→0 is a free resolution of a module M, we will say that the Successive Matrices Condition (SMC) holds at level iif the two succesive morphisms ϕi, ϕi+1 have matrices verifying: 1Taking into account that computing a complete free resolution can be a problem of great complexity, this option is very interesting. 3 1. The matrix of ϕi+1 has no part in Oin some column j, i.e. all the elements in the j-th column are in the (left) ideal generated by ∂1,... ,∂n. 2. The matrix of ϕihas no constants in the in the j-th row. That is, for each Pin the j-th row P(1) is a function hwith h(0) = 0. Algorithm 2.3. INPUT: A local equation f= 0 of a divisor D; 1. Compute a set of generators {s1,... ,sr}of Syz(f1,... ,fn, f). The ideal e I(α) log Dis generated by the elements sj      ∂1 . . . ∂n −α     ∈ D. 2. Compute a free resolution of M=f M(α) log D 0→ Drsϕs −→ · · · → Dr2ϕ2 −→ Dr1ϕ1 −→ D π −→ M→0. OUTPUT: IF SMC holds THEN O[∗D]6=f M(α) log D. We need a lemma to justify the algorithm. It explains the role of the SMC. The idea is obtaining an element in Kerϕi+1 that is not in Imϕi. Lemma 2.4. Let Dbe a divisor and 0→ Drsϕs −→ · · · → Dr2ϕ2 −→ Dr1ϕ1 −→ D π −→ f M(α) log D→0 (∗) a free resolution of f M(α) log Dthat verifies SMC at level i. Then Exti D(f M(α) log D,O)6= 0. Proof. To obtain the Ext groups, we have to apply the functor HomD(−,O) to (∗). Using that HomD(Dr,O)≃ Or we obtain the complex 0→ O ϕt 1 −→ Or1ϕt 2 −→ Or2→ · · · ϕt s −→ O ϕt s −→ Ors→0, where ϕt idenotes the morphism with matrix the transposed of ϕi. The derivatives now act naturally. Then Exti D(f M(α) log D,O) = Kerϕt i+1/Imϕt i. If the matrix of ϕi+1 has no part in Oin the j-th row, then e= (0,... ,1,... ,0) –where 1 is in the j-th position– is in Kerϕt i+1, as the derivatives applied to 1 are zero. This element can not be in Imϕt iif the matrix has no constants. Applying the operators of the matrix it is not possible to obtain elements of degree 0. 4 We have the key to state the main result of this section: the correctness of 2.3. Proposition 2.5. Let Dbe a divisor with a free resolution of f M(α) log Dthat verifies SCM at some level. Then O[∗D]6=f M(α) log D. Proof. Evident from 2.4 and 1.1. A special case of SCM appears when you have a resolution of type 0→ Drnϕn −→ · · · → Dr2ϕ2 −→ Dr1ϕ1 −→ D π −→ M→0 of length n. Then Extn D(M, O) = Orn Imϕt n , and SCM means, at level n, that you can find in the matrix of ϕna row with no constants. Remark 2.6. Of course, natural generalizations of the SCM condition has to do with finding explicit elements in some Kerϕt i+1 with special properties. It is not that easy in general! Nevertheless the results of [17] can be applied in this situation as follows: Ext0 An(C)(R[∗D], R)6= 0 ⇒Ext0 D(O[∗D],O)6= 0, where Ris the ring of polynomials and R[∗D] is its localization with respect to the equation fof D. 3 Application to the Spencer case. In this section we explain how to apply the sufficient condition to a special case in which a tailored free resolution is provided. Definition 3.1. ([15]) The divisor Dis said to be free at the point p∈Dif the Op-module Der(log D)pis free. The divisor Dis called free if it is free at each point p∈D. Smooth divisors and normal crossing divisors are free. By [15] any reduced germ of plane curve D⊂C2is a free divisor. By Saito’s criterium [15], D≡(f= 0) ⊂Cnis free at a point pif and only if there exist nvector fields δi=Pn j=1 aij∂j,i= 1,... ,n, such that det(aij) = uf where uis a unit in Op. Here ∂jis the partial derivative ∂ ∂xjand aij is a holomorphic function in Op. Definition 3.2. We say that a free divisor Dis of Spencer type if the complex D ⊗O∧•Der(log D)→Mlog D→0 (introduced in [4]) is a (locally) free resolution of Mlog Dand if this last D-module is holonomic. 5 There are analogous resolutions for the family of modules f M(k) log D. For this family of divisors, the solution complex Sol(Mlog D) (that is, the complex RHomD(Mlog D,O)) is naturally quasi-isomorphic to Ω•(log D) (as we pointed in [10] as a deduction of [4]). On the other hand, a duality theorem proved in [10] has important consequences comparing f Mlog Dand O[⋆D], namely Theorem 3.3. ([18, 9]) In dimension 2, the morphism φ1 Dis an isomorphism if and only if Dis a quasi-homogeneous plane curve. Theorem 3.4. [10] Suppose the divisor D⊂Cnis free and locally quasi-homogeneous. Then the morphism φ1 Dis an isomorphism (so, f Mlog Dand O[⋆D]are isomorphic as D- modules). The methods presented in section 2 give us computational tools to check the comparison between f Mlog Dand O[⋆D] . Remark 3.5. Once you have the duality of [10], you also have a strategy to study the Logarithmic Comparison Theorem (LCT), that is, the complex Ω•(⋆D) of meromorphic differential forms and the complex Ω•(log D) are quasi isomorphic. You have to travel round the following chain of isomorphisms: Ω•(⋆D)≃DR(O[⋆D]) ≃DR(D/AnnD(1/f)) ≃DR(f Mlog D)≃ ≃DR((Mlog D)∗)≃Sol(Mlog D)≃Ω•(log D), where for each coherent D–module Mwe denote by DR(M) the de Rham complex of M (see [13]). Remark 3.6. There are two interesting experimental suggestions: •We don’t know examples of free divisors with integer roots of their b-function less than -1. •We only know free divisors of Spencer type. Finally, we have the following result: Proposition 3.7. Let D≡(f= 0) be a Spencer divisor. Let δ1,... ,δnbe a basis of Der(log D)with δi=Pn j=1 aij∂jfor 1≤j≤n. If Pn j=1 ∂j(aij) = 0, then f Mlog Dand O[⋆D]are not isomorphic. 6 Proof. The last matrix of the Spencer free resolution of f Mlog Dis of a very special type . Its elements (due to the duality formulas of [10]) are of the form δi+ n X j=1 ∂j(aij). So, applying lemma 2.4 Extn D(f Mlog D,O)6= 0,so f Mlog Dand O[⋆D] are not isomorphic. Thus there is no possible quasi-isomorphism between DR(D/AnnD(1/f)) and DR(f Mlog D). 4 Examples In the following examples, the computation of syzygies among polynomials have been made with CoCoA (see [11]). The computations of syzygies in the Weyl Algebra, global b-functions and ideals of type AnnD(1/fα) have been made with kan/sm1, [16], that is, using the algorithms of [14]. 4.1 Example 1: D≡(x(x2−y3)(x2−zy3) = 0) We will treat here the divisor D⊂C3whose local equation at (0,0,0) is given by f= 0 with f=x(x2−y3)(x2−zy3). This divisor is (globally) free and δ1, δ2, δ3form a (global) basis of Der(logD), where δ1=3 2x∂x+y∂y δ2= (y3z−x2)∂z δ3= (−1 2xy2)∂x−1 3x2∂y+ (y2z2−y2z)∂z, whose coefficients verify that  3 2x y 0 0 0 y3z−x2 −1 2xy2−1 3x2y2z2−y2z =−1 2f. To begin with, we have to follow two steps: •Step 1: Verify that Mlog Dis holonomic2. The interest of this question is evident: if Mlog Dis not holonomic, the computation of its dual could not be managed as we do. •Step 2: Compute a free resolution of Mlog Dwith Gr¨obner basis computation of syzygies. Check if Dhas a free resolution of Spencer type. If this happens then duality holds by [10]. 2This computation could be made with [16] 7 The example verifies these properties: 1. The module Syz(δ1, δ2, δ3) is generated by the syzygies obtained from the commutators [δi, δj]. We have Syz(δ1, δ2δ3) = hs12,s13,s23iwhere s12 = (−δ2, δ1−3,0) s13 = (−δ3,0, δ1−2) s12 = (0,−δ3−y2z, δ2). 2. On the other hand, the module Syz(s12,s13,s23) is generated by the element r: r= (−y2z2∂z+y2z∂z+1 2xy2∂x−y2z+1 3x2∂y, y3z∂z−x2∂z,−y∂y−3 2x∂x+ 5). This is the element required to have the Spencer type resolution so, as we have said, duality holds. We calculate the b-function of f. Its least integer root is -1, so O[⋆D]≃ D · 1/f ≃AnnD(1/f). To finish, we check that e Ilog D=AnnD(1/f). 4.2 Example 2: D≡(x4+y4+z4+x2y2z2= 0) ⊂C3 This is divisor is not free. A set of generators of f Mlog Dis δ1= (1/8x2y3z2+y) + (−1/32x3y3z2−1/16xyz4−1/4xy)∂x+ +(−1/32x2y4z2+ 1/8x2z2−1/4y2)∂y+ (−1/4yz)∂z, δ2= (−1/16x4y4z+z) + (1/64x5y4z+ 1/32x3y2z3−1/4xz)∂x+ +(1/64x4y5z−1/16x4yz −1/4yz)∂y+ (1/8x2y2−1/4z2)∂z, δ3= (−1/16x4y3z3−1/8x2y5z)+ +(1/64x5y3z3+ 1/32x3y5z+ 1/32x3yz5+ 1/16xy3z3)∂x+ +(1/64x4y4z3+ 1/32x2y6z−1/16x4z3−1/8x2y2z−1/4z3)∂y+ +(1/8x2yz2+ 1/4y3)∂z, δ4= (−1/16xy4z4+x) + (1/64x2y4z4+ 1/32y2z6+ 1/8y2z2−1/4x2)∂x+ +(1/64xy5z4−1/16xyz4−1/4xy)∂y+ (−1/4xz)∂z, δ5= (1/8x2y2z+ 1/4z3)∂x+ (−1/8xy2z2−1/4x3)∂z, δ6= (1/8xy5z2+ 1/4x3y3)+ +(−1/32x2y5z2−1/16x4y3−1/16y3z4−1/8x2yz2−1/4y3)∂x+ +(−1/32xy6z2−1/16x3y4+ 1/8xy2z2+ 1/4x3)∂y. 8 The free resolution is huge, but anyway computable with kan/sm1. It is of type 0→ D ϕ3 −→ Drϕ2 −→ Dsϕ1 −→ D π −→ M→0 To use proposition 2.4 you can check that there the elements in the last matrix ϕ3has no constants. So Ext3 D(f Mlog D,O) = O/im(ϕ3)t6= 0. You have O[∗D]6=f Mlog D. 4.3 Example 3: D≡(x4+y5+xy4+zx6= 0) ⊂C3 In this example we detect that we have, in fact, a (non-Euler homogeneous) product that is a free divisor3. The basis of Der(log D) is δ1= (x6z2+ 5/4x5yz2−2x4z−x2−5/4xy)∂x+ +(5/4x5yz2+ 3/2x4y2z2−5/2x3yz −1/2x2y2z−3/4xy −y2)∂y+ +(2x3z2−2xz)∂z, δ2= (−4/5x4yz −8/5x3y2z−3/4x2y3z+ 25/4x4z+ 125/16x3yz −xy2−1/4y3+ +125/16xy)∂x+ (−x3y2z−39/20x2y3z−9/10xy4z+ 125/16x3yz + 75/8x2y2z− −3/4y3+ 1/4x2−5/16xy + 25/4y2)∂y+ (−8/5xyz −6/5y2z+ 25/2xz)∂z, δ3= (−3/10x3y2z−3/8x2y3z+ 25/8x4z+ +125/32x3yz + 8/25x3−1/8y3+ 125/32xy)∂x+ (−3/8x2y3z−9/20xy4z+ +125/32x3yz + 75/16x2y2z+ 2/5x2y−1/50xy2+ 1/40y3+ 1/8x2−5/32xy+ +25/8y2)∂y+ (−3/5y2z+ 25/4xz + 16/25)∂z. ¿From the monomial 16/25∂zwe deduce by the Flow Theorem that Dis a product. It is of Spencer type and the third matrix can be used to show that Ext3 D(f Mlog D,O)6= 0 so O[∗D]6=f Mlog D. 4.4 Example 4: D≡((x+y)(xz +y)(x4+y5+xy4) = 0) ⊂C3 In this last example the divisor D⊂C3has as a local equation at (0,0,0) the form f= (x+y)(xz +y)(x4+y5+xy4) = 0. 3By the way, this situation is impossible in dimension 2. 9