Explicit Comparison Theorems for D -modules

Explicit Comparison Theorems for D-modules F. J. CASTRO-JIM´ENEZ AND J. M. UCHA-ENR´IQUEZ Departamento de ´Algebra, Universidad de Sevilla, Apdo. 1160, E-41080 Sevilla, Spain We prove in an explicit way a duality formula between two A2-modules Mlog and M f log associated to a plane curve and we give an application of this duality to the comparison between M f log and the A2-module of rational functions along the curve. We treat the analytic case as well. 1. Introduction Let R=C[x1, . . . , xn] be the ring of complex polynomials in nvariables. Denote by ∂i=∂ ∂xithe partial derivative with respect to xiand by An=Rh∂1, . . . , ∂nithe complex Weyl algebra of order n. Let f∈Rbe a non-zero polynomial. The ring of quotients Rf={g fm|g∈R, m ≥0} is a left An-module by considering ∂i(g fm) = ∂i(g) fm−mg∂i(f) fm+1 for 1 ≤i≤n. From the existence of the Bernstein polynomial (Bernstein, 1972), Rfis a finitely generated Anmodule. More precisely, there exists an integer k≥1 such that Rf=An1 fk(i.e. Rfhas a single generator 1 fkas left An-module). The integer kis related to the roots of the (polynomial) b-function of f(Bernstein, 1972). AC-derivation of R,δ=Piai∂iis said to be logarithmic w.r.t. fif δ(f) = Piai∂i(f) =af for some a∈R. We denote by Der(R, log f) the R-module of logarithmic derivations w.r.t. fand by Ilog f(or simply Ilog) the left ideal of Angenerated by Der(R, log f). The quotient module An/Ilog is denoted by Mlog. The set of differential operators δ+a∈An, for δ∈Der(R, log f) verifying δ(f) = af, generates a left ideal in Anthat will be denoted by e Ilog. The quotient module An/e Ilog is denoted by f Mlog. From e Ilog ⊂AnnAn(1/f) we deduce a natural morphism from f Mlog to An1 fand by composing with the embedding An1 f⊂Rfwe obtain a natural morphism φ : Mflog → Rf . Most results in this paper are concerned with the case n = 2. The main goal is to prove in a explicit way a duality formula between Mlog and Mflog (in the sense of Anmodules). The duality between Mlog and Mflog extends to the category of analytic Dmodules (Theorem 3.1) and we give an application of this duality to the comparison between Mflog and Rf (Section 4). To achieve these goals we calculate explicit free resolutions of both Mlog and Mflog. For Mlog the free resolution is the Spencer logarithmic resolution of Calder´on (1997) (see also Calder´on, 1999). Finally we show that some results can be generalized to special examples in dimension 3. 2. Preliminaries: Analytic D D D-modules For each p= (p1, . . . , pn)∈Cndenote by Opthe ring of germs of holomorphic functions in p. The ring Opis isomorphic to the ring C{x1−p1, . . . , xn−pn}of convergent power series in a neighborhood of pand we have R⊂ Op. Instead of the rings Rand Anwe can consider Opand Dp=Oph∂1, . . . , ∂nithe ring of linear differential operators with coefficients in Op. In fact Anis a subring of Dpand the elements in Dpcan be written as finite sums Pαaα∂αwhere α= (α1, . . . , αn)∈Nn,aα∈ Opand ∂α=∂α1 1· · · ∂αn n. More generally, we can consider on X=Cnthe sheaf DXof linear differential operators with holomorphic coefficients. Let us fix a point p∈Cn. For each f∈R(or more generally f∈ Op) we denote by Der(Op,log f) the Op-module of C-derivations of Oplogarithmic w.r.t. f(i.e. Cderivations δsuch that δ(f) = af for some a∈ Op). According to Saito (1980) we say that fis free at pif there exists a family {δ1, . . . , δn} ⊂ Der(Op,log f), δi=Pjcij∂j, such that the following condition holds: det((cij)) = upf, for some up∈ Op, up(p)6= 0.(∗) If fis free at pthen Der(Op,log f) is a free Op-module of rank nwith basis {δ1, . . . , δn} verifying the condition (∗) (Saito, 1980). As the inclusion R⊂ Opis flat, for each element f∈R, free at p, there exists a basis ∆ = {δ1, . . . , δn}of Der(Op,log f) with coefficients cij in R. Such a basis can be computed in an algorithmic way by considering a finite system Sof generators of the R-module SyzR(∂1(f), . . . , ∂n(f), f) of syzygies among (∂1(f), . . . , ∂n(f), f). Each syzygy Piai∂i(f)+mf = 0 produces the logarithmic derivation Piai∂i. Indeed, we have Der(R, log f)≃SyzR(∂1(f), . . . , ∂n(f), f). If there is no family of nderivations in Sholding (∗), then fis not free at p. Remark. Suppose now f∈Rfree at every point in Cn. Then Der(R, log f) is a locally free R-module, so it is free (using the theorem of Quillen–Suslin). The basis of this free R-module could be obtained in the general case applying, for example, the algorithms of Logar and Sturmfels (1992). However, in dimension 2 we have an alternative way of computing a basis of the free C[x1, x2]-module Der(C[x1, x2],log f) using the Hilbert–Burch theorem (e.g. Eisenbud, 1994). If fdefines a smooth plane curve, there is nothing to calculate. Suppose fis not smooth. We apply the Hilbert–Burch theorem to the ideal Jgenerated by the homogenized polynomials h(f), h(∂1(f)), h(∂2(f)) in S=C[x0, x1, x2] because the variety V(J) has dimension 0 in the projective plane P2(C). More precisely, Jhas a minimal free resolution (that can be computed explicitly) of the form 0−→ S2A −→ S3−→ J−→ 0, where Ais the matrix whose rows are a set of generators of the module of syzygies SyzS(h(f), h(∂1(f)), h(∂2(f))). Dehomogenizing (making x0= 1) the matrix Aproduces the matrix whose rows generate the module SyzR(f, ∂1(f), ∂2(f)). These two rows must be a basis. We think that this argument could be generalized to dimension n. From now on we assume p= 0 in Cnand we will write D0=Dand O0=O. We consider on D(resp. on An) the filtration by the order of the differential operators. The order of P=Pαaα∂αis the maximum value of |α|=α1+· · · +αnfor aα6= 0. The graded associated ring is the polynomial ring gr(D) = O[ξ] (resp. gr(An) = R[ξ]) where ξ= (ξ1, . . . , ξn). The principal symbol of P∈ D (resp. P∈An) is the element of gr(D) (resp. gr(An)) defined by σ(P) = P|α|=daαξαwhere dis the order of P. For each left ideal Iin D(resp. in An) we denote by gr(I) the ideal of gr(D) (resp. gr(An)) generated by the family σ(P) for P∈I. The characteristic variety of An/I is the algebraic set of C2n(denoted by Ch(An/I)) defined by the ideal gr(I)⊂gr(An). In the analytic case the characteristic variety of D/I is the germ of subvariety in U×Cn defined by the ideal gr(I)⊂gr(D); here Uis a small neighbourhood of the origin in Cnsuch that the coefficients of the elements of a finite system of generators of Iare holomorphic functions on U. A left An-module An/I is said to be holonomic if dim Ch(An/I) = nand we have a similar definition for D-modules. Remark. For each f∈Rwe can consider the (left) ideal Ilog an generated by Der(O,log, f) in Dand the (left) ideal e Ilog an generated by the family δ+afor vector fields δsuch that δ(f) = af with a∈ O. We denote Mlog an =D/Ilog an and f Mlog an =D/e Ilog an . By flatness of the extension R⊂ O we have the equalities Ilog an =DIlog and e Ilog an =De Ilog. Then Mlog an =Mlog ⊗ROand f Mlog an =f Mlog ⊗RO. Given a left holonomic An-module M, the dual module of M(denoted by M∗) is the left An-module associated to the right An-module Extn An(M, An) (Bj¨ork, 1979). We have the analogous definition for left holonomic D-modules. 3. Duality In this section we suppose n= 2 and fa reduced polynomial in R. For each p∈C2 let us denote by Opthe ring of germs of holomorphic functions in the neighborhood of p. By Saito (1980, 1.7) Der(Op,log f) is Op-free of rank 2 for all p∈C2and so, according to the results in 2, Der(R, log f) is R-free of rank 2. Let {δ1, δ2}be a basis of Der(R, log f) (and hence a basis of Der(O,log f)). Let us write δ1=c11∂1+c12∂2, δ2=c21∂1+c22∂2 for some polynomials cij ∈R. According to the first remark of Section 2 we can suppose that det((cij)) = c11 c12 c21 c22 =f. According to Calder´on (1997) (see also Calder´on, 1999, Corollary 4.2.2), {σ(δ1), σ(δ2)} is a regular sequence in gr(D) and then also in gr(A2). In particular hσ(δ1), σ(δ2)i= gr(Ilog) = gr(e Ilog), where hσ(δ1), σ(δ2)idenote the ideal of gr(A2) generated by {σ(δ1), σ(δ2)}. From this equality one can deduce Ch(f Mlog) = Ch(Mlog) and dim(Ch(A2/Ilog)) = 2. So, both modules Mlog and f Mlog are holonomic. From the last remark of Section 2 we obtain Ch(Mlog an ) = Ch(Mlog) and Ch(f Mlog an ) = Ch(f Mlog). Now we will compute free resolutions of Mlog and f Mlog. Remember we have an explicitly computed basis {δ1, δ2}of Der(R, log f), δi=ci1∂1+ ci2∂2, with det((cij)) = fand explicitly computed polynomials miverifying δi(f) = mif. Let us write [δ1, δ2] = α1δ1+α2δ2for some (explicitly computed) α1, α2in R. From Calder´on (1997) (see also Calder´on, 1999) a free resolution of Mlog is 0−→ A2 ψ2 −→ A2 2 ψ1 −→ A2−→ Mlog −→ 0 where ψ2is defined by the matrix (−δ2−α1, δ1−α2) and ψ1by δ1 δ2. So, one has the following proposition. Corollary 3.1. Ext2 A2(Mlog, A2)≃A2/J where Jis the right ideal of A2generated by {δ1−α2, δ2+α1}. Proposition 3.1. A free resolution of f Mlog is 0−→ A2 φ2 −→ A2 2 φ1 −→ A2−→ f Mlog −→ 0 (∗∗) where φ2is defined by the matrix (−δ2−m2−α1, δ1+m1−α2), and φ1by δ1+m1 δ2+m2. Proof. It is easy to prove that (∗∗) is a complex of A2-modules. To check its exactness, it is enough to consider the order filtration on that complex and to verify the exactness of the resulting complex (see Bj¨ork, 1979, Chapter 2, Lemma 3.13). We are using here the same argument of Calder´on (1997), (see also Calder´on, 1999, 4.1.3). The graded associated complex to (∗∗) is precisely 0−→ gr(A2)M1 −→ gr(A2)2M2 −→ gr(A2)−→ gr(f Mlog)−→ 0, where the matrices are M1= (−σ(δ2), σ(δ1)), M2=σ(δ1) σ(δ2), which is exact because {σ(δ1), σ(δ2)}is a regular sequence in gr(A2) (see Calder´on, 1999). 2 Theorem 3.1. We have (Mlog)∗≃f Mlog and the same result holds in the analytic case. Proof. According to 3.1 it is enough to prove the equalities −δT 1+α2=δ1+m1,−δT 2−α1=δ2+m2, where ()Tmeans the corresponding adjoint operator. We denote by Cthe matrix (cij ) and by δ(resp. ∂) the vector (δ1, δ2) (resp. (∂1, ∂2)). We can write [δ1, δ2] = (α1, α2)δt= (α1, α2)C∂t where ()tmeans transpose. On the other hand, [δ1, δ2] = (δ1(c21)−δ2(c11), δ1(c22)−δ2(c12))∂t. From the last two equalities and multiplying by Adj(C)twe obtain (α1, α2)f= (δ1(c21)−δ2(c11), δ1(c22)−δ2(c12))Adj(C)t. It follows that (α1, α2) = (m2+∂1(c21) + ∂2(c22), m1−∂1(c11)−∂2(c12)) using δi(f) = (ci1∂1+ci2∂2)(c11c22 −c12c21) = mif. The same method can be applied to establish that (Mlog an )∗≃f Mlog an .2 Remark. In dimension n, if Mlog admits an analogous free resolution as in 3.1, we have a proof for the last theorem that generalizes the ideas above. See Castro and Ucha (2000). 4. An Application: Comparing Modules In this section n= 2. A polynomial f∈Ris said to be quasi-homogeneous if there exists w= (w1, w2)∈N2 such that f(xw1 1, xw2 2) is an homogeneous polynomial and wi>0 for i= 1,2. The duality formula has an interesting application in order to compare f Mlog to Rf and f Mlog an to O[1/f].We need two previous technical propositions to give the theorem. Proposition 4.1. If fis a quasi-homogeneous (reduced) polynomial, then e Ilog = AnnA2(1/f)and e Ilog an = AnnD(1/f) Proof. By flatness we only have to consider the first case. We have •Let sbe an indeterminate and let us denote A2[s] = A2⊗CC[s]. Let α0be the smallest root of the global b-function of f. If α /∈α0+1+Nthen AnnA2(fα) = {P(α)|P(s)∈AnnA2[s](fs)}. See Kashiwara (1976, 6) or Saito et al. (2000, 5.3.13) for a proof. •AnnA2[s](fs) = hχ−s, ∂1(f)∂2−∂2(f)∂1iwhere χ(f) = f. See Yano (1978, 2.24). •If fis a plane curve then the local b-function has no integer roots less than −1 (Varchenko, 1982). From this fact, as the global b-function is the least common multiple of the b-functions localized at any point (Mebkhout and Narv´aez-Macarro, 1991), then the global b-function has the same property. We deduce that AnnA2(1/f) = hχ+ 1, ∂1(f)∂2−∂2(f)∂1i, where χis an Euler vector field associated to the quasi-homogeneous curve f. Clearly, these elements of the annihilator generate the ideal e Ilog.2 Proposition 4.2. If fis not a quasi-homogeneous (reduced) plane curve, then Ext2 D(f Mlog an ,O)6= 0. Proof. The proof of this statement contains, as an essential ingredient, a re-reading of the demonstration of Calder´on et al. (1999, Theorem 3.7). By proposition 3.1, a free resolution of f Mlog an is 0−→ D φ2 −→ D2φ1 −→ D −→ f Mlog an −→ 0, where φ2is the matrix (−δ2−m2−α1, δ1+m1−α2). Now we apply to the complex above the functor HomD(−,O). Hence, Ext2 D(f Mlog,O)≃ O/Img(φ∗ 2). Here φ∗ 2denotes the associated mapping to φ2by applying the functor Hom. To guarantee that this vector space has dimension greater than zero, it is enough to show that a pair of functions h1, h2∈ O such that (−δ2−m2−α1, δ1+m1−α2)h1 h2= 1,(∗∗∗) does not exist, that is to say, that 1 /∈Img(φ∗ 2). Let us take δ1=c11∂1+c12∂2. As m1−α2=∂1(c11) + ∂2(c12), (from the proof of 3.1) we will show either c11 and c12 have no linear parts, or after derivation these linear parts become 0. Of course fhas no quadratic part: in that case, because of the classification of the singularities in two variables, fwould be equivalent to a quasi-homogeneous curve x2 1+ xk+1 2, for some k. Then we can suppose that f=fn+fn+1 +· · · =X k≥n fk=X k≥nX i+j=k aijxi 1xj 2, where n≥3 and fn6= 0. We will write δ1=c11∂1+c12∂2=δ1 0+δ1 1+· · · =X k≥0X i+j=k+1 (β1 ijxi 1xj 2∂1+γ1 ijxi 1xj 2∂2), where the linear part δ1 0is (x1x2)A0(∂1∂2)t, and A0is a 2 ×2 matrix with complex coefficients. If A0= 0, we have finished. Otherwise, the possibilities of the Jordan form of A0are A0=λ10 0λ2, A0=λ10 1λ1. As δ1is not an Euler vector (because fis not quasi-homogeneous), we deduce: •If we take the first Jordan form, then (see the cited demonstration of Calder´on et al., 1999) fn=xp 1xq 2and δ0=qx1∂1−px2∂2. After a sequence of changes of coordinates we have that f=xp 1xq 2with p+q=n≥3, that contradicts that fis reduced. •For the second Jordan form with λ16= 0, it has to be fn= 0, that contradicts that fhas its initial part of degree n. •For the second option with λ1= 0 we have δ1 0=x2∂1and, in this situation, the linear part of c11 is x2. If we precisely apply ∂1, we obtain 0. In a similar way, we prove the same for m2+α1. So the minimal order of monomials in (−δ2−m2−α1)(h1)+(δ1+m1−α2)(h2) is greater or equal to 1, for any h1, h2∈ O. So (∗ ∗ ∗) has no solution. 2 Theorem 4.1. The natural morphism f Mlog an ψ −→ O[1 f]is an isomorphism if and only if fis a quasi-homogeneous (reduced) polynomial. Proof. If fis quasi-homogeneous then e Ilog an = AnnD(1/f) because of Proposition 4.1 and therefore ψis an isomorphism. Reciprocally, if ψis an isomorphism, then Ext2 D(O[1/f],O)≃Ext2 D(f Mlog an ,O). Because of a result of Mebkhout (1989), Ext2 D(O[1/f],O) = 0 and, if we take into account proposition 4.2, we obtain that fhas to be quasi-homogeneous. 2 Remark. Another argument could be used in the second part of the last proof applying a result of Torrelli (1998, 3.2.2.3): if fis not a quasi-homogeneous (reduced) plane curve then AnnD(1/f) cannot be generated by elements of degree one in ∂and then AnnD(1/f)6=e Ilog an . Remark. In the polynomial case we have an analogous theorem to 4.1. Suppose first that f Mlog is isomorphic to Rf. Then by the last remark of Section 2 follows f Mlog an ≃ O[1/f] so fis a quasi-homogeneous polynomial in R. Reciprocally if fis quasi-homogeneous we use Proposition 4.1. 5. An Explicit Example in Dimension 3 Let R=C[x, y, z]. In Calder´on (1997, 4.1.3), the condition of being Koszul-free (that is, the principal symbols of the generators of Ilog an form a regular sequence) is a sufficient condition to assure the existence of the free resolution of Mlog an . We illustrate in this section that this condition is not necessary to have the duality formula. We will consider the surface defined by Calder´on (1997) with h=xy(x+y)(xz +y) = 0. We obtain in this case that: •AnnA3(1/h) = e Ilog. •f Mlog ≃(Mlog)∗. and the results are valid in the analytic case as well. The calculation is as follows: 1. We can compute a basis of Der (R, log h) with a set of generators of the syzygies among h, ∂h ∂x ,∂h ∂y ,∂h ∂z . We obtain δ1=x∂x+y∂y δ2=xz∂z+y∂z δ3=x2∂x−y2∂y−xz∂z−yz∂z with δ1(h) = 4h, δ2(h) = xh, δ3(h) = (2x−3y)h, and  x y 0 0 0 xz +y x2−y2−xz −yz =h. 2. The global b-function of hin A3is b(s) = (4s+ 5)(2s+ 1)(4s+ 3)(s+ 1)3. This polynomial has no integer roots smaller than −1, so Rh≃A3 1 h. 3. We check that AnnA3(1/h) is equal to e Ilog using Groebner bases obtained from the corresponding sets of generators. The computations of the b-function and the annihilating ideal of hshave been made using the algorithms of Oaku (1997), implemented in Maekawa et al. (2000). The same Groebner basis computation shows that Mlog =A3/Ilog (where Ilog = (δ1, δ2, δ3)) is holonomic. 4. We calculate a free resolution of the module Mlog. The first module of syzygies is generated in this case by the relations deduced from the expressions of the [δi, δj] with i6=j: [δ1, δ2] = δ2 [δ1, δ3] = δ3 [δ2, δ3] = −xδ2. The second module of syzygies is generated by only one element s= (s1, s2, s3): s1=−y2∂y+x2∂x−zy∂z−zx∂z−x s2=−y∂z−xz∂z s3=y∂y+x∂x−2. The computation of this free resolution is performed using Groebner bases. 5. With a similar procedure to the one used in 3.1 we obtain that (Mlog)∗is the left A3-module associated with the right A3-module A3/(s1, s2, s3)A3. Then (Mlog)∗≃A3/(st 1, st 2, st 3). 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