A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors

arXiv:1201.3594v4 [math.AG] 25 Jun 2015 A duality approach to the symmetry of Bernstein-Sato polynomials of free divisors Luis Narváez Macarro∗ Departamento de Álgebra & Instituto de Matemáticas (IMUS) University of Sevilla June 2015 Abstract In this paper we prove that the Bernstein-Sato polynomial of any free divisor for which the D[s]-module D[s]hsadmits a Spencer logarithmic resolution satisfies the symmetry property b(−s−2) = ±b(s). This applies in particular to locally quasi-homogeneous free divisors (for instance, to free hyperplane arrangements), or more generally, to free divisors of linear Jacobian type. We also prove that the Bernstein-Sato polynomial of an integrable logarithmic connection Eand of its dual E∗with respect to a free divisor of linear Jacobian type are related by the equality bE(s) = ±bE∗(−s−2). Our results are based on the behaviour of the modules D[s]hsand D[s]E[s]hsunder duality. Keywords: Bernstein-Sato polynomials, free divisors, logarithmic differential operators, Spencer resolutions, Lie-Rinehart algebras, logarithmic connections. MSC: 14F10, 32C38 Introduction In [16] Granger and Schulze proved that the Bernstein-Sato polynomial of any reductive prehomogeneous determinant or of any regular special linear free divisor satisfies the equality b(−s−2) = ±b(s). Their proof is based on Sato’s fundamental theorem for irreducible reductive prehomogeneous spaces. This symmetry property has been also checked for many other examples of linear (see for instance [16] and [32]) and non-linear free divisors (e.g. quasi-homogeneous plane curves and the examples in [27]). In this paper we prove the above symmetry property for free divisors for which the D[s]-module D[s]hsadmits a logarithmic Spencer resolution (see Theorem (4.1) for a precise statement). This hypothesis holds for any free divisor of linear Jacobian type, and so for any locally quasi-homogeneous free divisor (for instance, free hyperplane arrangements or discriminants of stable maps [22, Corollary 6.13] in Mather’s “nice dimensions” [24]; “nice dimensions” are those dimensions of source and target manifolds for which stable proper mappings are dense in the proper mappings). ∗Partially supported by MTM2010-19298, P12-FQM-2696, MTM2013-46231-P and FEDER. 1 The main ingredient of the proof is the explicit description of the D[s]-dual of D[s]hsby means of the logarithmic duality formula in [7, 8]. Let us mention that for any quasi-homogeneous germ h: (Cd,0) →(C,0) with isolated singularity, its reduced Bernstein-Sato polynomial eb(s) = b(s) s+1 satisfies the equality eb(s) = ±eb(−s−d). This result and ours suggest that both are extremal cases of a whole family of “pure” cases where symmetry properties occur with other intermediate shiftings (see Question (6.2)). One can expect even that in the “non-pure” cases, the factors of the Bernstein-Sato polynomial which break the symmetry appear as minimal polynomials of the action of son other D[s]-modules attached to our singularity (see for instance the examples in [28, §3]), possibly related with the microlocal structure. Let us now comment on the content of the paper. In section §1 we recall the different conditions and hypotheses on free divisors we will use throughout the paper. In section §2 we recall the logarithmic Bernstein construction and we study the hypotheses we will need later to prove our main results. In section §3 we apply the duality formula in [8] to describe the D[s]-dual of D[s]hϕ(s), where ϕis a C-algebra automorphism of C[s], under the hypotheses studied in section §2. In section §4 we prove the symmetry property b(−s−2) = ±b(s)under the above hypotheses. The idea of the proof is the following: once we know that the D[s]-dual of D[s]hs(resp. of D[s]hs+1) is concentrated in degree 0and is isomorphic to D[s]h−s−1(resp. to D[s]h−s−2), we can compute the D[s]-dual of the exact sequence 0→D[s]hs+1 →D[s]hs→Q:= (D[s]hs)/D[s]hs+1→0 and deduce that the D[s]-dual of Qis concentrated in degree 1and is isomorphic to D[s]h−s−2/D[s]h−s−1. From here the symmetry property comes up. At the end of the section we give some applications to the logarithmic comparison problem and a characterization of the logarithmic comparison theorem for Koszul free divisors. In section §5 we generalize the above results to the case of integrable logarithmic connections with respect to free divisors of linear Jacobian type. In section §6 we have included some open questions dealing with the relationship between the results of [16] and ours, and with the symmetry properties of (reduced) Bernstein-Sato polynomials in the non-free case. Finally, and for the ease of the reader, we have included an Appendix A with a detailed proof of the duality formula (A.32), and the needed notions and results about Lie-Rinehart algebras. This material includes a simple proof of the associativity law (see Theorem (A.11) and Corollary (A.14) ), the original proof in [8] being unpleasant. I would like to thank Francisco Castro, Michel Granger and Mathias Schulze for useful discussions and comments. I would also like to thank the referees for their comments. 1 Notations and linearity conditions In this paper Xwill denote a complex manifold of pure dimension d,D⊂Xa hypersurface (= divisor), OX[⋆D]the sheaf of meromorphic functions along D, 2 OX(D)the sheaf of meromorphic functions along Dwith poles of order ≤1and DXthe sheaf of linear differential operators with coefficients in OX. On DX[s] we will consider two filtrations: the one induced by the usual order filtration DX[s]i=Di X[s], and the total order filtration Fi TDX[s] = ⊕p+q=iDp Xsq,i≥0. The associated graded rings to both filtrations are isomorphic to (gr DX)[s], but the corresponding gradings on the last sheaf are different. We will also denote by J D⊂OXthe Jacobian ideal of D⊂X, i.e. the coherent ideal of OXwhose stalk at any p∈Xis the ideal generated by h, h′ x1,...,h′ xd, where h∈OX,p is any reduced local equation of Dat pand x1,...,xd∈OX,p is a system of local coordinates centered at p. We recall that Dis a free divisor, in the sense of K. Saito [31], if the coherent OX-module Der C(−log D)of logarithmic vector fields with respect to Dis locally free (of rank d). In such a case we will denote by V X= OX[Der C(−log D)] ⊂DXthe sheaf of logarithmic differential operators with respect to D[4]. (1.1) Definition. (Cf. [37, §7.2]) Let Abe a commutative ring and I⊂Aan ideal. We say that Iis of linear type if the canonical (surjective) map of graded A-algebras SymA(I)→Rees(I)is an isomorphism. In the above definition, if I= (a1,...,ar)and (si1,...,sir),i∈L, is a system of generators of the syzygies of a1,...,ar, to say that the ideal I is of linear type is equivalent to saying that any homogeneous polynomial F(ξ1,...,ξr)∈A[ξ]such that F(a1,...,ar) = 0 is a linear combination with coefficients in A[ξ]of the linear forms si1ξ1+···+sirξr,i∈L. (1.2) Example. ([26, Théorème 1]; see also [6, Proposition 2.4]) An ideal generated by a regular sequence is of linear type. (1.3) Definition. (See [9, Definitions 1.11, 1.14].) (a) We say that the divisor Dis of linear Jacobian type at p∈Dif J D,p ⊂OX,p is of linear type. We say that Dis of linear Jacobian type if it is so at any p∈D. (b) We say that the divisor Dis of differential linear type at p∈Dif for some (and hence for any) reduced local equation h∈OX,p of Dat p, the ideal annDX,p[s]hsis generated by order 1 operators (with respect to the usual or to the total order filtration). We say that Dis of differential linear type if it is so at any p∈D. The following Proposition is proven in [9, Proposition 1.15]. (1.4) Proposition. Any divisor of linear Jacobian type is of differential linear type. (1.5) Definition. (a) We say that the divisor Dis strongly Euler homogeneous if for any p∈Dand for some (and hence for any) reduced local equation h∈OX,p of Dat pthere is a germ of vector field χat pvanishing at psuch that χ(h) = h. (b) We say that the divisor Dis locally quasi-homogeneous if for any p∈D there is a system of local coordinates x= (x1,...,xd)centered at psuch that the germ (D, p)has a reduced weighted homogeneous defining equation (with strictly positive weights) with respect to x. 3 The following theorem has been proven in [6, Theorem 5.6]. (1.6) Theorem. Any locally quasi-homogeneous free divisor is of linear Jacobian type. We do not know any example of a free divisor of linear Jacobian type which is not locally quasi-homogeneous. (1.7) Remark. There is also the notion of Euler homogeneity. Namely, we say that the divisor Dis Euler homogeneous at a point p∈Dif there is a reduced local equation h∈OX,p of Dat pand a germ of vector field χat p (not necessarily vanishing at p) such that χ(h) = h. In that case we also say that his Euler homogeneous. It is clear that if Dis Euler homogeneous at p, then it is also Euler homogeneous at any point q∈Dclose enough to p. Notice that not any local reduced equation of an Euler homogeneous divisor is Euler homogeneous. Notice also that for any divisor D⊂X, which might not be Euler homogeneous, the divisor D′=D×C⊂X′=X×Cis always Euler homogeneous. Nevertheless, a divisor D⊂Xis strongly Euler homogeneous if and only if D′=D×C⊂X′=X×Cis strongly Euler homogeneous. Let us also notice that a divisor D⊂Xis of linear Jacobian type at p∈Dif and only D′=D×C⊂X′=X×Cis of linear Jacobian type at (p, 0) ∈D′. (1.8) Example. Any smooth hypersurface is of linear Jacobian type. More generally, any quasi-homogeneous (with strictly positive weights) isolated singularity is of linear Jacobian type, since any reduced equation hbelongs to the ideal generated by the partial derivatives and so the Jacobian ideal is generated by the regular sequence h′ x1,...,h′ xd(see Example (1.2)). (1.9) Proposition. If the divisor Dis of linear Jacobian type, then it is strongly Euler homogeneous. Proof. Let h∈OX,p be a reduced local equation of (D, p)and x= (x1,...,xd) a system of local coordinates centered at p. We recall the argument in [9, Remark 1.26 (a)] for the ease of the reader. Since hbelongs to the integral closure of the ideal I= (h′ x1,...,h′ xd)(cf. [33, §0.5, 1]), there is a homogeneous polynomial F∈OX,p[s, ξ1, . . ., ξd]of degree m > 0such that F(h, h′ x1,...,h′ xd) = 0 and F(s, 0,...,0) = sm. Let δi= d X j=1 aij ∂ ∂xj ,1≤i≤n be a system of generators of Der(−log D)pand let us write δi(h) = αih. In other words, (−αi, ai1,...,aid),1≤i≤n, is a system of generators of the syzygies of h, h′ x1,...,h′ xd. From our hypothesis, the polynomial Fmust be a linear combination of the polynomials −αis+ai1ξ1+···+aidξd,1≤i≤m with coefficients in OX,p[s, ξ]. Putting ξ1=···=ξd= 0 we deduce that at least one of the αimust be a unit, i.e. h∈(h′ x1,...,h′ xd)and his Euler homogeneous. Once we know that his Euler homogeneous, let us prove that his strongly Euler homogeneous by induction on the ambient dimension d. The case d= 1 is 4 obvious. For d > 1, if the Euler vector field χ(χ(h) = h) does not vanish at p, we can integrate it and prove that (D, X, p)≃(D′×C,Cd−1×C,(0,0)), where (D′,0) ⊂(Cd−1,0) is a germ of a divisor. First, we deduce that D′is of linear Jacobian type, and second, from the induction hypothesis, that D′is strongly Euler homogeneous and so Dis strongly Euler homogeneous too (see Remark (1.7)). Q.E.D. Let us recall that a free divisor Dis said to be Koszul ([4, Definition 4.1.1]) at p∈Dif for some (and hence any) basis δ1,...,δdof Der C(−log D)p, the sequence σ(δ1), . . . , σ(δd)is regular in gr DX,p. It turns out that this property is equivalent to being holonomic in the sense of Saito [15, Theorem 7.4]. The following definition is inspired by [16, Definition 7.1], which only applies to the case of linear free divisors (see also Proposition 7.2 and the subsequent remark in [16]. (1.10) Definition. Assume that Dis a free divisor. We say that Dis strongly Koszul at p∈Dif for some (and hence any) basis δ1,...,δdof Der C(−log D)p and for some (and hence any) reduced equation h∈OX,p of (D, p), the sequence h, σ(δ1)−α1s,...,σ(δd)−αds, with δi(h) = αih, is regular in gr DX,p[s](since the sequence is formed by homogeneous elements, its regularity does not depend on the order). (1.11) Proposition. Assume that Dis a free divisor and p∈D. The following properties are equivalent: (a) Dis of linear Jacobian type at p. (b) Dis strongly Koszul at p. Proof. Let x1, . . . , xd∈O:= OX,p be a system of local coordinates centered at p,h∈Oa reduced local equation of Dat pand J=J D,p = (h, h′ x1,...,h′ xd). Let {δi=Pd j=1 aij ∂ ∂xj}1≤i≤dbe a basis of Der (−log D)p, and let us write δi(h) = αihand σi:= σ(δi) = Pd j=1 aijξj∈gr DX,p =O[ξ]. The family {(−αi, ai1,...,aid)}1≤i≤dis a basis of the syzygies of h, h′ x1,...,h′ xd. (a) ⇒(b): From Proposition (1.9) we know that his Euler homogeneous, i.e. h∈(h′ x1,...,h′ xd), and we can take α1=···=αd−1= 0 and αd= 1. In other words, {(ai1,...,aid)}1≤i≤d−1)is a basis of the syzygies of h′ x1,...,h′ xd. Let ϕ:O[ξ]−→ Rees(J) = O[h′ x1t,...,h′ xdt]be the surjective map of Oalgebras defined by ϕ(ξi) = h′ xit. Since Jis an ideal of linear type, the kernel of ϕis generated by the σi,1≤i≤d−1. So dim O[ξ] (σ1,...,σd−1)= dim Rees(J) = d+ 1 and σ1,...,σd−1is a regular sequence in O[ξ]. On the other hand, since ker ϕ= (σ1,...,σd−1)is a prime ideal and h /∈ ker ϕ, we deduce that h, σ1,...,σd−1is also a regular sequence in O[ξ]and, indeed h, σ1,...,σd−1, σd−sis a regular sequence in O[ξ, s] = gr DX,p[s]and D is strongly Koszul at p. 5 (b) ⇒(a): Assume that Xis a small enough open neighborhood of p,K(1) = (σ1−α1s,...,σd−αds)⊂OX[s, ξ1,...,ξd]and let Kbe the kernel of the canonical graded surjective map Φ : OX[s, ξ1,...,ξd]−→ Rees(J D), s 7→ ht, ξi7→ h′ xit. (1) The homogeneous components of K(1) and Kare coherent OX-modules. Since Dis of linear Jacobian type at any smooth point, we deduce that K/K(1) is supported by the singular locus of D. In particular, for any homogeneous polynomial F∈Kpthere is an N > 0such that hNF∈K(1) p, but h, σ1− α1s,...,σd−αdsis a regular sequence and so F∈K(1) p. We deduce that Kp=K(1) pand Dis of linear Jacobian type at p. Q.E.D. (1.12) Corollary. Assume that Dis a free divisor and p∈D. The following properties are equivalent: (a) Dis strongly Koszul at p. (b) Dis Euler homogeneous at pand for any reduced Euler homogeneous equation h∈OX,p of (D, p)and any basis δ1,...,δdof Der C(−log D)pwith δ1(h) = ···=δd−1(h) = 0 and δd(h) = h, the sequence h, σ(δ1),...,σ(δd−1) is regular in gr DX,p. (c) There is a reduced equation h∈OX,p of (D, p)and a basis δ1,...,δdof Der C(−log D)pwith δ1(h) = ··· =δd−1(h) = 0 and δd(h) = hsuch that the sequence h, σ(δ1),...,σ(δd−1)is regular in gr DX,p. Proof. It is a straightforward consequence of Propositions (1.11) and (1.9). Q.E.D. Let us notice that property (c) in the above corollary appeared as condition (c’) in [35, Corollary 1.8]. The following notion was introduced in [28, page 257] and was called “(GK)”. (1.13) Definition. Assume that Dis a free divisor. We say that Dis weakly Koszul at p∈Dif for some (and hence any) basis δ1,...,δdof Der C(−log D)p and some (and hence any) reduced local equation h∈OX,p of (D, p), the sequence σ(δ1)−α1s,...,σ(δd)−αds, with δi(h) = αih, is regular in gr DX,p[s]. We say that Dis weakly Koszul if it is so at any p∈D. (1.14) Proposition. For a free divisor, the following implications hold: (a) strongly Koszul ⇒Koszul. (b) Koszul ⇒weakly Koszul. Proof. The first implication is a consequence of Proposition (1.11) and [9, Proposition 1.27]. The second one comes from [28, Proposition 2.2.14], [9, Proposition 1.22]. Q.E.D. 6 The example x1x2(x1+x2)(x1+x3x2) = 0 is a weakly Koszul free divisor which is not Koszul [28, Example 3.1] and any non-quasihomogeneous plane curve is a Koszul free divisor non-strongly Koszul (Proposition 2.3.1 in [28]). (1.15) Remark. Let Dbe a free divisor and h∈OX,p a reduced local equation of (D, p). If there is a germ of vector field χat psuch that χ(h) = h(i.e. h is Euler homogeneous), then Dis weakly Koszul at pif and only if for some (and hence any) basis δ1,...,δd−1of germs of vector fields vanishing on h, the sequence σ(δ1), . . . , σ(δd−1)is regular in gr DX,p. 2 Logarithmic–meromorphic comparison for Bernstein modules From now on we assume that h: (Cd,0) →(C,0) is a reduced local equation of a germ of a free divisor (D, 0) ⊂(Cd,0). Let us write for short O=OCd,0,D= DCd,0and V=V Cd,0=O[Der C(−log D)0]⊂D. We consider the logarithmic Bernstein module O[s]hs[9, §1.6], which is a V[s]-submodule of the Bernstein D[s]-module O[s, h−1]hs[2]. Obviously O[s]hsis generated by hsover V[s]and annV[s]hsis the left V[s]-ideal generated by the Lie-Rinehart algebra over (C,O) (see Appendix A) Θh:= {δ−αs |δ∈Der (−log D)0, δ(h) = αh} ⊂ V[s]. The following result generalizes [35, Proposition 4.4] to the non-Euler homogeneous case and completes Proposition (1.11). (2.1) Proposition. With the above hypotheses, the following properties are equivalent: (a) (D, 0) is of differential linear type and weakly Koszul. (b) (D, 0) is strongly Koszul (or equivalently, of linear Jacobian type). Proof. (b) ⇒(a): It is a consequence of Propositions (1.11), (1.4) and (1.14). (a) ⇒(b): We follow Torrelli’s argument in 3 ⇒4 of [35, Proposition 4.4]. Let δ1,...,δdbe a basis of Der C(−log D)0with δi(h) = αihand let us write K= annD[s]hs. It is clear that Θhis freely generated as O-module by δ1− α1s,...,δd−αds. Since (D, 0) is of differential linear type, we have K= D[s]Θh. Since σ(δ1)−α1s,...,σ(δd)−αdsis a regular sequence in gr D[s] = grFT(D[s]), we deduce that σFT(K)is the ideal of grFT(D[s]) generated by σ(δ1)−α1s,...,σ(δd)−αds. We know that the characteristic variety f W= V(σFT(K)) ⊂C×T∗Cdof D[s]hsis irreducible of dimension d+ 1 ([20, §5], [38, Proposition 2.3]). In fact I(f W) = ker Φ, where Φhas been defined in (1). Since Φ(h)6= 0 we deduce that dim V(h, σ(δ1)−α1s,...,σ(δd)−αds) = dim(W∩V(h)) = dand so h, σ(δ1)−α1s,...,σ(δd)−αdsis a regular sequence. Q.E.D. Let us denote by Sp• Θh,V[s]=V[s]⊗U(Θh)Sp• Θh, where the complex Sp• Θhis defined in (A.18). From Proposition 1.21 in [9], we know that Sp• Θh,V[s]becomes 7 aV[s]-free resolution of O[s]hswith the augmentation ε0: Sp0 Θh,V[s]=V[s]→ O[s]hs,ε0(P) = Phs. The proof of the following proposition is clear. (2.2) Proposition. Under the above hypotheses, the following properties are equivalent: (a) The canonical map D[s] L ⊗V[s](O[s]hs)−→ D[s]hsis an isomorphism in the derived category of left D[s]-modules. (b) The divisor Dis of differential linear type at 0and the complex D[s]⊗V[s] Sp• Θh,V[s]is exact in degrees 6= 0. (2.3) Proposition. Any germ of free divisor (D, 0) ⊂(Cd,0) of differential linear type and weakly Koszul at 0satisfies the equivalent properties of Proposition (2.2). Proof. To prove that the complex D[s]⊗V[s]Sp• Θh,V[s]is exact in degrees 6= 0, we filter it in such a way that its graded complex is the Koszul complex associated with the sequence σ(δ1)−α1s,...,σ(δd)−αdswith δ1,...,δda basis of Der C(−log D)0and δi(h) = αih(see [9, Proposition 1.18]). Q.E.D. The following corollary is a particular case of [9, Theorem 3.1]. (2.4) Corollary. Under the above hypotheses, if (D, 0) ⊂(Cd,0) is a germ of a free divisor of linear Jacobian type, then the equivalent properties of Proposition (2.2) hold. Proof. It is clear from Proposition (2.1). Q.E.D. (2.5) Definition. For any polynomial q(s)∈C[s]we define: (1) The q(s)-Bernstein module as the free O[s, h−1]-module O[s, h−1]hq(s)with basis hq(s)endowed with the left D[s]-module structure given by δ·(ahq(s)) = δ(a) + q(s)δ(h)h−1ahq(s) for any δ∈DerC(O). (2) The logarithmic q(s)-Bernstein module as the left V[s]-submodule O[s]hq(s) of the q(s)-Bernstein module O[s, h−1]hq(s). It is clear that O[s]hq(s)is generated by hq(s)over V[s]and annV[s]hq(s)is the left V[s]-ideal generated by the (C,O)-Lie-Rinehart algebra Θh,q(s):= {δ−αq(s)|δ∈Der (−log D)0, δ(h) = αh}. For any C-algebra map ϕ:C[s]→C[s]let us also call ϕits trivial extensions to O[s],O[s, h−1],D[s]and V[s]. For any q(s)∈C[s]the map ϕ:ahq(s)∈O[s, h−1]hq(s)7→ ϕ(a)hϕ(q(s)) ∈O[s, h−1]hϕ(q(s)) (resp. ϕ:ahq(s)∈O[s]hq(s)7→ ϕ(a)hϕ(q(s)) ∈O[s]hϕ(q(s))) is linear over ϕ:D[s]→D[s](resp. over ϕ:V[s]→V[s]): ϕP(s)hq(s)=ϕ(P(s))hϕ(q(s)), P(s)∈D[s]. 8 In particular, ϕD[s]hq(s)⊂D[s]hϕ(q(s)). For any left D[s]-module M, let us call ϕ∗(M) := D[s]⊗ϕMthe scalar extension associated with ϕ:D[s]→D[s]. In ϕ∗(M)one has (Pϕ(Q)) ⊗m= P⊗(Qm)for m∈Mand P, Q ∈D[s]. In a similar way we define ϕ∗(M) := V[s]⊗ϕMfor any left V[s]-module M. Since ϕΘh,q(s)= Θh,ϕ(q(s)), the map eϕ:ϕ∗O[s]hq(s)=V[s]⊗ϕO[s]hq(s)−→ O[s]hϕ(q(s)) induced by ϕ:O[s]hq(s)→O[s]hϕ(q(s)) is an isomorphism of left V[s]-modules: V[s]⊗ϕO[s]hq(s)≃V[s]⊗ϕV[s]/V[s]·Θh,q(s)≃ V[s]/V[s]·ϕΘh,q(s)≃V[s]/V[s]·Θh,ϕ(q(s)) ≃O[s]hϕ(q(s)). It is clear that ϕannD[s]hq(s)⊂annD[s]hϕ(q(s)). If moreover ϕis an automorphism, this inclusion becomes an equality. This shows the following lemma. (2.6) Lemma. If ϕ:C[s]→C[s]is an automorphism, then the map eϕ:ϕ∗D[s]hq(s):= D[s]⊗ϕD[s]hq(s)−→ D[s]hϕ(q(s)) induced by ϕ:D[s]hq(s)→D[s]hϕ(q(s)) is an isomorphism of left D[s]-modules. (2.7) Proposition. Assume that ϕ:C[s]→C[s]is an automorphism of C-algebras. Then, the following properties are equivalent to the properties of Proposition (2.2): (a’) The canonical map D[s] L ⊗V[s]O[s]hϕ(s)−→ D[s]hϕ(s)is an isomorphism in the derived category of left D[s]-modules. (b’) annD[s]hϕ(s)is the left D[s]-ideal generated by Θh,ϕ(s)and the complex D[s]⊗V[s]Sp• Θh,ϕ(s),V[s]is exact in degrees 6= 0, where Sp• Θh,ϕ(s),V[s]is defined in a completely similar way to Sp• Θh,V[s]. Proof. Since ϕis an automorphism, the functors ϕ∗are exact. On the other hand we obviously have ϕ∗D[s]⊗V[s]−≃D[s]⊗V[s]ϕ∗(−)and so ϕ∗D[s] L ⊗V[s]O[s]hs≃D[s] L ⊗V[s]ϕ∗(O[s]hs)≃D[s] L ⊗V[s]O[s]hϕ(s). The equivalence between (a’) and property (a) in Proposition (2.2) comes from Lemma (2.6). The equivalence between (a’) and (b’) comes from the fact that Sp• Θh,ϕ(s),V[s]is a free resolution of the left V[s]-module O[s]hϕ(s). Let us also notice that Sp• Θh,ϕ(s),V[s]≃ϕ∗Sp• Θh,V[s]. Q.E.D. 9 explained in [9, § 3.1], we consider the logarithmic Bernstein-Kashiwara module E[s]hsinside the meromorphic connection E[s, h−1]hs. Corollary 3.2 in [9] tells us that the canonical map DX[s] L ⊗V X[s]E[s]hs→DX[s]E[s]hs is an isomorphism in Db f(D[s]). More generally, for any q(s)∈C[s]we consider the q(s)-Bernstein module associated with E E[s]hq(s):= E[s]⊗O[s]O[s]hq(s)⊂E[s, h−1]hq(s):= E[s, h−1]⊗O[s]O[s, h−1]hq(s) as in Definition (2.5). In the same way as in Lemma (2.6) and Proposition (2.7) we prove that for any automorphism of C-algebras ϕ:C[s]→C[s]the canonical map ϕ∗D[s]E[s]hq(s):= D[s]⊗ϕ(D[s]E[s]hs)→D[s]E[s]hϕ(q(s)) (3) induced by ϕ:O[s, h−1]hq(s)→O[s, h−1]hϕ(q(s)) is an isomorphism of left D[s]- modules. Also, the canonical map D[s] L ⊗V[s]E[s]hϕ(s)−→ D[s]E[s]hϕ(s) is an isomorphism in the derived category of left D[s]-modules. As in Corollary (3.6) we obtain a canonical isomorphism DD[s]E[s]hϕ(s)≃D[s]E∗[s]h−ϕ(s)−1. Recall that the Bernstein-Sato polynomial of Eis defined as the minimal polynomial of the action of son the quotient QE:= D[s]E[s]hs/D[s]E[s]hs+1 and it is denoted by bE(s)[9, Remark 3.5]. The existence of a non-zero bE(s)is a straightforward consequence of the existence of non-trivial Bernstein-Sato functional equations with respect to sections of holonomic D-modules ([21, Theorem 2.7]; see also [25]). Now we are ready to state and prove the announced extension of Theorem (4.1) to the case of arbitrary logarithmic connections. (5.1) Theorem. Let (D, 0) ⊂(Cd,0) be a germ of a free divisor of linear Jacobian type with reduced equation h: (Cd,0) →(C,0) and Ea germ at 0of integrable logarithmic connection with respect to D. Then the Bernstein-Sato polynomials of Eand of its dual E∗are related by the equality bE(s) = ±bE∗(−s−2). Proof. We proceed as in the proof of Theorem (4.1). Let us consider the exact sequence of left D[s]-modules 0→D[s]E[s]hs+1 →D[s]E[s]hs→QE→0. 16 By applying the duality functor Dwe obtain a triangle D(QE)→D(D[s]E[s]hs)→DD[s]E[s]hs+1+1 → in which the second arrow corresponds to the inclusion D[s]E∗[s]h−s−1→ D[s]E∗[s]h−s−2,D(QE)is concentrated in degree 1and there is an exact sequence of left D[s]-modules 0→D[s]E∗[s]h−s−1→D[s]E∗[s]h−s−2→D1(QE)→0. Let ϕ:C[s]→C[s]be the automorphism of C-algebras determined by ϕ(s) = −s−2and let us consider the exact sequence of left D[s]-modules 0→D[s]E∗[s]hs+1 →D[s]E∗[s]hs→QE∗→0. From (3) we deduce an isomorphism ϕ∗(QE∗)≃D1(QE)and so the minimal polynomial of the action of son D1(QE)is ϕ(bE∗(s)) = bE∗(−s−2). On the other hand, the action of son D1(QE)is annihilated by bE(s)and we conclude that bE(s)is a multiple of bE∗(−s−2). In a symmetric way we deduce that bE∗(s)is a multiple of bE(−s−2), or equivalently bE∗(−s−2) is a multiple of bE(s), and so bE(s) = ±bE∗(−s−2). Q.E.D. 6 Open questions The starting point of this paper has been [16] and the observed effect of duality on the Bernstein-Sato polynomial of some examples of integrable logarithmic connections with respect to quasi-homogeneous plane curves [29]. In [16], the authors proved that the Bernstein-Sato polynomial of any regular special linear free divisor (in particular, any reductive linear free divisor), and of any reductive prehomogeneous determinant (these are the non-reduced version of linear free divisors) have the symmetry property b(s) = ±b(−s−2) (see Theorems 3.5 and 5.5 and the definition of the involved notions in [16]). These results and our theorem (4.1) overlap, but they are logically independent. On one hand, the results in [16] only apply to invariants of some prehomogeneous vector spaces. On the other hand, our theorem (4.1) cannot cover either the case of non-reduced reductive prehomogeneous determinants, since it only applies to reduced equations, or the case of reductive linear free divisor, since there are examples of such divisors which are not of differential linear type. Namely, D. Andres and J. Martín-Morales have recently studied the example D={h= 0} ⊂ M3,4=C3×4in [15, Proposition 7.12] by using the techniques in [1] and have found differential operators of order 2annihilating hswhich are not in ann(1) D[s]hs=D[s] annV[s]hs. All the examples of free divisors given in [27] are of linear Jacobian type and so they are covered by Theorem (4.1). (6.1) Question. Is there a common generalization of Theorem (4.1) and Theorems 3.5 and 5.5 in [16]? 17 (6.2) Question. The reduced Bernstein-Sato polynomial eb(s) = b(s) s+1 of any quasi-homogeneous polynomial h:Cd→Cwith an isolated singularity at the origin satisfies the symmetry eb(s) = ±eb(−s−d). This is a consequence of a celebrated result of Malgrange [23], namely that the roots of eb(s)for any isolated singularity are the eigenvalues of the connection on the saturated Brieskorn lattice which equals the usual one precisely in the quasi-homogeneous case. In this case the spectrum and the roots of eb(s)are (up to an overall shift) the same, hence the symmetry of the spectrum gives the symmetry of the roots of eb(s). In the non-quasi-homogeneous isolated singularity case, the spectrum is different from the roots of eb(s), so that the symmetry of the spectrum does not imply the symmetry of the roots of eb(s). Let us notice that in the quasi-homogeneous isolated singularity case, the roots of eb(s)can be explicitly listed in terms of the weights of variables (cf. [38, Corollary 3.9]) and the symmetry can be checked directly. Also, R. Arcadias and the author observed that this symmetry can be obtained by using D-module duality theory. The intersection of Theorem (4.1) and the symmetry property for quasihomogeneous isolated singularities is the case of quasi-homogeneous plane curves. Quasi-homogeneous isolated singularities are of linear Jacobian type and their Jacobian ideal are a complete intersection, and so Cohen-Macaulay. On the other hand, the Jacobian ideal of a singular free divisor is also Cohen-Macaulay of codimension 2. By means of M. Saito’s formula for the reduced b-function of the ThomSebastiani join of two germs, we can give, for any e= 2,...,d, non-trivial examples of irreducible germs h: (Cd,0) →(C,0) such that the following properties hold: (a) the Jacobian ideal Jis of linear type; (b) Jis Cohen-Macaulay; and (c) the equality ebh(s) = ±ebh(−s−e)holds for e=codimension of the singular locus of {h= 0}. Nevertheless, (a) + (b) 6⇒ (c) as shown in Example (6.3). The question of finding general criteria on himplying property (c) and extending the known extreme cases of quasi-homogeneous isolated singularities (e=d) and free divisors of linear Jacobian type (e= 2) seems interesting and remains open. I owe Kari Vilonen the suggestion of checking the following example. (6.3) Example. Let Xbe the vector space of square n×ncomplex matrices and h= det : X→C. It is well known that the Bernstein-Sato polynomial of his given by bh(s) = (s+ 1)(s+ 2) ···(s+n)and so ebh(s) = (s+ 2) ···(s+n). The Jacobian ideal Jof his generated by all the (n−1) ×(n−1) minors of the generic matrix (xij). In particular the singular locus of D=h−1(0) consists of matrices of rank ≤n−2,dim Dsing =n2−4and e= codim Dsing = 4. We know that Jis of linear type [19] and Cohen-Macaulay [3, page 25]. However, the equality ebh(−s−4) = ±ebh(s)only holds for n= 2. (6.4) Question. Corollary (4.2) applies to locally quasi-homogeneous free divisors after [6, Theorem 5.6]), in particular to free hyperplane arrangements. However, the Bernstein-Sato polynomial of a non-free hyperplane arrangement does not satisfy in general the symmetry b(s) = ±b(−s−2). For instance, for 18 h=x1x2x3(x1+x2+x3) = 0 we have bh(s) = (s+ 1)3(s+ 3/2)(s+ 3/4)(s+ 5/4), and bh(s)6=±bh(−s−2). In fact, this symmetry property fails in many other examples of non-free hyperpane arrangements. Can we characterize the hyperplane arrangements whose Bernstein-Sato polynomial satisfy the above symmetry property? Appendix A This appendix contains some basic notions and results about Lie-Rinehart algebras and their modules, a simple proof of the associativity law in [8] (see Theorem (A.11)) and a detailed proof of the duality theorem in [8] (see Theorem (A.32)). To be brief, we have included only the statements and results strictly needed for the proof of the cited results and of Propositions (A.15) and (A.16). There are variants of all these results which are left up to the reader (see Remark (A.17)). Let k→Abe a homomorphism of commutative rings. Let us denote by Derk(A)the A-module of k-linear derivations λ:A→A, which is a left sub-Amodule of Endk(A)closed by the bracket [−,−]. A Lie-Rinehart algebra over (k, A)(or a (k, A)-Lie algebra in [30]) is an A-module Lendowed with a k-Lie algebra structure and an A-linear map ρ: L→Derk(A), called anchor map, which is also a morphism of Lie algebras and satisfies [λ, aλ′] = a[λ, λ′] + ρ(λ)(a)λ′ for λ, λ′∈Land a∈A. To simplify, we write λ(a) := ρ(λ)(a)for λ∈Land a∈A. Let (L, ρ),(L′, ρ′)be two Lie-Rinehart algebras over (k, A). A map of LieRinehart algebras f:L→L′is an A-linear map which is also a k-Lie algebra map and such that ρ′◦f=ρ. It is clear that Derk(A)is a Lie-Rinehart algebra over (k, A)with the identity as anchor map, and that for any Lie-Rinehart algebra (L, ρ),ρis a map of LieRinehart algebras. Let Lbe a Lie-Rinehart algebra over (k, A). A left L-module is an A-module Mendowed with a k-bilinear action (λ, m)∈L×M→λm ∈Msuch that (aλ)m=a(λm),[λ, λ′]m=λ(λ′m)−λ′(λm), λ(am) = a(λm) + λ(a)m for all a∈A,λ, λ′∈Land m∈M. The A-module Abecomes a left L-module with the action (λ, a)∈L×A7→ λ(a)∈A. A right L-module is an A-module Q, where the multiplication by elements of Ais written on the right, endowed with a k-bilinear action (q, λ)∈Q×L→ qλ ∈Qsuch that q(aλ) = (qa)λ, q[λ, λ′] = (qλ)λ′−(qλ′)λ, (qa)λ= (qλ)a−qλ(a) for all a∈A,λ, λ′∈Land q∈Q. 19 Let Lbe a Lie-Rinehart algebra over (k, A)and U=U(L)its enveloping (or universal) algebra (see [30]). It is endowed with an injective ring map A≃ U0֒→Uwith k→Ucentral, and a left A-linear map L→U. Moreover, Uis generated as a ring by Aand the image of L, and it carries a canonical filtration (Ur)r≥0(Uris generated as left or right A-module by all products of elements of Lof length ≤r) such that gr Uis a commutative A-algebra. From the universal property of U(see [17, pgs. 63-64]) we have the following: If Mis an A-module, to give a left L-module structure on Mis equivalent to extending its A-module structure to a left U-module structure. Similarly, if Qis an A-module, to give a right L-module structure on Qis equivalent to extending its A-module structure to a right U-module structure. The surjection p∈U7→ p·1∈Ainduces a canonical isomorphism of left U-modules U/U ·L≃A. (A.1) Example. (a) If Xis a complex smooth manifold, p∈X,k=Cand A=OX,p is the ring of germs at pof holomorphic functions, then the enveloping algebra of the Lie-Rinehart algebra DerC(OX,p)is the ring DX,p of germs at p of linear differential operators with holomorphic coefficients in X. (b) If Xis a complex smooth manifold, D⊂Xis a free divisor, p∈D, k=C,A=OX,p and L=Der (−log D)pis the Lie-Rinehart algebra of germs of logarithmic vector fields with respect to D, then the enveloping algebra of L is the ring of germs of logarithmic differential operators DX,p(−log D), which coincides with the subring of DX,p generated by OX,p and Der (−log D)p(see [4, prop. 2.2.5]). (A.2) Internal operations: In what follows, M, M1, M2,... will denote left U-modules and Q, Q1, Q2,... right U-modules. It is well known that the Amodules M1⊗AM2,HomA(M1, M2)and HomA(Q1, Q2)(resp. Q1⊗AM1and HomA(M1, Q1)) have natural left (resp. right) U-module structures (cf. [18, §2]). (A.3) The natural A-linear maps M1⊗AM2≃M2⊗AM1, A ⊗AM2≃M2≃ HomA(A, M2),[M1⊗AM2]⊗AM3≃M1⊗A[M2⊗AM3]are left U-linear, and the natural A-linear maps Q1⊗AA≃Q1≃HomA(A, Q1),[Q1⊗AM1]⊗AM2≃ Q1⊗A[M1⊗AM2]are right U-linear. The proof of the following lemmas is straightforward. (A.4) Lemma. The natural isomorphisms of A-modules HomA(M1⊗AM2, M3)≃HomA(M1,HomA(M2, M3)),(4) HomA(Q1⊗AM1, Q2)≃HomA(M1,HomA(Q1, Q2)) (5) are left U-linear. (A.5) Lemma. The natural A-linear map Q1⊗AHomA(Q1, Q2)→Q2is right U-linear. Moreover, it is an isomorphisms if Q1is a free A-module of rank 1. If M1is a projective A-module of finite rank, we denote M∗ 1= HomA(M1, A). (A.6) Lemma. Assume that M1is a projective A-module of finite rank. Then, the natural map M∗ 1⊗AM2→HomA(M1, M2)is an isomorphism of left Umodules. 20 (A.7) Lemma. The natural A-linear map M1→HomA(Q1, Q1⊗AM1)is left U-linear. Moreover, it is an isomorphism if Q1is a free A-module of rank 1. (A.8) Lemma. For any left U-modules Mand M′(resp. for any right Umodules Qand Q′), a map h∈HomA(M, M′)(resp. h∈HomA(Q, Q′)) is U-linear if and only if λh = 0 for all λ∈L. Consequently there are canonical isomorphisms HomU(M, M′)≃HomU(A, HomA(M, M′)),HomU(Q, Q′)≃ HomU(A, HomA(Q, Q′)). (A.9) Lemma. The U-linear isomorphism (4) in Lemma (A.4) and Lemma (A.8) induce a k-linear isomorphism β: HomU(M1⊗AM2, M3)−→ HomU(M1,HomA(M2, M3)). (A.10) Lemma. The U-linear isomorphism (5) in Lemma (A.4) and Lemma (A.8) induce a k-linear isomorphism γ: HomU(Q1⊗AM1, Q2)≃ −→ HomU(M1,HomA(Q1, Q2)). Assume now that L0→Lis a map of Lie-Rinehart algebras over (k, A), which induces a ring map U0=U(L0)→U=U(L)between their enveloping algebras. Given a right U-module Qand a left U0-module M0, we know that Q⊗AM0 and Q⊗A[U⊗U0M0]have natural right module structures over U0and U, respectively, and that the map σ:Q⊗AM0→Q⊗A[U⊗U0M0], σ(q⊗m) = q⊗[1 ⊗m] is U0-linear. The following theorem gives the associativity law needed in the proof of the duality theorem (A.32). The proof we give here simplifies the original proof in [8]. (A.11) Theorem. The U-linear map τ: [Q⊗AM0]⊗U0U→Q⊗A[U⊗U0M0] induced by σis an isomorphism of right U-modules. Proof. It is enough to prove that for any right U-module Q′, the induced map τ∗: HomU(Q⊗A[U⊗U0M0], Q′)−→ HomU([Q⊗AM0]⊗U0U, Q′) is an isomorphism, and for that we consider the following commutative diagram HomU([Q⊗AM0]⊗U0U, Q′)τ∗ ←−−−− HomU(Q⊗A[U⊗U0M0], Q′) α∗ 1  y≃γ  y≃by (A.10) HomU0(Q⊗AM0, Q′) HomU(U⊗U0M0,HomA(Q, Q′)) ≃by (A.10)  yγ0≃  yα∗ 2 HomU0(M0,HomA(Q, Q′)) HomU0(M0,HomA(Q, Q′)). 21 where α1:Q⊗AM0−→ [Q⊗AM0]⊗U0Uis the natural (right) U0-linear map defined by α1(ξ) = ξ⊗1for ξ∈Q⊗AM0and α2:M0−→ U⊗U0M0is the natural (left) U0-linear map defined by α2(m) = 1 ⊗mfor m∈M0. Q.E.D. Let us notice that Q⊗AUhas two right U-module structures: the first one comes by scalar extension A→Ufrom the A-module structure on Q(we forget here the right U-module structure on Q): (q⊗p)◦p′:= q⊗(pp′), q ∈Q, p, p′∈U, and the second one comes by (A.2) from the right U-module structure on Qand the left U-module structure on U: (q⊗p)⋆ a := (qa)⊗p=q⊗(ap), q ∈Q, p ∈U, a ∈A, (q⊗p)⋆ λ := (qλ)⊗p−q⊗(λp), q ∈Q, p ∈U, λ ∈L. It is clear that (ξ◦p)⋆ a = (ξ ⋆ a)◦p,(ξ◦p)⋆ λ = (ξ ⋆ λ)◦pfor any ξ∈Q⊗AU, p∈U,a∈Aand λ∈L, and so (ξ◦p)⋆ p′= (ξ ⋆ p′)◦pfor any ξ∈Q⊗AU, p, p′∈U. Let us also notice that both structures induce by scalar restriction two different A-module structures on Q⊗AU: for the first one (q⊗p)◦a=q⊗(pa), and for the second one (q⊗p)⋆ a = (qa)⊗p=q⊗(ap). (A.12) Corollary. Under the above conditions, there is a unique U-linear map τ: (Q⊗AU, ◦)→(Q⊗AU, ⋆)such that τ(q⊗1) = q⊗1for all q∈Q. Moreover, τis an isomorphism, τ(ξ ⋆ p) = τ(ξ)◦pfor all ξ∈Q⊗AUand for all p∈U, and τis involutive. Proof. Let us consider the A-linear map σ:Q→Q⊗AUdefined by σ(q) = q⊗1, where we consider on Q⊗AUthe A-module structure induced by the second right U-module structure. The map σinduces a map of right U-modules τ: (Q⊗AU, ◦)→(Q⊗AU, ⋆) with τ(q⊗p) = (q⊗1) ⋆ p. By applying Theorem (A.11) to L0= 0,U0=A and M0=Awe deduce that τis an isomorphism. For any q∈Qand any a∈Awe have τ(q⊗a) = τ((q⊗1)◦a) = (q⊗1)⋆a = q⊗a, and so τ2(q⊗a) = q⊗a. Let (Ur)r≥0be the canonical filtration on U:Uris the left (or right) Amodule generated by Aand Lr. Let (Γr)r≥0be the filtration on Q⊗AUinduced by (Q⊗AUr)r≥0. It is clear that Γr◦Us⊂Γr+s. It is not difficult to prove that τ(ξ⋆p) = τ(ξ)◦pby induction on deg ξ+deg p. The involutivity of τcomes from the fact that τ2(q⊗p) = τ((q⊗1) ⋆ q) = (q⊗1)◦p=q⊗p. Q.E.D. (A.13) Corollary. If Mis a free left U-module and Qis a right U-module which is free over A, then Q⊗AMis a free right U-module. Proof. Since Mis free, M≃ ⊕i∈IUfor some set I, and Q⊗AM≃ ⊕i∈I(Q⊗A U, ⋆), but (Q⊗AU, ⋆)≃(Q⊗AU, ◦)is a free right U-module because Qis a free A-module. Q.E.D. 22 (A.14) Corollary. Assume that Qis a right U-module which is free over A. Then, for any upper bounded complex M• 0of left U0-modules there is a canonical isomorphism τ: [Q⊗AM• 0] L ⊗U0U→Q⊗A[U L ⊗U0M• 0] in the upper bounded derived category of right U-modules. Proof. Let L•→M• 0be a free resolution. Since Qis a free A-module, we have an isomorphism Q⊗AL•≃ −→ Q⊗AM• 0in the upper bounded derived category of right U0-modules, and we can apply Corollary (A.13) to deduce that Q⊗AL•is an upper bounded complex of free right U0-modules. So we have [Q⊗AM• 0] L ⊗U0U≃[Q⊗AL•] L ⊗U0U≃[Q⊗AL•]⊗U0U and Q⊗A[U L ⊗U0M0]≃Q⊗A[U⊗U0L•]. To conclude, we apply Theorem (A.11). Q.E.D. Let Mbe a left U-module and N0an A-module. The module U⊗AN0is, by scalar extension, a left U-module: p(q⊗n) = (pq)⊗n,p, q ∈U, n ∈N. Let us denote by [U⊗AN0]⊗′ AMthe tensor product with respect to the A-module structure on U⊗AN0induced by its left U-module structure: a([q⊗n]⊗′m) = (a[q⊗n]) ⊗′m= [(aq)⊗n]⊗′m= [q⊗n]⊗′(am) for a∈A, q ∈U, n ∈N0, m ∈M. It has a left U-module structure given by (A.2): λ([q⊗n]⊗′m) = [(λq)⊗n]⊗′m+[q⊗n]⊗′(λm), λ ∈L, q ∈U, n ∈N0, m ∈M. The map σ′:N0⊗AM→[U⊗AN0]⊗′ AM, σ′(n⊗m) = [1 ⊗n]⊗′m is A-linear and induces a U-linear map τ′:U⊗A[N0⊗AM]→[U⊗AN0]⊗′ AM with τ′(p⊗[n⊗m]) = p·σ′(n⊗m). (A.15) Proposition. Under the above hypotheses, the map τ′:U⊗A[N0⊗AM]→[U⊗AN0]⊗′ AM is an isomorphism of left U-modules. Proof. It can be done in a completely similar way to the proof of theorem (A.11), but we use lemma (A.9) instead of lemma (A.10). Q.E.D. In the special case where N0=A, we obtain a reacher statement. The left U-modules U⊗AMand U⊗′ AMare endowed with compatible right U-module structures (A.2): (q⊗m)a=q⊗(am) = (qa)⊗m, (q⊗m)λ= (qλ)⊗m−q⊗(λm), (q⊗′m)p= (qp)⊗′m(remember that (aq)⊗′m=q⊗′(am)) for a∈A, λ ∈L, p, q ∈U, m ∈M. 23 (A.16) Proposition. The left U-linear map τ′:U⊗AM→U⊗′ AMdefined by τ′(p⊗m) = p·(1 ⊗′m)is also right U-linear and so it is an isomorphism of (U;U)-bimodules. Proof. The fact that τ′is an isomorphism of left U-modules comes from Proposition (A.15). The fact that τ′is right U-linear can be proven in a similar way to Corollary (A.12). Q.E.D. (A.17) Remark. Proposition (A.15) is a particular case of a second associativity law U⊗U0[N0⊗AM]≃[U⊗U0N0]⊗′ AMwith L0= 0 and U0=A. Actually, there is a third associativity law [Q0⊗AM]⊗U0U≃[Q0⊗U0U]⊗AM. Both laws and their corresponding corollaries can be proved in a similar way to Theorem (A.11). For that one needs some variants of Lemmas (A.4) – (A.10). We do not need these results in this paper and we skip their proof. From now on, we assume that Lis a Lie-Rinehart algebra over (k, A)which is a free A-module of rank d. By the Poincaré-Birkhoff-Witt theorem [30, th. 3.1] we know that Sym L≃gr U. Let us denote ΩL=L∗= HomA(L, A),Ωn L=VnΩL≡HomA(VnL, A)for n= 0,...,d, and ωL= Ωd L. (A.18) For each left U-module Ewe define the Cartan-Eilenberg-ChevalleyRinehart-Spencer Sp• L(E)complex associated with Eas [9, 1.1.2]: Sp−n L(E) = U⊗A n ^L⊗AE, n = 0,...,d where the left U-module structure comes exclusively from the first factor Uof the tensor product, the differential d−n: Sp−n L(E)→Sp−n+1 L(E)is given by: d−1(P⊗λ⊗e) = (Pλ)⊗e−P⊗(λe), d−n(P⊗(λ1∧ · · · ∧ λn)⊗e) = n X i=1 (−1)i−1Pλi⊗(λ1∧ · · · b λi· · · ∧ λn)⊗e − n X i=1 (−1)i−1P⊗(λ1∧ · · · b λi· · · ∧ λn)⊗(λie) +X 1≤i<j≤n (−1)i+jP⊗([λi, λj]∧λ1∧ · · · b λi···c λj· · · ∧ λn)⊗e, 2≤n≤d, and the augmentation is P⊗e∈U⊗AE= Sp0 L(E)7→ d0(P⊗e) := P·e∈E. In the case E=Athe above complex coincides with the complex defined in [30, §4], which will be simply denoted by Sp• L. Let us denote by g Sp• Land g Sp• L(E)the augmented complexes Sp• L→Aand Sp• L(E)→Erespectively. We quote the following result (cf. [9, Proposition 1.7]): (A.19) Proposition. Under the above hypotheses, if Eis free over A, then Sp• L(E)is a free resolution of the U-module E. For any left U-module M, the complex HomU(Sp• L, M)is canonically isomorphic to the “de Rham” complex Ω• L(M), where Ωn L(M) := HomA(VnL, M)≡ 24 HomA(VnL, A)⊗AM= Ωn L⊗AM,n= 0,...,d, and the differential d: Ωn−1 L(M)→Ωn L(M)is given by (dα)(λ1∧ · · · ∧ λn)) = n X i=1 (−1)i−1λiα(λ1∧ · · · b λi· · · ∧ λn) +X 1≤i<j≤n (−1)i+jα([λi, λj]∧λ1∧ · · · b λi···c λj· · · ∧ λn). For any λ∈Land any nwe have the Lie derivative Lλ: Ωn L(M)→Ωn L(M) (cf. [30], prop. 6.3) defined by (Lλα)(λ1∧ · · · ∧ λn) = λ·α(λ1∧ · · · ∧ λn)− n X i=1 α(λ1∧ · · · ∧ [λ, λi]∧ · · · ∧ λn). The proof of the following proposition can be found in [18], prop. 2.8 and th. 2.10 (see also [7, Proposition 3.1]). It is a generalization of the well known corresponding results in D-module theory. (A.20) Proposition. Under the above hypotheses, the following properties hold: (1) The action (α, λ)∈ωL×L7→ −Lλα∈ωLdefines a right U-module structure on ωL. (2) The complex Ω• L(U)can be augmented through the map α⊗p∈Ωd L⊗AU7→ αp ∈ωL= Ωd Land it becomes a free resolution of the right U-module ωL. (A.21) Corollary. The complex of right U-modules RHomU(A, U)is concentrated in degree dand its d-cohomology Extd U(A, U)is canonically isomorphic to the right U-module ωL. Proof. It is a consequence of the above propositions and the canonical isomorphisms RHomU(A, U)≃HomU(Sp• L, U)≃Ω• L(U). Q.E.D. (A.22) Definition. The right L-module ωLis called the dualizing module of the Lie-Rinehart algebra L. (A.23) Proposition. With the above notations, we have a canonical isomorphism of left U-modules g Sp• L(E)≃g Sp• L⊗′ AEwhere the tensor product ⊗′ Ais taken with respect to the A-module structure on each Sp−n Linduced by its left U-module structure. Proof. From Proposition (A.15) there are canonical isomorphisms α−n: Sp−n L(E) = U⊗A n ^L⊗AE!≃ U⊗A n ^L!⊗′ AE= Sp−n L⊗′ AE for n= 0,...,d. To prove the commutativity of the diagrams U⊗An VL⊗AEα−n −−−−→ U⊗A n VL⊗′ AE d−n  y  yd−n⊗′Id U⊗An−1 VL⊗AEα−n+1 −−−−→ U⊗A n−1 VL⊗′ AE 25