On Euler-Homogeneity for free divisors

arXiv:2305.11277v1 [math.AG] 18 May 2023 On Euler-homogeneity for free divisors Abraham del Valle Rodr´ıguez Abstract In 2002, it was conjectured that a free divisor satisfying the so-called Logarithmic Comparison Theorem must be strongly Euler-homogeneous and it was proved for the two-dimensional case. Later, in 2006, it was shown that the conjecture is also true in dimension three, but, today, the answer for the general case remains unknown. In this paper, we use the decomposition of a singular derivation as the sum of a semisimple and a topologically nilpotent derivation that commute in order to deal with this problem. By showing that this decomposition preserves the property of being logarithmic, we are able to give alternative proofs for the low-dimensional known cases. 1 Introduction Let Xbe a complex analytic manifold of dimension nand D⊂Xbe a divisor. Let Ω• X(∗D) be the complex of meromorphic differential forms with poles along Dand Ω• X(log D)⊂Ω• X(∗D) the subcomplex of logarithmic differential forms. We say that the Logarithmic Comparison Theorem (LCT) holds for Dif the inclusion Ω• X(log D)֒→Ω• X(∗D) is a quasi-isomorphism. By Grothendieck’s Comparison Theorem [9, Theorem 2], if we write U=X\Dand j:U ֒→X, then the morphism Ω• X(∗D)→Rj∗CU is a quasi-isomorphism. So, if LCT holds for D, then Ω• X(log D) computes the cohomology of its complement. We denote by DerX(−log D) the OX-module of logarithmic derivations (or vector fields) along D, which is the dual of Ω1(log D). Locally, a derivation δ∈DerX,x belongs to DerX,x(−log D) if δ(f)∈ hfifor any reduced local equation fof Dat x. In this case, we will say that δis a logarithmic derivation for f. Definition 1.1. (See [15].) Dis called a free divisor if DerX(−log D) (or, equivalently, Ω1 X(log D)) is a locally free OX-module. Let x∈Xand consider the stalk DerX,x(−log D). Given a set of derivations S={δ1,...,δn}in DerX,x(−log D), we call A= (δi(xj))i,j the Saito matrix with respect to S(and to the coordinate system). Denoting by ¯ δ= (δ1,...,δn)tand ¯ ∂= (∂1,...,∂n)t, we have that ¯ δ=A¯ ∂. If Dis free, then DerX,x(−log D) is free as an OX,x-module for every x∈X. By Saito’s criterion [15, Theorem 1.8], the freeness of DerX,x(−log D) is equivalent to the existence of some δ1,...,δn∈DerX,x(−log D) such that if f∈ OX,x is a reduced local equation around x, then there exists a unit u∈ OX,x with f=udet(A), where Ais the Saito matrix with respect to δ1,...,δn. In this case, {δ1,...,δn}is a basis of DerX,x(−log D) and we will also say that fis free. 1 Definition 1.2. A germ of holomorphic function f∈ OX,x is called strongly Euler-homogeneous at xif there exists a δ∈mX,x DerX,x such that δ(f) = f(where mX,x denotes the maximal ideal of the local ring OX,x). A divisor Dis called strongly Euler-homogeneous if, for each x∈D, some (and hence, any) reduced local equation of Dat xis strongly Euler-homogeneous. In [4], the authors proposed the following conjecture and proved it in the case n= 2: Conjecture 1.1. If Dis a free divisor in a complex analytic manifold Xof dimension nthat satisfies the Logarithmic Comparison Theorem, then it is strongly Euler-homogeneous. Later, in 2006, M. Granger and M. Schulze proved in [8, Theorem 1.6] that Conjecture 1.1 also holds for n= 3. They did it by proving the so-called formal structure theorem, by which they were able to find a basis of logarithmic derivations formed by diagonal and nilpotent derivations in a suitable system of formal coordinates that satisfy additional properties. The main objective of this article is to give a new proof of the conjecture for these cases with the aim of better understanding the conjecture. We expect that this new perspective could provide us some hints to approach the conjecture in higher dimension. Throughout this paper, we will deal with both formal and convergent objects. Some definitions for a convergent series can be extended in a natural way to formal ones, so Section 2is destinated to present those definitions and see the relations between them. In Section 3we give the definitions of semisimple and topologically nilpotent derivations, whose importance will become clear later on. These concepts, introduced by R. G´erard and A. Levelt in [7] generalize, and make intrinsic the ones given by K. Saito in [14]. They allow them to prove an analogue to the Jordan decomposition theorem, which states that a singular formal derivation can be decomposed in a unique way as the sum of two commuting derivations, where one of them is semisimple and the other is topologically nilpotent. Our main result in this section is that if a singular derivation is logarithmic, then so do its semisimple and topologically nilpotent parts, which allows us to give a basis of the logarithmic derivations in which each element has one of these properties. In Section 4we show that if LCT holds for a free divisor D, then at least one of the logarithmic derivations must have non-zero trace. This was first noticed in [4], but here we give an alternative and more conceptual proof based on D-module theory. Finally, in Section 5, we use the central result in Section 3to show that every non-Euler derivation of a certain type of plane curve must be topologically nilpotent. As a corollary, using the result in Section 4, we get an alternative proof of the conjecture in dimension n= 2. We also give an example that shows that this is no longer true in higher dimension. However, these results allow us, by reproducing part of the arguments given in [8], to give another proof for the case n= 3. While proving the conjecture for n= 2, the authors state in [4] that, for a certain kind of plane curves, there always exists a basis of the logarithmic derivations such that one of them has no linear part. Although this happens to be true, it seems to be not so immediate as they claim, so in Appendix Awe clarify the proof of this statement. We prove that there are only two possibilities and later, after some work, we are able to discard one of them, reaching the desired conclusion. The author would like to thank his PhD advisors Luis Narv´aez Macarro and Alberto Casta˜no Dom´ınguez for their support and guidance while writing this paper. He is supported by a Junta de Andaluc´ıa PIF fellowship num. PREDOC 00485 and also partially supported by PID2020- 114613GB-I00. 2 2 Preliminaries As we will see in Section 3, the Jordan decomposition of a derivation requires to consider formal objects instead of just convergent ones. That is why we need to give some definitions for both convergent and formal series. We are interested in how these properties behave when a convergent series is seen as a formal one. Let us introduce some relevant notation. Consider the local ring of convergent power series O:= OCn,0=C{x1,...,xn}with m=hx1,...,xniits maximal ideal and its completion (with respect to the m-adic topology) b O=C[[x1,...,xn]], which is the ring of formal power series. When working with free divisors, we will always consider reduced equations (i.e. with no repeated factors) in O. This property is preserved when they are seen as elements of b O. Proposition 2.1. Let f∈ O be a reduced convergent power series. Then, it is also reduced as a formal power series. Proof. Since fis reduced, the local ring O/hfiis reduced. The ring Ois excellent [11, Theorem 102], so it is a Nagata ring [11, Theorem 78] and so does O/hfi, since this property is preserved under quotients. But the completion of a local reduced Nagata ring is also reduced [2, AC IX 4.4. Corollaire 1], so b O/hfiis reduced, meaning fis reduced as a formal power series. Let us denote by Der the O-module of C-derivations of O, which is free with a basis given by the partial derivatives {∂1,...,∂n}, and d Der its m-adic completion, which coincides with the b O-module of C-derivations of b Oand is free (as an b O-module) with the same basis. Every derivation δ∈Der can be uniquely decomposed as a sum δ=P∞ i=−1δi, where δi= Pn j=1 aij ∂jwith aij being homogeneous of degree i+ 1 for all j= 1,...,n. Moreover, δ0, which is called the linear part of δ, can be written as xA∂, where Ais a constant matrix and x= (x1,...,xn). The same is true for a formal derivation δ∈d Der. We say δ∈Der (resp. δ∈d Der) is singular if δ−1= 0 or, equivalently, δ∈mDer (resp. δ∈md Der = b md Der). For a convergent power series f∈ O (or a formal power series f∈b O), we will denote by Derfthe O-module ( b O-module) of convergent (formal) logarithmic derivations. Its m-adic completion will be denoted by d Derfand will consist of those derivations δ∈d Der such that δ(f)∈b Of. If f∈ O, we can consider it as a formal power series and the formal Derf(which is an b O-module) is precisely the m-adic completion of the O-module Derf. As in Definition 1.2, we will say that a formal power series f∈b Ois strongly Euler-homogeneous at 0 if there exists a formal (singular) derivation δ∈md Der such that δ(f) = f. Let us see the relation with the convergent definition. Proposition 2.2. Let f∈ O. Then, fis strongly Euler-homogeneous at 0as a convergent power series if and only if so is it as a formal power series. Proof. The direct implication is clear, since mDer ⊂md Der and O ⊂ b O. For the other one, let us note that if fis strongly Euler-homogeneous at 0 as a formal power series, then f∈ b mh∂if, i = 1,...,ni. But, by faithful flatness, O ∩ b mh∂if, i = 1,...,ni=mh∂if, i = 1,...,ni, so f∈mh∂if, i = 1,...,ni. This means that there exists δ∈mDer such that δ(f) = fand fis strongly Euler-homogeneous at 0 as a convergent power series. Another important concept that will be used throughout the paper is that of product. First, we give the geometric definition (which justifies its name) and then we establish its algebraic translation. 3 Definition 2.1. Let (D, x) be a germ of a divisor in a complex analytic manifold Xof dimension n. We will say that (D, x) is a product if there exists some germ of divisor (D′,0) ⊆(Cn−1,0) such that (D, x) is biholomorphic to (D′,0) ×(C,0). If D=V(f) with f∈ OX,x is a product, we will also say that fis a product (at x). The following equivalence is proved in a more general setting in [15, Lemma 3.5]: Lemma 2.1. Let fbe a local equation of a divisor D⊂Xat a point x∈D. Then, fis a product at xif and only if it admits a non-singular logarithmic derivation, that is, if DerX,x(−log D)is not contained in mxDerX,x. Remark. Note that, for f∈ O and with the previous notation, fis a product (at 0) if and only if Derf6⊂ mDer. From now on, we will adopt this as our definition. Definition 2.2. Let f∈b O. We say that fis a product if Derf6⊂ md Der. For a convergent power series, this definition is equivalent to the previous one: Proposition 2.3. Let f∈ O. Then, fis a product as a convergent power series if and only if so is it as a formal power series. Proof. Let us suppose that fis not a product as a convergent series, so that Derf⊂mDer. Taking completions we get d Derf⊂md Der. But d Derfis the formal Derf, so fis not a product as a formal series. Reciprocally, if fis not formally a product, then Derf⊂d Derf⊂md Der. By faithful flatness, md Der ∩Der = mDer, so Derf⊂md Der ∩Der = mDer and fis not convergently a product. The following lemma is well-known, (see for example [8, Lemmata 7.3 and 7.4]), and it will allow us to just consider the case in which the local equation of the divisor is not a product. Lemma 2.2. Let f∈ O. Then, fis a product if and only if there exists a coordinate change such that f=ug for some unit uand some convergent power series gin n−1variables. In this case, freeness and strongly Euler-homogeneity are equivalent for fand g. Moreover, the divisor defined by fsatisfies LCT if and only if so does the divisor defined by gin Cn−1. 3 Semisimple and nilpotent derivations It was K. Saito who, in order to prove the equivalence between strong Euler-homogeneity and local quasihomogeneity for isolated singularities [14, Theorem 4.1], first introduced the notions of a semisimple differential operator and a niloperator. Given a singular derivation δ, K. Saito says that δis semisimple if it only has linear part and its associated matrix is diagonalizable. This definition does depend on the coordinate system and so it is not intrinsic. He says that δis a niloperator if the associated matrix of δ0is nilpotent. Note that this definition does not depend on the coordinate system. R. G´erard and A. Levelt introduce in [7] intrinsic definitions of semisimplicity and nilpotency and those will be the ones that will be used in our discussion. They first define the category Vof complete filtered k-vector spaces as the category whose •objects are k-vector spaces Vequipped with a decreasing filtration F={FiV}i∈Nsuch that every vector space V/FiVis finite dimensional, Ti∈NFiV={0}and Vis complete for the topology defined by F(where Fis a fundamental system of neighbourhoods of 0). 4 •morphisms T:V→Ware k-linear maps such that T(FiV)⊂FiWfor all i∈N. The classical definitions of semisimple and nilpotent endomorphisms are then replaced by these new ones. Definition 3.1. Let V∈Obj(V), a morphism T:V→Vis called: •semisimple if every closed vector space Hstable by Tadmits a supplementary subspace H′ (that is, V=H⊕H′) that is also stable by T. •topologically nilpotent if for all x∈V, the sequence {Tnx}n∈Ntends to 0 (that is, for all i∈Nthere exists n0∈Nsuch that for n≥n0, T nx∈FiV). They state an analogue of the classical Jordan decomposition theorem for the morphisms of this category. Then, they apply it to the case of formal power series and its singular derivations [7, Th´eor`emes 1.5, 2.2 and 2.3]. Although the theorem is valid for every algebraically closed field of characteristic 0, we will formulate it in the case k=C. Theorem 3.1 (Jordan decomposition).Let δbe a singular derivation of b O. Then, there exist two commuting singular derivations δS,δNsuch that δSis semisimple, δNis topologically nilpotent and δ=δS+δN. Moreover, this decomposition is unique and there exists a regular system of parameters of b Oin which δSis diagonal (i.e. δSis of the form Pn i=1 λixi∂i) and the matrix of the linear part of δNis nilpotent. Remark. K. Saito already proved that there exists a coordinate system in which this decomposition holds [14, Theorem 3.1]. Theorem 3.1 shows that, actually, the decomposition holds intrinsically and that, in a particular coordinate system, it takes the desired form by K. Saito. Let us note that a derivation δ∈Der is singular if and only if δ(mk)⊆mkfor all k∈N, so they induce a C-linear map δ(k):O/mk→ O/mkin each quotient. The same property holds for formal singular derivations δ∈md Der in the quotients b O/b mk. The following proposition relates the properties of these maps with those of δ. Proposition 3.1. A derivation δ∈md Der is topologically nilpotent if and only if, for every k∈N, the induced C-linear map δ(k):b O/b mk→b O/b mkis nilpotent. Proof. Suppose δis topologically nilpotent. Let us fix k∈Nand let [a1],...,[am] be a basis of the C-vector space b O/b mk. By definition, for each i∈ {1,...,m}there exists ni∈Nsuch that δni(ai)∈b mk. Let n= max{ni}, so δn(ai)∈b mkfor all i= 1,...,mand (δ(k))n([ai]) = [δn(ai)] = 0. Since (δ(k))nis zero when applied to a basis, it must be the zero map, so δ(k)is nilpotent. Reciprocally, let f∈b Oand k∈N. Since by hypothesis δ(k)is nilpotent, there exists n0∈Nsuch that (δ(k))n0= 0. So δn0(f)∈b mkand δn(f)∈b mkfor all n≥n0. Remark. By [7, Proposition 1.3], we also know that a singular derivation δis semisimple if and only if δ(k)is semisimple (or diagonalizable, since these two concepts are equivalent for finite dimensional complex vector spaces) for all k∈N. In particular, if δ=xA∂ with Asemisimple (definition of semisimplicity for K. Saito), after a linear coordinate change, Ais diagonal and every induced map δ(k)has a diagonal matrix with respect to the canonical basis, so δis semisimple. Corollary 3.1. A derivation δ∈md Der is topologically nilpotent if and only if the matrix of δ0is nilpotent. In particular, singular derivations with no linear part are topologically nilpotent. 5 Proof. If the matrix of δ0is nilpotent, by [8, Lemma 2.4], every induced map δ(k)is nilpotent and we deduce that δis topologically nilpotent by Proposition 3.1. Reciprocally, if δis topologically nilpotent, then δ(2) is nilpotent. The matrix of δ0, call it A, coincides with that of the restriction of δ(2) to b m/b m2with respect to the basis {[x1],...,[xn]}. Since the restriction is still a nilpotent map, its associated matrix, A, is nilpotent. Remark. This result states that the definitions of topologically nilpotent derivation by G´erard- Levelt and niloperator by K. Saito are equivalent. Motivated by Corollary 3.1 and for the sake of simplicity, in what follows, we will remove the word “topologically” when referring to a topologically nilpotent derivation. As we might expect, it is not true that if δ∈mDer with δ0semisimple, then δis semisimple. Just consider δ=x∂x+y2∂y. We have that δ0=x∂xis semisimple, y2∂yis nilpotent and [x∂x, y2∂y] = 0, so δS=x∂xand, by uniqueness, δcannot be semisimple. Corollary 3.2. Let δbe a singular derivation of b O. If we denote by δS,0and δN,0the linear parts of δSand δN, respectively, then δ0=δS,0+δN,0is the Jordan decomposition of δ0, that is, δS,0=δ0,S and δN,0=δ0,N , where δ0,S and δ0,N are, respectively, the semisimple and nilpotent parts of δ0. Proof. By the previous discussion about the restriction of the induced maps to b m/b m2, the matrices of δS,0and δN,0are, respectively, semisimple and nilpotent. Thus, δS,0is semisimple and δN,0is nilpotent. Besides, [δS,0, δN,0] = [δS, δN]0= 0 since δSand δNcommute. So δ0= (δS+δN)0= δS,0+δN,0is written as the sum of a semisimple and a nilpotent derivation that commute. By uniqueness, this must be the Jordan decomposition of δ0. Corollary 3.3. Let f∈b O,f6= 0, and let δ∈md Der be a nilpotent derivation such that δ(f) = bf for some b∈b O. Then b∈b m. Proof. Since δis singular, δ(f)∈b m, so if fis a unit, then it is clear that b∈b m. If f∈b m, let k be such that f∈b mk\b mk+1. It is easy to see by induction that δn(f)≡bn 0f(mod b mk+1) for all n∈N, where b0is the constant coefficient of b. By Proposition 3.1,δ(k+1) is nilpotent, so there exists nwith (δ(k+1))n= 0. But then 0 = (δ(k+1))n([f]) = [δn(f)] = bn 0[f]. Since f /∈b mk+1, [f]6= 0, so b0= 0 and b∈b m. Let us prove that the decomposition of a singular derivation into a semisimple and a nilpotent part preserves the property of being logarithmic. This seems very natural, but we have not found it in the literature. Proposition 3.2. Let f∈b Oand let δbe a singular logarithmic derivation for f. Then δSand δNas in Theorem 3.1 are also (singular) logarithmic derivations for f. Proof. Since δ=δS+δN, we have that the induced map in b O/b mkis δ(k)=δ(k) S+δ(k) N. We know that δ(k) Sis semisimple, δ(k) Nis nilpotent (by the previous proposition) and [δ(k) S, δ(k) N] = [δS, δN](k)= 0(k), so δ(k) Sand δ(k) Nare precisely the semisimple and nilpotent part of δ(k). Each quotient b O/b mkis a C- vector space of finite dimension, so δ(k) Sand δ(k) Ncan be written as polynomials in δ(k)by the Jordan- Chevalley decomposition [1, Cap. VII, Th. 1]. But since δ(f) = af with a∈b O,δ(k)([f]) = [a][f], and taking into account that δ(k)is also a derivation, (δ(k))n([f]) ∈ h[f]ifor all n. So a polynomial in δ(k)maps [f] to an element in h[f]iand we conclude that δ(k) S([f]), δ(k) N([f]) ∈ h[f]i. This means 6 that δS(f), δN(f)∈ hfi+b mkfor all k∈N. But Tk≥1hfi+b mk=hfisince every ideal in a Noetherian local ring is closed for the b m-adic topology [12, Theorem 8.14], so δS(f), δN(f)∈ hfi and δS, δNare logarithmic. Corollary 3.4. If f∈ O is free and Derf⊂mDer (that is, every logarithmic derivation is singular), then there exists a basis of d Derfformed by semisimple and nilpotent derivations. Moreover, fis a formal unit multiplied by the determinant of the Saito matrix with respect to that basis. Proof. Since fis free, Derf⊂mDer is free as an O-module and d Derf⊂md Der is free as an b O- module. Let {δ1,...,δn}be an b O-basis of d Derf. By Proposition 3.2,{δ1,S, δ1,N ,...,δn,S, δn,N }is a generating set of d Derfand, by Nakayama’s Lemma, we can extract a basis Cof d Derffrom this set. For the second part, if Ais the Saito matrix with respect to a convergent basis Band Bis the change-of-basis matrix from Cto B(whose columns are the coordinates of the elements of Cwith respect to B), then C=BtAis the Saito matrix of C. By Saito’s Criterion, there exists a unit usuch that f=udet(A). Since Bis invertible, det(B) is a (formal) unit, so det(A) = det(B)−1det(C) and f=udet(B)−1det(C), where udet(B)−1is a formal unit. Remark. Although we will only use the preceding result in the case of free divisors, the first assertion is also true for non-free divisors if we replace the word “basis” by “generating set”. 4 A necessary condition for LCT in free divisors In [6], the authors show, by constructing two different double complexes and comparing the four associated spectral sequences, that if LCT holds for a free divisor Din Cn, then, for each basis {δ1,...,δn}of Der(−log D), the morphism d1:ˇ Hn−1(V\ {0},OX)→ˇ Hn−1(V\ {0},OX)n [g]7→ ([δ1(g)],...,[δn(g)]) must be injective, where ˇ Hn−1(V\ {0},OX) denotes ˇ Cech cohomology of a sufficiently small Stein neighbourhood Vof 0 with values in OXwith respect to the open cover {Vi={xi6= 0}, i = 1,...,n}. It can be identified with the set of Laurent series with strictly negative powers in every variable. We define the trace of a singular derivation δ, and denote it tr(δ), as the trace of its linear part. Note that this definition is valid for both convergent and formal derivations. Moreover, the trace of a singular convergent derivation coincides with the trace of its corresponding derivation in the completion, since m/m2∼ =b m/b m2. It is easy to see [8, Lemma 7.5] that, if δis singular, then δ 1 x1·...·xn=tr(δ) x1·...·xn. So if LCT holds for Dand it is not a product at 0, then some of the δimust have non-zero trace, as otherwise the class of Qn i=1 x−1 iwould be in the kernel of d1. In this section, we will show that we can reach the same conclusion in an alternative way by using D-module theory. Let D=O[∂1,...,∂n] be the ring of linear differential operators and consider the left ideal Dhx1,...,xni. The following lemma describes the singular derivations lying in this ideal. Lemma 4.1. A singular derivation η=Pn i=1 ai∂ibelongs to the left ideal Dhx1,...,xniif and only if tr(η) = 0. 7 Proof. Let ci=∂ai ∂xi (0), that is, the coefficient of the term xiin ai, so that tr(η) = Pn i=1 ci. By the relation [∂i, ai] := ∂i·ai−ai∂i=∂i(ai) and taking into account that ∂i·ai∈ Dhx1,...,xnisince ai∈m, we have that η=−Pn i=1 cimodulo Dhx1,...,xni. Thus, ηwill belong to Dhx1,...,xniif and only if tr(η) = Pn i=1 ci= 0. In [5, Corollary 4.2], Calder´on and Narv´aez proved that the following statements are equivalent: (1) Dsatisfies LCT. (2) The natural morphism ρ:D L ⊗V0O(D)→ O(∗D) induced by the inclusion O(D)⊂ O(∗D) is an isomorphism in the derived category of bounded complexes of coherent left D-modules, where V0=D(−log D) denotes the ring of logarithmic differential operators, O(∗D) is the O-module of meromorphic functions with poles along Dand O(D) is the submodule of meromorphic functions with poles of order at most 1. We are now able to prove the desired result. Theorem 4.1. If Dis a free divisor in Cnsatisfying LCT and f∈ O is a reduced local equation of Dat 0such that it is not a product, then there exist derivations of Derfwith non-zero trace. Proof. Consider a basis B={δ1,...,δn}of Derf⊂mDer and suppose that every derivation of B has zero trace. By Lemma 4.1,Dhδ1,...,δni ⊂ Dhx1,...,xniand we have a surjective morphism of holonomic D-modules D Dhδ1,...,δni−→ D Dhx1,...,xni−→ 0. By applying the duality functor D(−) = RHomD(−,D)[n]left we get an injective morphism 0−→ DD Dhx1,...,xni−→ DD Dhδ1,...,δni. It is a well-known fact that D Dhx1,...,xniis self-dual (that is, its dual is isomorphic to itself) and that D L ⊗V0O∼ =DD L ⊗V0O(D)[5, Corollary 3.1.2]. Thus, if Dsatisfies LCT, then D L ⊗V0O∼ =D(O(∗D)), meaning D L ⊗V0Ois concentrated in degree 0 and D L ⊗V0O=D ⊗V0O∼ =D ⊗V0 V0 V0hδ1,...,δni∼ =D Dhδ1,...,δni. Applying Dto both sides we get DD Dhδ1,...,δni∼ =DD L ⊗V0O∼ =O(∗D). Finally, we have an injective morphism 0−→ D Dhx1,...,xni−→ O(∗D). 8 But D/Dhx1,...,xnicannot be a submodule of O(∗D) since the first one has O-torsion and the second one does not, so we conclude that at least one derivation of Bmust have non-zero trace. A similar statement holds even if we remove the condition of not being a product: Corollary 4.1. If Dis a free divisor in Cnsatisfying LCT and f∈ O is a reduced local equation of Dat 0, then there exist singular derivations of Derfwith non-zero trace. Proof. The case in which fis not a product is already proved in Theorem 4.1, so we may assume that fis a product. Then, applying inductively Lemma 2.2, we can write f=ug for a certain unit uand a convergent power series gin m < n variables such that it is not a product. Since LCT also holds for the divisor D′defined by gin Cm, by Theorem 4.1, we can find singular logarithmic derivations for gwith non-zero trace. These derivations are also logarithmic for f, so we are done. 5 LCT and Euler-homogenenity for free divisors We begin this section with a technical lemma that is valid in an arbitrary number of variables, although we will only use it in the cases n= 2 and n= 3. Lemma 5.1. Let δ=Pn i=1 λixi∂ibe a logarithmic diagonal derivation for a formal equation f∈b Osuch that δ(f) = cf for some c∈b O. Then, there exist a formal unit uand g∈b Osuch that f=ug and δ(g) = c0g, where c0is the value of cat the origin. Proof. Let us write c=Pαcαxα, where α= (α1,...,αn) runs in Nnand xα=xα1 1·...·xαn n. Note that if αverifies α·λ:= Pn j=1 αjλj6= 0 and we set aα= exp cαxα α·λ, then aαverifies δ(aα) = cαxα·aα. Now, set u=Y α·λ6=0 aα= exp  X α·λ6=0 cαxα α·λ . Note that uis a well-defined formal power series because b:= Pα·λ6=0 cαxα α·λvanishes at the origin, so we can compose the two series and the result is another formal power series that does not vanish at the origin. Thus, uis a formal unit and δ(u) = δ(eb) = ebδ(b) = u·X α·λ6=0 cαxα. Then, we can write f=ug with g=u−1fand δ(g) = δ(u−1)f+u−1δ(f) = −u−1· X α·λ6=0 cαxα f+u−1cf = X α·λ=0 cαxα!g. Let us see that c′:= Pα·λ=0 cαxαmust be c0and we will have the result: Note that δ(c′) = 0, so if we decompose gas a sum of eigenvectors of δ(what can be done in a unique way by [14, Lemma 2.3]), g=Pµ∈Cgµwith δ(gµ) = µgµ, we have δ(g) = X µ∈C µgµ=c′g=X µ∈C c′gµ, 9