Gevrey solutions of irregular hypergeometric systems in two variables

arXiv:0811.3390v1 [math.AG] 20 Nov 2008 Gevrey solutions of irregular hypergeometric systems in two variables M.C. Fern´andez-Fern´andez and F.J. Castro-Jim´enez ∗ Departamento de ´ Algebra Universidad de Sevilla 20 November 2008 Abstract We describe the Gevrey series solutions at singular points of the irregular hypergeometric system (GKZ system) associated with an affine plane monomial curve. We also describe the irregularity complex of such a system with respect to its singular support and in particular we prove, using elementary methods, that this irregularity complex is a perverse sheaf as assured by a theorem of Z. Mebkhout. Introduction To every row integer matrix A= (a1a2), with positive and relatively prime entries, and every complex parameter β∈Cwe can associate the hypergeometric system MA(β) defined by the following two linear partial differential equations: ∂ ∂x1a2 (ϕ)−∂ ∂x2a1 (ϕ) = 0 a1x1 ∂ϕ ∂x1 +a2x2 ∂ϕ ∂x2 −βϕ = 0. General hypergeometric systems have been introduced by I.M. Gel’fand, M.I. Graev, M.M. Kapranov and A.V. Zelevinsky ([6], [7], [8]) and their analytic solutions, at a generic point in Cn, have been widely studied (see e.g. [7], [8], [1], [19], [18]). In this work we explicitly describe the Gevrey solutions –at singular points– of the hypergeometric system MA(β) associated with A= (a1a2) and β∈C. To this end we will use the Γ–series introduced in [8] and also used in [19] in a very useful and slightly different form. We use these Γ–series to describe the Gevrey filtration of the irregularity complex of MA(β) with respect to the coordinate axes and in particular with respect to the singular support of the system. Despite the simplicity of the equations defining these hypergeometric systems the study of its Gevrey solutions is quite involved. In [5] the study of the Gevrey solutions of the hypergeometric system associated with any affine monomial curve in Cnis reduced to the two dimensional case by using deep results in D–module theory. This justifies our separated treatment for the two variables case. ∗Both authors partially supported by MTM2007-64509 and FQM333. The first author is also supported by the FPU Grant AP2005-2360, MEC (Spain). e.mail addresses: [email protected],[email protected] 1 The behavior of Gevrey solutions of a hypergeometric system (and more generally of any holonomic D–module) is closely related to its irregularity complex as proved by Z. Mebkhout and by Y. Laurent and Z. Mebkhout ([16], [15], [12], [13]). For any hypergeometric system in two variables we will describe its irregularity complex without using any of the deep results in the above references. In particular we will prove (see Conclusions) that the irregularity complex is a perverse sheaf. The paper has the following structure. In Section 1 we recall the definition of Gevrey series. In Section 2 we recall the definition of Γ–series and summarize the description of the holomorphic solutions of MA(β) at a generic point of C2. Section 3 is devoted to the definition –due to Z. Mebkhout [16]– of IrrY(MA(β)), the irregularity complex of the system MA(β) with respect to its singular support Yin C2. In Section 4 we prove that the germ of the irregularity complex IrrY(MA(β)) at the origin is zero. In Section 5 we first prove that the complex IrrY(MA(β))pfor p∈Y,p6= (0,0), is concentrated in degree 0 and then we describe a basis of its 0-th cohomology group. We also prove, by elementary methods, that the irregularity complex IrrY(MA(β)) is a perverse sheaf on Y. This is a very particular case of a theorem of Z. Mebkhout [16, Th. 6.3.3]. Some related results can be found in [17], [20] and also in [14] and [10]. The second author would like to thank N. Takayama for his very useful comments concerning logarithm-free hypergeometric series and for his help, in April 2003, computing the first example of Gevrey solutions: the case of the hypergeometric system associated with the matrix A= (1 2) (i.e. with the plane curve x2−y= 0). The authors would like to thank L. Narv´aez-Macarro and J.M. Tornero for their very useful comments and suggestions. 1 Gevrey series Let us write X=C2with its structure of complex manifold, OX(or simply O) the sheaf of holomorphic functions on Xand DX(or simply D) the sheaf of linear differential operators with coefficients in OX. The sheaf OXhas a natural structure of left DX–module. Let Zbe a hypersurface (i.e. a plane curve) in Xwith defining ideal IZ. We denote by OX|Zthe restriction to Zof the sheaf OX (and we will also denote by OX|Zits extension by 0 on X). Recall that the formal completion of OX along Zis defined as Od X|Z:= lim ←− k OX/Ik Z. By definition Od X|Zis a sheaf on Xsupported on Zand it has a natural structure of left DX–module. We will also denote by Od X|Zthe corresponding sheaf on Z. We denote by QZthe quotient sheaf defined by the following exact sequence 0→ OX|Z−→ O d X|Z−→ QZ→0. The sheaf QZhas then a natural structure of left DX–module. Assume Y⊂Xis a smooth curve and that it is locally defined by x2= 0 for some system of local coordinates (x1, x2) around a point p∈Y. Let us consider a real number s≥1. A germ f=X i≥0 fi(x1)xi 2∈ O d X|Y ,p is said to be a Gevrey series of order s(along Yat the point p) if the power series ρs(f) := X i≥0 1 i!s−1fi(x1)xi 2 is convergent at p. 2 The sheaf Od X|Yadmits a natural filtration by the sub-sheaves Od X|Y(s) of Gevrey series of order s, 1 ≤s≤ ∞ where by definition Od X|Y(∞) = Od X|Y. So we have Od X|Y(1) = OX|Y. We can also consider the induced filtration on QY, i.e. the filtration by the sub-sheaves QY(s) defined by the exact sequence: 0→ OX|Y−→ O d X|Y(s)−→ QY(s)→0 (1.1) Definition 1.1. Let Ybe a smooth curve in X=C2and let pbe a point in Y. The Gevrey index of a formal power series f∈ O d X|Y ,p with respect to Yis the smallest 1≤s≤ ∞ such that f∈ O d X|Y(s)p. 2 The hypergeometric system associated with an affine monomial plane curve We denote by A2(C) or simply A2the complex Weyl algebra of order 2, i.e. the ring of linear differential operators with coefficients in the polynomial ring C[x] := C[x1, x2]. The partial derivative ∂ ∂xiwill be denoted by ∂i. Let A= (a b) be an integer nonzero row matrix and β∈C. Let us denote by EA(β) the linear differential operator EA(β) := ax1∂1+bx2∂2−β. The toric ideal IA⊂C[∂] := C[∂1, ∂2] associated with Ais generated by the binomial ∂b′ 1−∂a′ 2where a′=a/d,b′=b/d and d= gcd(a, b). The algebraic plane curve defined by IAis then an affine monomial plane curve. The left ideal A2IA+A2EA(β)⊂A2is denoted by HA(β) and it is called the hypergeometric ideal associated with (A, β). The (global) hypergeometric module associated with (A, β) is by definition (see [6], [8]) the quotient MA(β) := A2/HA(β). To the pair (A, β) we can also associate the corresponding analytic hypergeometric DX–module, denoted by MA(β), which is the quotient of DXmodulo the sheaf of left ideals in DXgenerated by HA(β). In this paper we will assume that A= (a b) is an integer row matrix with 0 < a < b and β∈C. We can assume without loss of generality that a, b are relatively prime. Nevertheless similar methods to the ones presented here can be applied to different kind of plane monomial curves (see Remark 5.15). The module MA(β) is the quotient of DXmodulo the sheaf of ideals generated by the operators P:= ∂b 1−∂a 2and EA(β) = ax1∂1+bx2∂2−β. Sometimes we will write E=E(β) = EA(β) if no confusion is possible. Although it can be deduced from general results ([6] and [1, Th. 3.9]) a direct computation shows that the characteristic variety of MA(β) is T∗ XX∪T∗ YXwhere Y= (x2= 0) and then the singular support of MA(β) is the axis Y. The module MA(β) is therefore holonomic. 2.1 Holomorphic solutions of MA(β)at a generic point By [8, Th. 2] and [1, Cor. 5.21] the dimension of the vector space of holomorphic solutions of MA(β) at a point p∈X\Yequals b. A basis of such vector space of solutions can be described, using Γ–series ([6], [8, Sec. 1] and [19, Sec. 3.4]) as follows. For v∈C2and u∈Z2let us denote Γ[v;u] := (v)u− (v+u)u+ if (v+u)u+6= 0 and Γ[v;u] := 0 otherwise. Here (z)α=Y i:αi>0 αi−1 Y j=0 (zi−j) 3 is the Pochhammer symbol, for any z∈Cnand any α∈Nn. For j= 0,...,b−1 let us consider vj= (j, β−ja b)∈C2 and the corresponding Γ–series φvj=xvjX m≥0 Γ[vj;u(m)] xb 1 xa 2m ∈xvjC[[x1, x−1 2]] with u(m) = (bm, −am)∈LA= KerZ(A), which defines a holomorphic function at any point p∈X\Y. This can be easily proven by applying d’Alembert ratio test to the series in xb 1 xa 2 ψ:= X m≥0 Γ[vj;u(m)] xb 1 xa 2m . Writing cm:= Γ[vj;u(m)] we have lim m→∞  cm+1 cm= lim m→∞ (am)a (bm)b= 0. Notice that φvj6∈ OX(X\Y) if β−ja b6∈ Z. 3 Gevrey solutions of MA(β) The definition of the irregularity (or the irregularity complex) of a left coherent DX–module Mhas been given by Z. Mebkhout [16, (2.1.2), page 98 and (6.3.7)]. In dimension 2 this definition is the following: Definition 3.1. (a) Let Zbe a curve in X. The irregularity of Malong Z(denoted by IrrZ(M)) is the solution complex of Mwith values in QZ, i.e. IrrZ(M) := RHomDX(M,QZ). (b) If Yis a smooth curve in Xand for each 1≤s≤ ∞, the irregularity of order sof Mwith respect to Yis the complex Irr(s) Y(M) := RHomDX(M,QY(s)). Since Od X|Y(∞) = Od X|Ywe have Irr(∞) Y(M) = IrrY(M). By definition, the irregularity of Malong Z(resp. Irr(s) Y(M)) is a complex in the derived category Db(CX) and its support is contained in Z (resp. in Y). In Sections 4 and 5 we will describe the cohomology of the irregularity complex IrrY(MA(β)), and moreover we will compute a basis of the vector spaces Hi(Irr(s) Y(MA(β)))p=Exti DX(MA(β),QY(s))p for p∈Y,i∈Nand 1 ≤s≤ ∞. Lemma 3.2. A free resolution of MA(β)is given by 0−→ D ψ1 −→ D2ψ0 −→ D π −→ MA(β)−→ 0 (3.1) where ψ0is defined by the column matrix (P, E)t,ψ1is defined by the row matrix (E+ab, −P)and πis the canonical projection. 4 Remark 3.3. For any left DX–module Fthe solution complex RHomDX(MA(β),F)is represented by 0−→ F ψ∗ 0 −→ F ⊕ F ψ∗ 1 −→ F −→ 0 where ψ∗ 0(f) = (P(f), E(f)) and ψ∗ 1(f1, f2) = (E+ab)(f1)−P(f2)for f, f1, f2local sections in F. 4 Description of IrrY(MA(β))(0,0) Theorem 4.1. With the previous notations we have Irr(s) Y(MA(β))(0,0) = 0 for all β∈Cand 1≤ s≤ ∞. In other words Exti(MA(β),QY(s))(0,0) = 0 for all β∈C,1≤s≤ ∞ and i∈N. Here the Ext groups are taken over the sheaf of rings DX. Previous result is related to [17, Th. 1]. Let us denote by VA(β, s) and WA(β, s) the vector spaces (X α∈N2 aαxα∈ O d X|Y(s)(0,0) :aα= 0 if Aα =β),(X α∈N2 aαxα∈ O d X|Y(s)(0,0) :aα= 0 if Aα 6=β) respectively. Notice that VA(β, s) = Od X|Y(s)(0,0) if and only if β6∈ aN+bN. Lemma 4.2. i) The C–linear map EA(β) : VA(β, s)→VA(β, s)is an automorphism for all 1≤s≤ ∞ and β∈C. In particular, if β /∈aN+bNthen EA(β)is an automorphism of Od X|Y(s)(0,0) for all 1≤s≤ ∞. ii) The C–linear map P:WA(β, s)→WA(β−ab, s)is surjective for all 1≤s≤ ∞ and β∈C. Proof. Part ii) is obvious. Let’s prove part i). We have EA(β) = ax1∂1+bx2∂2−βthen for f=Pα∈N2fαxα∈C[[x1, x2]] we have EA(β)(f) = X α∈N2 fα(Aα −β)xα. This implies that EA(β) is an automorphism of VA(β, ∞). It is also clear that EA(β) is an automorphism of VA(β, 1). For any 1 < s < ∞we have ρsEA(β) = EA(β)ρsand then EA(β) is an automorphism of VA(β, s) (see Section 1 for the definition of ρs). Corollary 4.3. EA(β)is an automorphism of the vector space QY(s)(0,0) for 1≤s≤ ∞ and β∈C. Proof. [Theorem 4.1] Let us simply write E:= EA(β). The complex Irr(s) Y(MA(β))(0,0) is represented by the germ at (0,0) of the following complex 0−→ QY(s)ψ∗ 0 −→ QY(s)⊕ QY(s)ψ∗ 1 −→ QY(s)−→ 0 where ψ∗ 0(f) = (P(f), E(f)) and ψ∗ 1(f1, f2) = (E+ab)(f1)−P(f2) for f, f1, f2germs in QY(s) (see Remark 3.3). In particular, we only need to prove the statement for i= 0,1,2. For i= 0,2 the statement follows from Corollary 4.3 Let us see the case i= 1. Let us consider (f, g)∈Ker(ψ∗ 1)(0,0) (i.e. (E+ab)(f) = P(g)). We want to prove that there exists h∈ QY(s)(0,0) such that P(h) = fand E(h) = g, where ( ) means modulo OX|Y,(0,0) =C{x}. From Corollary 4.3 we have that there exists a unique h∈ QY(s)(0,0) such that E(h) = g. Since PE = (E+ab)Pand (E+ab)(f) = P(g) we have: (E+ab)(f) = P(g) = P(E(h)) = (E+ab)(P(h)). Since for all β∈C,E+ab =EA(β−ab) is an automorphism of QY(s)(0,0) (see Corollary 4.3) we also have f=P(h). So (f, g) = (P(h), E(h)) ∈Im(ψ∗ 0)(0,0). 5 Remark 4.4. From Theorem 4.1 and the long exact sequence of cohomology associated with the exact sequence (1.1) we have Exti(MA(β),OX|Y)(0,0) ≃ Exti(MA(β),Od X|Y(s))(0,0) for 1≤s≤ ∞,i∈Nand β∈C. In fact we have the following two propositions. Proposition 4.5. With the previous notations we have Exti(MA(β),Od X|Y(s))(0,0) = 0 for all β6∈ aN+bN,1≤s≤ ∞ and i∈N. Proof. The proof is similar to the one of Theorem 4.1 because E=EA(β) is an automorphism of Od X|Y(s))(0,0). Proposition 4.6. With the previous notations we have dimC(Exti(MA(β),Od X|Y(s))(0,0)) = 1if i= 0,1 0if i≥2 for all β∈aN+bNand 1≤s≤ ∞. Moreover, Exti(MA(β),Od X|Y(s))(0,0) is generated by a polynomial φvqwhen i= 0 and by the class of (0, φvq)when i= 1 (see the proof of Lemma 5.1 for the definition of φvq). Proof. By Remark 3.3 it is enough to consider i= 0,1,2. Let’s treat first the case i= 2. Let h∈ O d X|Y(s)(0,0) and write h=h1+h2with h1∈VA(β−ab, s) and h2∈WA(β−ab, s). From Lemma 4.2 there exist f∈VA(β−ab, s) and g∈WA(β, s) such that (E+ab)(f) = h1and P(g) = −h2. Then (E+ab)(f)−P(g) = h1+h2=h. Let’s see now that the Ext0has dimension 1. Assume that h∈ O d X|Y(s)(0,0) satisfies P(h) = E(h) = 0 and let’s write h=h1+h2with h1∈VA(β, s) and h2∈WA(β, s). We have E(h2) = 0 and then E(h) = E(h1) = 0 implies h1= 0 because of Lemma 4.2. Now, from P(h) = P(h2) = 0 we get h2=λφvqfor some λ∈C(see Proposition 5.4). Finally, let’s prove that the Ext1has dimension 1. Let’s consider (f, g)∈ O d X|Y(s)(0,0) such that (E+ab)(f) = P(g). Let’s write f=f1+f2, g =g1+g2with f1∈VA(β−ab, s), f2∈WA(β−ab, s), g1∈ VA(β, s) and g2∈WA(β, s). As (E+ab)(f2) = 0 we have (E+ab)(f) = (E+ab)(f1) = P(g1)+P(g2). This implies P(g2) = 0 since (E+ab)(f1) and P(g1) belong to VA(β−ab, s). By Lemma 4.2 there exists h1∈VA(β, s) such that E(h1) = g1. We also have (E+ab)(f1−P(h1)) = (E+ab)(f1)−PE(h1) = 0 and again by Lemma 4.2 we have (f1, g1) = (P(h1), E(h1)). By Lemma 4.2 there exists h2∈WA(β, a) such that P(h2) = f2. So, (f2, g2)−(P(h2), E(h2)) = (0, g2) = λ(0, φvq) for some λ∈Csince P(g2) = 0 (see Proposition 5.4). 5 Description of IrrY(MA(β))pfor p∈Y,p6= (0,0) We will compute a basis of the vector space Exti(MA(β),QY(s))pfor 1 ≤s≤ ∞,i∈N,p∈Y, p6= (0,0). In this section we are writing p= (ǫ, 0) ∈Ywith ǫ∈C∗. We are going to use Γ–series following ([6], [8, Section 1]) and in the way they are handled in [19, Section 3.4]. We will consider the family vk= (β−kb a, k)∈C2for k= 0,...,a−1. They satisfy Avk=βand the corresponding Γ–series are φvk=xvkX m≥0 Γ[vk;u(m)]x−bm 1xam 2∈xvkC[[x−1 1, x2]] 6 where u(m) = (−bm, am) for m∈Z. Although φvkdoes not define in general any holomorphic germ at (0,0) it is clear that it defines a germ φvk,p in Od X|Y ,p for k= 0,1,...,a−1. Let us write x1=t1+ǫand remind that ǫ∈C∗. We have φvk,p = (t1+ǫ)β−bk axk 2X m≥0 Γ[vk;u(m)](t1+ǫ)−bmxam 2. Lemma 5.1. 1. If β∈aN+bNthen there exists a unique 0≤q≤a−1such that φvqis a polynomial. Moreover, the Gevrey index of φvk,p ∈ O d X|Y ,p is b afor 0≤k≤a−1and k6=q. 2. If β6∈ aN+bNthen the Gevrey index of φvk,p ∈ O d X|Y ,p is b afor 0≤k≤a−1. Proof. The notion of Gevrey index is given in Definition 1.1. Let assume first that β∈aN+bN. Then there exists a unique 0 ≤q≤a−1 such that β=qb +aN. Then for m∈Nbig enough β−qb a−bm is a negative integer and the coefficient Γ[vq;u(m)] is zero. So φvqis a polynomial in C[x1, x2] (and then φvq,p(t1, x2) is a polynomial in C[t1, x2]) since for β−qb a−bm ≥0 the expression xvqx−bm 1xam 2 is a monomial in C[x1, x2]. Let us consider an integer number kwith 0 ≤k≤a−1. Assume β−bk a6∈ N. Then the formal power series φvk,p(t1, x2) is not a polynomial. We will see that its Gevrey index is b/a. It is enough to prove that the Gevrey index of ψ(t1, x2) := X m≥0 Γ[vk;u(m)](t1+ǫ)−bmxam 2=X m≥0 Γ[vk;u(m)] xa 2 (t1+ǫ)bm is b/a, i.e. the series ρs(ψ(t1, x2)) = X m≥0 Γ[vk;u(m)] (am)!s−1xa 2 (t1+ǫ)bm is convergent for s=b/a and divergent for s < b/a. Considering ρs(ψ(t1, x2)) as a power series in (xa 2/(t1+ǫ)b) and writing cm:= Γ[vk;u(m)] (am)!s−1 we have that lim m→∞  cm+1 cm= lim m→∞ (bm)b (am)as and then by using the d’Alembert’s ratio test it follows that the power series ρs(ψ(t1, x2)) is convergent for b≤as and divergent for b > as. Proposition 5.2. We have dimCExt0(MA(β),Od X|Y)p=afor all β∈C,p∈Y\ {(0,0)}. Proof. Recall that p= (ǫ, 0) with ǫ∈C∗. The operators defining MA(β)pare (using coordinates (t1, x2)) P=∂b 1−∂a 2and Ep(β) := at1∂1+bx2∂2+aǫ∂1−β. We will simply write Ep=Ep(β). First of all, we will prove the inequality dimCExt0(MA(β),Od X|Y)p≤a. 7 Assume that f∈C[[t1, x2]], f6= 0, satisfies Ep(f) = P(f) = 0. Then choosing ω∈R2 >0such that aω2> bω1, we have in(−ω,ω)(Ep) = aǫ∂1and in(−ω,ω)(P) = ∂a 2, where in(−ω,ω)(−) stands for the initial part with respect to the weights weight(xi) = −wi,weight(∂i) = wi. Then (see [19, Th. 2.5.5]) ∂1(inω(f)) = ∂a 2(inω(f)) = 0. So, inω(f) = λlxl 2for some 0 ≤l≤a−1 and some λl∈C. This implies the inequality. Now, remind that φvk,p = (t1+ǫ)β−bk axk 2X m≥0 Γ[vk;u(m)](t1+ǫ)−bmxam 2 and that the support of such a formal series in C[[t1, x2]] is contained in N×(k+aN) for k= 0,1,...,a−1. Then the family {φvk,p |k= 0,...,a−1}is C-linearly independent and they all satisfy the equations defining MA(β)p. Proposition 5.3. If β /∈aN+bNthen Ext0(MA(β),Od X|Y(s))p=Pa−1 k=0 Cφvk,p if s≥b a 0if s < b a for all p= (ǫ, 0) ∈C∗× {0}. Proof. From the proof of Proposition 5.2 and Lemma 5.1 it follows that any linear combination Pa−1 k=0 λkφvk,p with λk∈Chas Gevrey index equal to b/a if β /∈aN+bN. Proposition 5.4. If β∈aN+bNthen Ext0(MA(β),Od X|Y(s))p=Pa−1 k=0 Cφvk,p if s≥b a Cφvqif s < b a for all p= (ǫ, 0) ∈C∗× {0}where qis the unique k∈ {0,1,...,a−1}such that β∈kb +aN. Proof. The proof is analogous to the one of Proposition 5.3 and follows from Lemma 5.1. Lemma 5.5. The germ of E:= EA(β)at any point p= (ǫ, 0) ∈C∗× {0}induces a surjective endomorphism on Od X|Y(s)pfor all β∈C,1≤s≤ ∞. Proof. We will prove that Ep:Od X|Y(s)(0,0) −→ O d X|Y(s)(0,0) is surjective (using coordinates (t1, x2). It is enough to prove that F:= ∂1+bx2u(t1)∂2−βu(t1) yields a surjective endomorphism on Od X|Y(s)(0,0), where u(t1) = (a(t1+ǫ))−1∈C{t1}. For s= 1, the surjectivity of Ffollows from Cauchy-Kovalevskaya theorem. To finish the proof it is enough to notice that ρs◦F=F◦ρsfor 1 ≤s < ∞. For s=∞ the result is obvious. Corollary 5.6. We have: i) Ext2(MA(β),Od X|Y(s))p= 0 for all p∈Y,p6= (0,0),β∈Cand 1≤s≤ ∞. ii) Ext2(MA(β),QY(s))p= 0 for all p∈Y,p6= (0,0),β∈Cand 1≤s≤ ∞. Proof. i) We first consider the germ at pof the solution complex of MA(β) as described in Remark 3.3 for F=Od X|Y(s). Then we apply that E+ab is surjective on Od X|Y(s)p(Lemma 5.5). ii) It follows from i) and the long exact sequence in cohomology associated with (1.1). 8 5.1 Computation of Ext0(MA(β),QY(s))pfor p∈Y,p6= (0,0) Lemma 5.7. Assume that f∈C[[t1, x2]] satisfies Ep(f) = 0. Then f=Pa−1 k=0 f(k)where f(k)=X m≥0 fk+am(t1+ǫ)β−bk a−bmxk+am 2 with fk+am ∈C. Proof. Let us sketch the proof. We know that in(−ω,ω)(Ep)(inω(f)) = 0 [19, Th. 2.5.5] for all ω= (ω1, ω2)∈R2 ≥0. If ω1>0 then in(−ω,ω)(Ep) = aǫ∂1and so, inω(f)∈C[[x2]] for all ωwith ω1>0. On the other hand, if w1= 0 then in(−ω,ω)(Ep) = Epand in particular Ep(in(0,1)(f)) = 0 and inω(in(0,1)(f)) ∈C[x2], for all ω∈R2 >0. There exists a unique (k, m) with k∈ {0,...,a−1}and m∈Nsuch that in(0,1)(f) = xam+k 2h(t1) for some h(t1)∈C[[t1]] with h(0) 6= 0. There exists fam+k∈C∗such that t1divides in(0,1)(f)−fam+k(t1+ǫ)β−bk a−bmxk+am 2∈C[[t1]]xam+k 2. But we have Ep(in(0,1)(f)−fam+k(t1+ǫ)β−bk a−bmxk+am 2) = 0. This implies that in(0,1)(f) = fam+k(t1+ǫ)β−bk a−bmxk+am 2. We finish by induction by applying the same argument to f−in(0,1)(f) since Ep(f−in(0,1)(f)) = 0. Let’s recall that Y= (x2= 0) ⊂X=C2and vk= (β−bk a, k) for k= 0,...,a−1. Remark 5.8. As in the proof of Lemma 5.1 if β∈aN+bNthen there exists a unique 0≤q≤a−1 such that β∈qb +aN. Let us write m0=β−qb a. The series φvqis in fact a polynomial in C[x1, x2]since for m0−bm ≥0the expression xvqx−bm 1xam 2 is a monomial in C[x1, x2]. Let us write m′the smallest integer number satisfying bm′≥m0+ 1 and e vq:= vq+u(m′) = (m0−bm′, q +am′). Let us notice that Ae vq=βand that e vqdoes not have minimal negative support (see [19, p. 132-133]) and then the Γ–series φf vqis not a solution of HA(β). We have φf vq=x f vqX m∈N;bm≥m0+1 Γ[ e vq;u(m)]x−bm 1xam 2. It is easy to prove that HA(β)p(φf vq,p)⊂ OX,p for all p= (ǫ, 0) ∈Xwith ǫ6= 0, and that φf vq,p is a Gevrey series of index b/a. Theorem 5.9. For all p∈Y\ {(0,0)}and β∈Cwe have dimC(Ext0(MA(β),QY(s))p) =    aif s≥b/a 0if s < b/a Moreover, we also have 9