Linear nested Artin approximation theorem for algebraic power series

arXiv:1511.09275v5 [math.AC] 29 Mar 2018 LINEAR NESTED ARTIN APPROXIMATION THEOREM FOR ALGEBRAIC POWER SERIES FRANCISCO-JES´ US CASTRO-JIM´ ENEZ, DORIN POPESCU, GUILLAUME ROND Abstract. We give an elementary proof of the nested Artin approximation Theorem for linear equations with algebraic power series coefficients. Moreover, for any Noetherian local subring of the ring of formal power series, we clarify the relationship between this theorem and the problem of the commutation of two operations for ideals: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals. In particular we prove that a Grothendieck conjecture about morphisms of analytic/formal algebras and Artin’s question about linear nested approximation problem are equivalent. 1. Introduction The aim of the paper is to investigate the nested Artin approximation problem for linear equations. Namely the nested Artin approximation problem is the following: if F(x, y) = 0 is a system of algebraic or analytic equations which are linear in y, with x= (x1,...,xn) and y= (y1,...,ym), and if y(x) is a formal power series solution F(x, y(x)) = 0 with the property that (1.1) yi(x) depends only on the variables x1,...,xσi for some integers σi, is it possible to find algebraic or analytic solutions satisfying (1.1)? D. Popescu [Po86] proved the nested Artin approximation theorem for any vector F(x, y) of algebraic power series (not necessarily linear in y). In this paper we give an elementary proof of this theorem when F(x, y) is linear in y. Moreover we provide a characterization for a certain class of germs of functions F(x, y), linear in y, to satisfy the nested Artin approximation property. From an example of A. Gabrielov [Ga71] we know that the answer to the nested Artin approximation problem is negative for the ring of convergent power series. 1991 Mathematics Subject Classification. Primary : 13B40, Secondary : 13J05, 14B12. Key words and phrases. Henselian rings, Algebraic power series rings, Nested Artin approximation property. F.-J. Castro-Jim´enez was partially supported by Ministerio de Econom´ıa y Competitividad (MTM2013-40455-P and Feder). D. Popescu was partially supported by the project ID-PCE-2011- 1023, granted by the Romanian National Authority for Scientific Research, CNCS - UEFISCDI. G. Rond was partially supported by ANR projects STAAVF (ANR-2011 BS01 009) and SUSI (ANR-12-JS01-0002-01). 1 2 F. J. CASTRO-JIM´ ENEZ, D. POPESCU, G. ROND In order to explain the situation let us consider the following theorem (proven by M. Artin in characteristic zero and by M. Andr´e in positive characteristic): Theorem 1.1. [Ar68][An75] Let kbe a complete valued field and let F(x, y)be a vector of convergent power series in two sets of variables xand y. Assume given a formal power series solution by(x)vanishing at 0, F(x, by(x)) = 0. Then, for any c∈N, there exists a convergent power series solution ey(x), F(x, ey(x)) = 0 which coincides with by(x)up to degree c, ey(x)≡by(x)modulo (x)c. Then M. Artin (see [Ar71, p.7]) asked, whether or not, given a formal solution by(x) = (by1(x),...,bym(x)) satisfying byj(x)∈kJx1,...,xσjK∀j for some integers σj∈ {1,...,n}, there exists a convergent solution ey(x) as in Theorem 1.1 such that eyj(x)∈k{x1,...,xσj} ∀j. Shortly after, A. Gabrielov [Ga71] gave an example showing that the answer to Artin’s question is negative in general. On the other hand since Theorem 1.1 remains valid if we replace convergent power series by algebraic power series (cf. [Ar69]) the question of M. Artin is also relevant in this context and in this case this question has a positive answer. Let us recall that a formal power series f(x)∈kJx1,...,xnKis called algebraic if it is algebraic over the ring of polynomials k[x1,...,xn]. The ring of algebraic power series is denoted by khx1, . . . , xni. Indeed after A. Gabrielov gave a negative answer to Artin’s question, D. Popescu showed that it has a positive answer in the case the ring of convergent power series is replaced by the ring of algebraic power series: Theorem 1.2. [Po86] Let kbe a field and F(x, y)be a vector of algebraic power series in two sets of variables xand y. Assume given a formal power series solution by(x) = (by1(x),...,bym(x)) vanishing at 0, F(x, by(x)) = 0. Moreover let us assume that byj(x)∈kJx1,...,xσjK,1≤j≤m, for some integers σj,1≤σj≤n. Then for any c∈Nthere exists an algebraic power series solution ey(x)such that for all j,eyj(x)∈khx1,...,xσjiand ey(x)−by(x)∈(x)c. Let us remark that if F(x, y) is a vector of polynomials in ywith coefficients in khxi we may drop the condition that by(x) vanishes at 0 by replacing F(x, y) (resp. by(x)) by F(x, y +by(0)) (resp. by(x)−by(0)). This result has a large range of applications (see [FB12], [Mir12] or [Sh10] for some recent examples). Its proof relies on an idea of Kurke from 1972 and the Artin approximation property of rings of type kJxKhzi based on the General N´eron Desingularization Theorem which is quite involved (see [Po86] or [Sw98]). The first goal of this paper is to provide a new and elementary proof of Theorem LINEAR NESTED ARTIN APPROXIMATION 3 1.2 for equations F(x, y) = 0 which are linear in y(see Theorem 3.1). This shows that Theorem 1.2 is really easier in the case F(x, y) is linear in y. Let us mention that in the case where there is only one nest (i.e. when there is a given k≤nsuch that σi=kor nfor every i) this has been already proven by E. Bierstone and P. Milman (see Theorem 12.6 [BM87]). In fact our proof is based on a reduction to this case. In the second part of this paper we investigate the relationship between Artin’s question and the following conjecture of A. Grothendieck (see [Gr60, p. 13-08]): If ϕ:C{x}/I −→ C{y}/J is an injective morphism of analytic algebras then the corresponding morphism bϕ:CJxK/ICJxK−→ CJyK/JCJyKis again injective. In fact the counterexample of A. Gabrielov to Artin’s question is built from a counterexample to the conjecture of A. Grothendieck he gave in [Ga71]. Even if it is obvious that the counterexample of Gabrielov to Grothendieck’s conjecture provides a negative answer to the question of M. Artin, the relationship between these two problems is not clear in general. The second goal of this paper is to clarify the relationship between Grothendieck’s conjecture and Artin’s question. We show in a general framework (i.e. not only for the rings of convergent power series or algebraic power series but for more general families of rings - cf. Definition 4.1) that Grothendieck’s conjecture is equivalent to the question of M. Artin in the case where F(x, y) is linear in y(see Theorem 4.9). Let us mention that it is well known that Grothendieck’s conjecture is equivalent to Artin’s question for some very particular F(x, y) which are linear in y(see [Be77] and [Ro08]) but, to the best of our knowledge, it was not known that they are equivalent for all F(x, y) linear in y. We also prove (see Theorem 4.9) that these two problems are equivalent to the problem of the commutation of two operations: the operation of replacing an ideal by its completion and the operation of replacing an ideal by one of its elimination ideals (see 4.2). Finally we mention that the question of Grothendieck has been widely studied in the case of convergent power series rings and it has been shown that the answer is positive for some particular cases (see for instance [AvdP70], [Ga73], [EH77], [Mil78], [Iz89] or [To90]). One of them, similar to our situation, is the case of a morphism ϕ:k{x}/I −→ k{y}/J where the images of the xiare algebraic power series and the ideals Iand Jare prime and generated by algebraic power series. For such morphisms it is shown that ϕis injective if and only if bϕis injective (it has been proven in several steps in [To76], [Be77], [Mil78] and [Ro09]). 2. Acknowledgements This research was started in the frame of the Jean Morlet Chair Artin Approximation in Singularity Theory, held at CIRM (Marseille, France) from January until June 2015. The Chair was held by Prof. Herwig Hauser and the Local Project Leader was the third author. We are grateful for the hospitality and support of CIRM during the main stage of this work. We would like to thank Profs. H. Hauser and M.E. Alonso for their very useful comments and suggestions. We also would like to thank E. Bierstone for having indicated to us the reference [BM87] where is proven a result similar to our Proposition 3.3. 4 F. J. CASTRO-JIM´ ENEZ, D. POPESCU, G. ROND 3. Linear Nested Artin Approximation for algebraic series We will prove the following linear version of Theorem 1.2: Theorem 3.1 (Linear Nested Artin Approximation Theorem).Let m, n, p be positive integers, Tbe a p×mmatrix with entries in khxi:= khx1,...,xni,b= (b1,...,bp)∈khxipand σ:{1,...,m} −→ {1,...,n}be a map. Let y= (y1,...,ym) be a vector of new variables. Then for any solution by(x)in kJx1,...,xσ(1)K× · · · × kJx1,...,xσ(m)K of the following system of linear equations (3.1) T y =b and for any integer cthere exists a solution y(x)in khx1,...,xσ(1)i × · · · × khx1,...,xσ(m)i such that y(x)−by(x)∈(x)ckJxKm. We begin by giving some intermediate results: Lemma 3.2. Let (A, m)be a complete normal local domain, x= (x1,...,xn)and y= (y1,...,ym). Let B=AJxKhyibe the algebraic closure of AJxK[y]in AJx, yK and f∈B. Then there exist gin the algebraic closure Ahy, ziof A[y, z]in AJy, zK, with z= (z1,...,zs)for some s∈N, and bz∈AJxKssuch that f=g(y, bz). Proof. By replacing fby f−f(0, y) we may assume that f∈(x)B. Note that Bis the Henselization of C=AJxK[y](m,x,y)by [Na62, 44.1] and so there exists some ´etale neighborhood of Ccontaining f. Using for example [Sw98, Theorem 2.5] there exists a monic polynomial Fin uover AJxK[y] and h∈(m, x, y)AJxKhyi such that F(h) = 0, (∂F/∂u)(h)6∈ (m, x, y) and f∈AJxK[y, h](m,x,y)∩AJxK[y,h], let us say f=P(y, h)/Q(y, h) for some P(y, u), Q(y, u)∈AJxK[y, u], Q(y, h)/∈ (m, x, y)∩AJxK[y, h]. Let us write Q(y, h) = X α,i (qαi +bwαi)yαhi where qαi ∈Aand bwαi ∈(m+ (x))AJxKfor every α,i. We set ˜ Q=X α,i (qαi +wαi)yαui for new indeterminates wαi. Since Q(y, h)/∈(m, x, y), bwαi ∈(m+(x))AJxKand h∈ (m, x, y)AJxKhyi, we have that q00 is a unit of A. So ˜ Qis invertible in Ahy, u, wα,ii. Moreover ˜ Q−1(y, h, bwαi) = Q(y, h)−1 by uniqueness of the inverse. Thus by adding the new wαi and the coefficients of Pfrom AJxKas new bw, we see that our lemma works for fas soon as it works for h. So we can replace fby h and assume f∈(m, x, y)AJxKhyi,F(f) = 0 and F′(f) := (∂F/∂u)(f)6∈ (m, x, y). Let us write F=Pα,j Fαj yαujfor some Fαj ∈AJxK. Set bzαi =Fαi −Fαi(0) ∈(x)AJxK,bz= (bzαi) and G:= G(y, u, z) = Pαi(Fαi(0) + zαi)yαuifor some new variables z= (zαi). We have G(y, u, bz) = F. Set G′= ∂G/∂u. As G(y, f, 0) ≡G(y, f, bz)≡F(f)≡0 modulo (m, x, y, u), LINEAR NESTED ARTIN APPROXIMATION 5 G′(y, f, 0) ≡G′(y, f, bz)≡F′(f)6≡ 0 modulo (m, x, y, u) we get G(y, 0, z)≡0, G′(y, 0, z)6≡ 0 modulo (m, y, z)Ahy, zi. By the Implicit Function Theorem there exists g∈(m, y, z)Ahy, zisuch that G(y, g, z) = 0. It follows that G(y, g(y, bz), z) = 0. But F=G(y, u, bz) = 0 has just one solution u=fin (m, x, y)Bby the Implicit Function Theorem and so f=g(y, bz).  The following result can be rephrased as a particular case of Theorem 12.6 [BM87], and the proof we give here, for the sake of completeness, follows essentially the same principle as the proof given in [BM87]. Proposition 3.3. We set x= (x1,...,xn)and y= (y1,...,ym)and let Mbe a submodule of khx, yip. Then kJxK(M∩khxip) = c M∩kJxKp where c M=kJx, yKMdenotes the (x, y)-adic completion of M. Moreover, if c∈N and bu=Pr i=1 bviωi∈c M∩kJxKpfor some ωi∈M,bvi∈kJx, yKthen there exist vic ∈khx, yisuch that vic ≡bvimodulo (x, y)ckJx, yK,uc=Pt i=1 vicωi∈M∩khxip and buis the limit of (uc)cin the (x)-adic topology. Proof. Of course we always have kJxK(M∩khxip)⊂c M∩kJxKp. So we only have to prove the opposite inclusion. Let ω1,...,ωrbe generators of Mand bu(x) be an element of c M∩kJxKp. Such an element bu(x) has the form (3.2) bu(x) = r X ℓ=1 bvℓ(x, y)ωℓ for some formal power series bvℓ(x, y). The components of Equation (3.2) provide a system of plinear equations as follows: (3.3) Tbv(x, y) = bu(x) where Tis a p×rmatrix with entries in khx, yiand bv(x, y) is the vector of entries bvℓ(x, y). The morphism kJxKhyi −→ kJx, yKbeing faithfully flat, for any integer cthere exists a solution ev(x, y)∈kJxKhyirof (3.3) such that ev(x, y)−bv(x, y)∈(x, y)ckJx, yKr. Indeed, choose v′(x, y)∈kJxKhyirsuch that v′(x, y)−bv(x, y)∈(x, y)ckJx, yK. By faithfully flatness the linear system bu(x) = r X ℓ=1 vℓωℓ, v′(x, y)−v=X |α|+|β|=c xαyβwα,β has a solution ev(x, y), ew(x, y) in kJxKhyisince it has one in kJx, yK. Thus from now on we may assume that bv(x, y)∈kJxKhyir. By Lemma 3.2 there exist a new set of variables z= (z1,...,zs), algebraic power series gℓ(y, z)∈khy, zi for 1 ≤ℓ≤rand formal power series bz1(x),...,bzs(x)∈(x)kJxKsuch that bvℓ(x, y) = gℓ(y, bz1(x),...,bzs(x)). 6 F. J. CASTRO-JIM´ ENEZ, D. POPESCU, G. ROND Then, by replacing vℓby gℓ(y, z) for ℓ= 1,...,r in the linear system of equations T·v=bu(x) we obtain a new system of (non linear) equations f(x, y, bu(x),bz(x)) = 0 where f(x, y, u, z) is a vector of algebraic power series. Let Idenote the ideal of khx, u, zigenerated by all the coefficients of the monomials in yin the expansion of the components of fas power series in (y1,...,ym). Let h1,...,htbe a system of generators of I. By assumption (bu(x),bz(x)) is a formal power series solution of the system (3.4) h1(x, u, z) = ···=ht(x, u, z) = 0. Thus by Artin Approximation Theorem for algebraic power series [Ar69], for any integer c≥0 there exists (eu(x),ez(x)) ∈khxip+ssolution of the system (3.4) with euκ(x)−buκ(x)∈(x)ckJxK,ezk(x)−bzk(x)∈(x)ckJxK∀κ, k. Thus (eu(x),ev(x, y)) is a solution of the system (3.3) where evℓ(x, y) = gℓ(y, ez1(x),...,ezs(x)) ∀ℓ. In particular eu(x)∈M∩khxipand M∩khxipis dense in c M∩khxi. Moreover by Taylor’s formula we have that evℓ(x, y)−bvℓ(x, y)∈(x, y)ckJx, yKfor 1 ≤ℓ≤r.  The next theorem is a key result to reduce the proof of Theorem 3.1 to the case of only one nest (i.e. when there is k≤nsuch that σi=kor nfor every i). This one nest case is Proposition 3.3. As we have previously said, the linear one nest case was proven by E. Bierstone and P. Milman (see Theorem 12.6 [BM87]). Theorem 3.4. Let M⊂khxipbe a finitely generated khxi-submodule and σ: {1,...,p} −→ {1, . . . , n}be a weakly increasing function. Then N=M∩(khx1,...,xσ(1)i × ...×khx1,...,xσ(p)i) is dense in N′= (kJxKM)∩(kJx1,...,xσ(1)K×...×kJx1,...,xσ(p)K). Moreover, if c∈Nand bu=Pt i=1 bviωi∈ N′for some ωi∈M,bvi∈kJxKthen there exist vic ∈khxisuch that vic ≡bvimodulo (x)ckJxK,uc=Pt i=1 vicωi∈ N and buis the limit of (uc)cin the (x)-adic topology. Proof. Apply induction on p, the case p= 1 being done in Proposition 3.3. Assume that p > 1. We may reduce to the case when σ(p) = nreplacing Mby M∩ khx1,...,xσ(p)ipif σ(p)< n. Let q:kJxKp→kJxKp−1 be the projection on the first p−1 components and q′:kJxKp→kJxK be the projection on the last component. Let bu= (bu1,...,bup)∈ N ′,and M1= q(M). Assume that bu=Pt i=1 bviωifor some bvi∈kJxK,ωi∈M. By the induction LINEAR NESTED ARTIN APPROXIMATION 7 hypothesis applied to M1and q(bu), for every c∈Nthere exists vic ∈khxiwith vic ≡bvimodulo (x)ckJxKsuch that u′ c= t X i=1 vicq(ωi)∈q(N) = M1∩(khx1,...,xσ(1)i × ...×khx1,...,xσ(p−1)i) and q(bu) is the limit of (u′ c)cin the (x)-adic topology. Now, let u′′ c=Pt i=1 vicq′(ωi)∈khx1,...,xni. We have u′′ c≡q′(bu) modulo (x)ckJxK. Then uc= (u′ c, u′′ c) = Pt i=1 vicωi∈ N since σ(p) = n,uc≡bumodulo (x)ckJxKpand buis the limit of (uc)cin the (x)-adic topology.  Proof of Theorem 3.1. First of all we may assume that σis weakly increasing after permuting the yi. If b= 0 then it is enough to apply Theorem 3.4 for the module Mof the solutions of T y = 0 in A=khxi. Suppose that b6= 0. Replace the system T y =bby the homogeneous system of linear polynomials T′y′:= T y −by0= 0 from A[y0, y]pwhere y′= (y0, y). A nested formal solution byof T y =bin kJxKm with byi∈kJx1, . . ., xσ(i)K, 1 ≤i≤minduces a nested formal solution (by0,by), by0= 1 of T′y′= 0 with σ(0) = σ(1). As above, for all c∈Nwe get a nested algebraic solution (y0(x), y(x)) of T′y′= 0 with yi(x)∈khx1,...,xσ(i)iand yi(x)≡ byimodulo (x)ckJxKfor all 0 ≤i≤m. In particular y0(0) = 1 6= 0 and y0(x) is a unit. Thus (y0(x)−1y1(x),...,y0(x)−1ym(x)) is an algebraic nested solution of T y −b= 0. Moreover, for all j≥1, we have: y0(x)−1yj(x)−byj(x) = (y0(x)−1−1)yj(x) + (yj(x)−byj(x)) ∈(x)c.  4. Linear nested approximation property In the second part of this paper we generalize the method used to prove Theorem 3.1 in order to show that the question of A. Grothendieck, for local subrings of the ring of formal power series, is equivalent to the nested Artin approximation property for linear equations. We begin by giving several definitions. Definition 4.1. Let kbe a field. An admissible family of rings is an increasing sequence of rings F= (Rn)n∈Nsatisfying the following properties: (1) For every integer n≥0 the ring Rnis a k-subalgebra of kJx1,...,xnK(in particular R0=k). (2) For every integer n≥0, k[x1,...,xn]⊂Rn. (3) For every integer n > 0 the ring Rnis a Noetherian local ring whose maximal ideal is generated by x1,. . . , xn. (4) For every integer nthe completion of Rnis kJx1,...,xnK. (5) For every integers m, n with 0 ≤m≤nwe have Rn∩kJx1,...,xmK=Rm. When an admissible family of rings is given, any element of a member of this family is called an admissible power series. 8 F. J. CASTRO-JIM´ ENEZ, D. POPESCU, G. ROND Sometimes we will emphasize the dependency of Rnon the variables (x1,...,xn) by writing Rn=khhx1,...,xnii for n∈N. Example 4.2.The following families of rings are admissible: •The rings of convergent power series over a valued field k. •The rings of algebraic power series over a field k. •The rings of formal power series. •The rings of germs of rational functions at 0 ∈kn,k[x1,...,xn](x1,...,xn). 4.1. Krull topology. Let (A, m) be a Noetherian local ring. The Krull topology of Ais the topology in which the ideals mcconstitute a basis of neighborhoods of the zero of A. For a A-module Mthe Krull topology of Mis the one in which the submodules mcMconstitute a basis of neighborhoods of the zero of M. The completion of A(resp. M) for the Krull topology is denoted by b A(resp. c M). We have the following lemma asserting that the topological closure of a finite module and its completion coincide: Lemma 4.3. ([SZ58, Corollary 2, p. 257]) If Nis a A-submodule of a finite A-module Mthen the closure of Nin c Mis b N=b AN. Definition 4.4. If Mis a A-module where (A, m) is a Noetherian local ring and E is a subset of M, we say that an element f∈Mmay be approximated by elements of Eif fis in the closure (for the Krull topology) of Ein M, i.e. if for every integer cthere exists fc∈Esuch that f−fc∈mcM. 4.2. Strong elimination property. One says that an admissible family of rings F= (khhx1,...,xnii)nhas the strong elimination property for ideals if for every two sets of variables xand yand every ideal Iof khhx, yii we have (4.1) (I∩khhxii)kJxK=b I∩kJxK where b Idenotes the ideal of kJx, yKgenerated by I. One says that the admissible family Fhas the strong elimination property for modules if for every two sets of variables xand y, every positive integer pand every khhx, yii-submodule Mof khhx, yiipwe have (4.2) kJxK(M∩khhxiip) = c M∩kJxKp where c Mdenotes the kJx, yK-submodule of kJx, yKpgenerated by M. Remark 4.5.Since I∩khhxii ⊂ b I∩kJxK(resp. M∩khhxiip⊂c M∩kJxKp, Lemma 4.3 shows that (4.1) (resp. (4.2)) is equivalent to say that the elements of b I∩kJxK (resp. c M∩kJxKp) may be approximated by elements of I∩khhxii (resp. M∩khhxiip). 4.3. Linear nested approximation property. We say that an admissible family of rings F= (khhx1,...,xnii)nhas the linear nested approximation property if the following property holds: For every positive integers m, n, p, every p×mmatrix Twith entries in khhxii := khhx1,...,xnii, every b= (b1,...,bp)∈khhxiipand every map σ:{1,...,m} −→ {1,...,n}we have the following: let y= (y1,...,ym) be a vector of new variables. Then the set of solutions y(x) in khhx1,...,xσ(1)ii × · · · × khhx1,...,xσ(m)ii LINEAR NESTED ARTIN APPROXIMATION 9 of the following system of linear equations (4.3) T y =b is dense in the set of formal solutions in kJx1,...,xσ(1)K× · · · × kJx1,...,xσ(m)K. 4.4. Strongly injective morphisms. Definition 4.6. Let ϕ:A−→ Bbe a morphism of local rings. We denote by bϕthe induced morphism b A−→ b B. One says that ϕis strongly injective if bϕis injective. Definition 4.7. We say that an admissible family of rings F= (khhx1,...,xnii)n has the strong injectivity property if for every integers nand mand every ideals I of khhx1,...,xnii and Jof khhy1,...,ymii, every injective morphism of local rings khhxii I−→ khhyii J is strongly injective. Remark 4.8.Definition 4.6 is not the classical one. In [AvdP70] a morphism ϕ: A−→ Bis called strongly injective if bϕ(b A)∩B=ϕ(A). This definition, which is the classical one, is stronger than the one we use in this paper. Nevertheless we will prove that if an admissible family of rings (khhx1,...,xnii)nhas the strong injectivity property then for any morphism of local rings ϕ:A=khhxii I−→ B=khhyii Jwe have bϕ(b A)∩B=ϕ(A) (see Corollary 4.10). The main result of this part is the following: Theorem 4.9. For an admissible family of rings F= (khhx1,...,xnii)nthe following properties are equivalent: (i) Fhas the strong elimination property for ideals. (ii) Fhas the strong elimination property for modules. (iii) Fhas the linear nested approximation property. (iv) Fhas the strong injectivity property. Corollary 4.10. Let F= (khhx1,...,xnii)nbe an admissible family having the strong injectivity property. Then for any morphism of local rings ϕ:A=khhxii I−→ B=khhyii J we have bϕ(b A)∩B=ϕ(A). In particular if bϕis surjective then ϕis surjective too. Proof. Clearly ϕ(A)⊂bϕ(b A)∩B. Let us prove the reverse inclusion. We can replace ϕby ϕ◦πwhere π:khhxii −→ khhxii Iis the natural quotient morphism. This allows us to assume that I= 0 and A=khhxii. Let b f∈kJxKsuch that bϕ(b f) = b∈B. Let us denote by ϕi(y) an admissible power series of khhyii which is the image of xiby ϕmodulo J, for i= 1,...,n. Let q1(y),...,qs(y) be 16 F. J. CASTRO-JIM´ ENEZ, D. POPESCU, G. ROND References [AvdP70] S. S. Abhyankar, M. van der Put, Homomorphisms of analytic local rings, J. Reine Angew. Math.,242, (1970), 26-60. [An75] M. 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