Clôture intégrale des idéaux et équisingularité

Lejeune-Jalabert, Monique; Teissier, Bernard

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 17 (2008) no. 4, pp. 781-859.

This text has two parts. The first one is the essentially unmodified text of our 1973-74 seminar on integral dependence in complex analytic geometry at the Ecole Polytechnique with J-J. Risler’s appendix on the Łojasiewicz exponents in the real-analytic framework. The second part is a short survey of more recent results directly related to the content of the seminar.

The first part begins with the definition and elementary properties of the $\overline{\nu }$ order function associated to an ideal $I$ of a reduced analytic algebra $A$. Denoting by ${\nu }_I\left(x\right)$ the largest power of $I$ containing the element $x\in A$, one defines ${\overline{\nu }}_I\left(x\right)={\text{lim}}_{i\to \infty }{\nu }_I\left(x^k\right)/k$. The second paragraph is devoted to the equivalent definitions of the integral closure of an ideal in complex analytic geometry, one of them being $\overline{I}=\{x\in A/{\overline{\nu }}_I\left(x\right)\ge 1\}$. The third paragraph describes the normalized blowing-up of an ideal and the fourth explains how to compute ${\overline{\nu }}_I\left(x\right)$ with the help of the normalized blowing-up of the ideal $I$. It contains the basic finiteness results of the seminar, such as the rationality of ${\overline{\nu }}_I\left(x\right)$ (which had been proved by Nagata in algebraic geometry, a fact of which we were not aware at the time), the definitions of the fractional powers of coherent sheaves of ideals and the proof of their coherency. Given a coherent sheaf $ℐ$ of ${𝒪}_X$-ideals on a reduced analytic space $X$ one can define for each open set $U$ of $X$ and $f\in \Gamma (U,{𝒪}_X)$ the number ${\overline{\nu }}_{ℐ}^U\left(f\right)$ as the infimum of the ${\overline{\nu }}_{{ℐ}_y}\left(f_y\right)$ for $y\in U$.

Then one defines for each positive real number $\nu$ the sheaf $\overline{{ℐ}^{\nu }}$ (resp. $\overline{{ℐ}^{\nu +}}$) associated to the presheaf

$$U\mapsto \{f\in \Gamma (U,{𝒪}_X)/{\overline{\nu }}_{ℐ}^U\left(f\right)\ge \nu \}$$

(resp.

$$U\mapsto \{f\in \Gamma (U,{𝒪}_X)/{\overline{\nu }}_{ℐ}^U\left(f\right)>\nu \}).$$

Finally one has the graded ${𝒪}_X/ℐ$-algebra

$${\overline{\text{gr}}}_{ℐ}{𝒪}_X=\underset{\nu \in {ℝ}_0}{⨁}\overline{{ℐ}^{\nu }}/\overline{{ℐ}^{\nu +}}.$$

One important result is then that this algebra is locally finitely generated and that locally there is a universal denominator $q$ in the sense that all nonzero homogeneous components of the graded algebra have degree in $\frac{1}{q}\mathbb{N}$.

In § 5 it is shown that one can compute $\overline{\nu }$ using analytic arcs $h:(ℂ,0)\to (X,x)$, and § 6 shows that Łojasiewicz exponents are the inverses of $\overline{\nu }$, which implies that they are rational.

Risler’s appendix shows how to use blowing-ups to compute Łojasiewicz exponents and prove their rationality in the real analytic case.

The complements, added for this publication, point to some developments directly related to the subject of the seminar:

The first one is the proof in the spirit of the seminar of the classical Łojasiewicz inequality $\left|\text{grad}\left(f\left(z\right)\right)\right|\ge C_1{\left|f\left(z\right)\right|}^{\theta }$ with $\theta <1$.

Then we point to later work which shows that in fact given an ideal $I$ and an element $f\in A$ the rational number ${\overline{\nu }}_I\left(f\right)$ can be seen as the slope of one of the sides of a natural Newton polygon associated to $I$ and $f$, which is in several ways a better indicator of the relations of the powers of $f$ with the powers of $I$ and has some useful incarnations. The third complement points to results of Izumi using $\overline{\nu }$ to characterize the Gabrielov rank condition for a morphism of analytic algebras, the fourth is a presentation of a generalization due to Ciuperča, Enescu and Spiroff of the rationality of $\overline{\nu }$ to the case of several ideals, where it becomes the rationality of a certain polyhedral cone.

The fifth comment presents the connection of $\overline{\nu }$ with the type of ideals, which was introduced by D’Angelo in complex analysis and used recently by Heier for the proof of an effective Nullstellensatz. In the middle 1980’s, A. Płoski, J. Chadzyński and T. Krasiński found methods of evaluation for the local and global Łojasiewicz exponents in inequalities of the form $\left|P\left(z\right)\right|\ge {C\left|z\right|}^{\theta }$ where either $P=(P_1,...,P_k)$ is a collection of analytic functions on ${ℂ}^n$ having an isolated zero at the origin and the inequality should be true for $\left|z\right|$ small enough, or $P$ is a collection of polynomials with finitely many common zeroes and the inequality should be true for $\left|z\right|$ large enough. The results on the type are of the same nature, because it follows from the seminar that the type is in fact a Łojasiewicz exponent.

The sixth comment points to results of Morales and others about the Hilbert function associated to the integrally closed powers $\overline{I^n}$ of a primary ideal in an excellent local ring and the associated graded algebra.

Finally we point to two different but not unrelated uses of what is in fact the main object of study in the seminar: the reduced graded ring ${\overline{\text{gr}}}_{I}A$ defined and studied in § 4. In [T5] the second author uses the fact that for the local algebra $𝒪$ of a plane analytic branch the algebra ${\overline{\text{gr}}}_m𝒪$ is the algebra of the semigroup associated to the singularity and is a complete intersection (a result due to the first author) to revisit the local moduli problem. The key is that the local analytic algebra $𝒪$ of every plane branch in the same equisingularity class has the same ${\overline{\text{gr}}}_m𝒪$ because it has the same semigroup, so that the branch is a deformation of the monomial curve corresponding to that algebra. In [Kn], Allen Knutson uses the same specialization to the “balanced normal cone" corresponding to ${\overline{\text{gr}}}_{I}A$ in intersection theory.

Each paragraph has its own bibliography. Unfortunately at the time of the seminar we were unaware of the beautiful results of Samuel, Rees and Nagata (see [Sa], [N], [R1], [R2], [R3] in the bibliography of the complements), of which it appears a posteriori that some parts of the seminar are translations into the complex analytic framework. The demand for this text over the years, however, and the fact that some mathematicians are led to rediscover some of its results, indicate that its publication is probably of some use.

[1] Bourbaki (N.).— Algèbre commutative, Chapitres 5 et 6,  Hermann.

[2] Grothendieck (A.).— Éléments de géométrie algébrique, IV, Publications de l’IHES, PUF. | Numdam | Zbl

[3] Rees (D.).— $a$-transforms of local rings and a theorem on multiplicities of ideals, Proceedings Camb. Philos., 57, 1, 8–17. | MR | Zbl

[4] Zariski (O.), Samuel (P.).— Commutative algebra, Vol.I, Chap. V et Vol. II  Appendice 4 Van Nostrand (1960). | MR | Zbl

[5] Bourbaki (N.).— Algèbre commutative, chapitres 3 et 4,  Hermann.

[6] Frisch (J.).— Points de platitude d’un morphisme d’espaces analytiques complexes, Inventiones, 4, 118-138 (1967). | EuDML | MR | Zbl

[7] Cartan (H.).— Familles d’espaces complexes et fondement de la géométrie analytique, Séminaire Henri Cartan, 13, 2, 1960–1961. | Zbl

[8] Lejeune (M.), Teissier (B.).— Contribution à l’étude des singularités du point de vue du polynôme de Newton, Thèse, Paris VII (1973). | MR

[9] Lejeune (M.), Teissier (B.).— Transversalité, polygone de Newton et installations, Astérisque 7.8 (1973). | Numdam | MR | Zbl

[10] Zariski (O.), Samuel (P.).— Commutative algebra, Van Nostrand, 1960. | MR | Zbl

[B-R] Bochnack (J.), Risler (J.-J.).— Sur les exposants de Łojasiewicz, Comment. Mat. Helvetici 50 (1975). | Zbl

[hiro] Hironaka (H.).— Introduction to real-analytic sets and real-analytic maps, Quaderni dei Gruppi di Ricerca Matematica del Consiglio Nazionale delle Ricerche. Istituto Matematico « L. Tonelli » dell’Università di Pisa, Pisa, 1973. | MR

[M] Milnor (J.).— Singular points of complex hypersurfaces, Annals of Math. Studies no. 61 Princeton University Press, (1968). | MR | Zbl

[R] Risler (J.-J.).— Le théorème des zéros en géométrie algébrique et analytique réelles, (1976). | Numdam | MR | Zbl

[Bö1] Böger (E.).— Einige Bemerkungen zur theorie der ganzalgebraischen Abhängigkeit in Idealen, Math. Ann., 185, 303-308 (1970). | MR | Zbl

[Bon] Bondil (R.).— Geometry of superficial elements, Ann. Fac. Sci. Toulouse Math. (6) 14, no. 2, 185-200 (2005). | Numdam | MR | Zbl

[Br] Brumfiel (G. W.).— Real valuation rings and ideals, Springer L.N.M., No. 959 (1981). | MR | Zbl

[C] Cassou-Noguès (P.).— Courbes de semi-groupe donné, Rev. Mat. Univ. Complut. Madrid 4, no. 1, 13-44 (1991). | MR | Zbl

[C-E-S] Ciuperča (C.), Enescu (F.), Spiroff (S.).— Asymptotic growth of powers of ideals, ArXiv : Math. AC/0610774. | MR

[C-K1] Chadzyński (J.), Krasiński (T.).— The Łojasiewicz exponent of an analytic function of two complex variables at an isolated zero, Singularities 1985, Banach Center publications 20, PWN Varsovie (1988). | Zbl

[C-K2] Chadzyński (J.), Krasiński (T.).— A set on which the local Łojasiewicz exponent is attained, Annales Polon. Math., 67, 297-301 (1997). | EuDML | MR | Zbl

[C-S] Huneke (C.), Swanson (I.).— Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge. xiv+431 (2006). | MR | Zbl

[D] D’Angelo (J. P.).— Real hypersurfaces, orders of contact, and applications, Annals of Math., (2), 115(3), 615-637 (1982). | MR | Zbl

[E-L] Ein (L.), Lazarsfeld (R.).— A geometric effective Nullstellensatz, Invent. Math., 137(2), 427-448 (1999). | MR | Zbl

[F1] Fekak (A.).— Interpretation algébrique de l’exposant de Łojasiewicz, Annales Polonici Mathematici, LVI, 2, 123-131 (1992). | MR | Zbl

[F2] Fekak (A.).— Exposants de Łojasiewicz pour les fonctions semi-algébriques, C.R.A.S. Paris, t. 310, Série 1, 193-196 (1990). | Zbl

[Fu] Fulton (W.).— Intersection Theory, Springer (1983). | MR | Zbl

[G1] Gaffney (T.).— Integral closure of modules and Whitney equisingularity, Invent. Math, 107, 301-322 (1992). | MR | Zbl

[G2] Gaffney (T.).— Polar multiplicities and equisingularity of map-germs, Topology, Vol. 32, No.1, 185-223 (1993). | MR | Zbl

[G3] Gaffney (T.).— Multiplicities and equisingularity of ICIS germs, Inventiones Math., 123, 209-220 (1996). | MR | Zbl

[G4] Gaffney (T.).— The theory of integral closure of ideals and modules : applications and new developments With an appendix by Steven Kleiman and Anders Thorup. NATO Sci. Ser. II Math. Phys. Chem., 21, New developments in singularity theory (Cambridge, 2000), 379-404, Kluwer Acad. Publ., Dordrecht, (2001). | MR | Zbl

[Ge] Gerstenhaber (M.).— On the deformation of rings and algebras, II. Ann. of Math. 84, 1-19 (1966). | MR | Zbl

[Gw] Gwoździewicz (J.).— The Lojasiewicz exponent at an isolated zero, Commentari Math.Helvetici 74, 364-375 (1999). | MR | Zbl

[G-K1] Gaffney (T.), Kleiman (S.L.).— Specialization of integral dependence for modules, Inv. Math., 137, no. 3, 541-574 (1999). | MR | Zbl

[G-K2] Gaffney (T.), Kleiman (S.).— $W_f$ and integral dependence, Real and Complex singularities (Sao Carlos, 1998) Chapman and Hall//CRC Res. Notes in Math., 412, 33-45, Chapman and Hall//CRC Boca Raton, Florida, (2000). | MR | Zbl

[GB] García Barroso (E.).— Sur les courbes polaires d’une courbe plane réduite, Proc. London Math. Soc. (3) 81, 1-28 (2000). | MR | Zbl

[GB-G] García Barroso (E.), Gwoździewicz (J.).— Characterization of jacobian Newton polygons of branches, Manuscrit, (2007).

[GB-P] García Barroso (E.), Płoski (A.).— On the Łojasiewicz numbers, C. R. Acad. Sci. Paris, Ser. I., 336, 585-588 (2003). | MR | Zbl

[GB-K-P1] García Barroso (E.), Krasiński (T.), Płoski (A.).— On the Łojasiewicz numbers, II, C. R. Acad. Sci. Paris, Ser. I., 341, 357-360 (2005). | MR | Zbl

[GB-K-P2] García Barroso (E.), Krasiński (T.), Płoski (A.).— The Łojasiewicz numbers and plane curve singularities, Ann. Pol. Math., 87, 127-150 (2005). | MR | Zbl

[G-T] Goldin (R.), Teissier (B.).— Resolving plane branch singularities with one toric morphism, in « Resolution of Singularities, a research textbook in tribute to Oscar Zariski”, Birkhäuser, Progress in Math. No. 18, 315-340 (2000). | MR | Zbl

[H] Heier (G.).— Finite type and the effective Nullstellensatz, ArXiv : Math/AG 0603666. | MR | Zbl

[Hi] Hickel (M.).— Solution d’une conjecture de C. Berenstein-A. Yger et invariants de contact à l’infini, Ann. Inst. Fourier (Grenoble), 51, No.3, 707-744 (2001). | Numdam | MR | Zbl

[Hu] Huneke (C.).— Tight closure and its applications, C.B.M.S. Lecture Notes 88, A.M.S., Providence (1996). | MR | Zbl

[Hu-S] Huneke (C.), Swanson (I.).— Integral closure of ideals, rings, and modules, London Mathematical Society Lecture Note Series, 336. Cambridge University Press, Cambridge, (2006). | MR | Zbl

[H-I-O] Hermann (M.), Ikeda (S.), Orbanz (U.).— Equimultiplicity and blowing up, an algebraic study, with an appendix by Boudewin Moonen, Springer Verlag, (1988). | MR | Zbl

[H-L] Heier (G.), Lazarsfeld (R.).— Curve selection for finite type ideals, ArXiv : Math/CV0506557.

[I1] Izumi (S.).— A measure of integrity for local analytic algebras, Publ. R.I.M.S., Kyoto University, 21, 4, 719-735 (1985). | MR | Zbl

[I2] Izumi (S.).— Gabrielov’s rank condition is equivalent to an inequality of reduced orders, Math. Annalen, 276, 81-87 (1986). | MR | Zbl

[I3] Izumi (S.).— Fundamental properties of germs of analytic mappings of analytic sets and related topics, Real and Complex singularities, Proceedings of the Australian-Japanese Workshop, University of Sidney 2005, L. Paunescu, A. Harris, T. Fukui, S. Koike, Editors. World Scientific, 109-123 (2007). | MR | Zbl

[K] Kleiman (S. L.).— Equisingularity, multiplicity, and dependance, Commutative algebra and algebraic geometry (Ferrara), 211-225, Lecture Notes in Pure and Applied Math., 206, Dekker, New York, (1999). | MR | Zbl

[K-T] Kleiman (S.L), Thorup (A.).— A geometric theory of the Buchsbaum-Rim multiplicity, J. Algebra, 167, (1), 168-231 (1994). | MR | Zbl

[Kn] Knutson (A.).— Balanced normal cones and Fulton-MacPherson’s intersection theory, Pure Appl. Math. Q. 2, no. 4, 1103-1130 (2006). | MR | Zbl

[L] Lazarsfeld (R.).— Positivity in algebraic geometry II, Ergebnisse der Mathematik vol. 49, Springer Verlag (2004). | MR | Zbl

[Le] Lenarcik (A.).— On the jacobian Newton polygon of plane curve singularities, Soumis. | Zbl

[M1] Morales (M.).— Le polynôme de Hilbert-Samuel associé à la filtration par les clôtures intégrales des puissances de l’idéal maximal pour une courbe plane. C. R. Acad. Sci. Paris Sér. A-B 289, no. 6, A401-A404 (1979). | MR | Zbl

[M2] Morales (M.).— Polynôme d’Hilbert-Samuel des clôtures intégrales des puissances d’un idéal $m$-primaire. Bull. Soc. Math. France 112, no. 3, 343-358 (1984). | EuDML | Numdam | MR | Zbl

[M3] Morales (M.).— Clôture intégrale d’idéaux et anneaux gradués Cohen-Macaulay. Géométrie algébrique et applications, I (La Rábida, 1984), 151-171, Travaux en Cours, 22, Hermann, Paris (1987). | MR | Zbl

[Mi] Milnor (J.).— Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton U.P. (1968). | MR | Zbl

[M-T-V] Morales (M.), Trung (N.), Villamayor (O.).— Sur la fonction de Hilbert-Samuel des clôtures intégrales des puissances d’idéaux engendrés par un système de paramètres. J. Algebra 129, no. 1, 96-102 (1990). | MR | Zbl

[Mc-N] McNeal (J. D.), Némethi (A.).— The order of contact of a holomorphic ideal in ${ℂ}^2$, Math. Z., 250, no. 4, 873-883 (2005). | MR | Zbl

[N] Nágata (M.).— Note on a paper of Samuel concerning asymptotic properties of powers of ideals, Mem. Coll. Sci. Univ. Kyoto, Series A, Math., 30, 165-175 (1957). | MR | Zbl

[No] Northcott (D. G.).— Lessons on Rings, Modules, and Multiplicities, University Press, Cambridge, (1968). | MR | Zbl

[N-R] Northcott (D. G.), Rees (D.).— Reductions of ideals in local rings, Proc. Camb. Phil. Soc., 50, 145-158 (1954). | MR | Zbl

[P] Philippon (P.).— Dénominateurs dans le théorème des zéros de Hilbert, Acta Arithm., 58 1, 1-25 (1991). | EuDML | MR | Zbl

[Pl1] Płoski (A.).— On the growth of proper polynomial mappings, Annales Polonici Math., XLV, 297-309 (1985). | EuDML | MR | Zbl

[Pl2] Płoski (A.).— Remarque sur la multiplicité d’intersection des branches planes, Bull. Pol. Acad. Sci. Math., 33 No. 11-12, 601-605 (1985). | MR | Zbl

[Pl3] Płoski (A.).— Multiplicity and the Łojasiewicz exponent, in : “Singularities", Banach Center publications, 353-364 (1988). | EuDML | Zbl

[Po] Popov (V. L.).— Contraction of the action of reductive algebraic groups, Math. USSR Sbornik, 58, no. 2, 311-335 (1987). | MR | Zbl

[P-T] Pham (F.), Teissier (B.).— Saturation Lipschitzienne d’une algèbre analytique complexe et saturation de Zariski, Preprint 1969. Fichier .pdf disponible sur http://people.math.jussieu.fr/~teissier/oldpapers.html | MR

[P-U-V] Polini (C.), Ulrich (B.), Vasconcelos (W. V.).— Normalization of ideals and Briançon-Skoda numbers, Math. Res. Lett. 12, no. 5-6, 827-842 (2005). | MR | Zbl

[R1] Rees (D.).— Valuations associated with ideals, Proc. London Math. Soc. (3) 6, 161-174 (1956). | MR | Zbl

[R2] Rees (D.).— valuations associated with ideals, II, J. London Math. Soc. 31, 221-228 (1956). | MR | Zbl

[R3] Rees (D.).— Degree functions in local rings, Proc. Camb. Phil. Soc., 57, 1-7 (1961). | MR | Zbl

[R4] Rees (D.).— A-transforms of local rings and a theorem on multiplicities of ideals, Proc. Camb. Phil. Soc., 57, 8-17 (1961). | MR | Zbl

[R5] Rees (D.).— Local birational Geometry, Actas del Coloquio internacional sobre Geometría algebraica, Madrid, Sept. (1965). | MR | Zbl

[R6] Rees (D.).— Multiplicities, Hilbert functions and degree functions, Commutative Algebra-Durham 1981, London Math. Soc. Lectures Notes 72 (Ed. R.Y. Sharp, University Press, Cambridge 1983), pp 170-178. | MR | Zbl

[R7] Rees (D.).— Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2), 24, 467-479 (1981). | MR | Zbl

[R8] Rees (D.).— Rings associated with ideals and analytic spread, Math. Proc. Camb. Phil. Soc., 89, 423-432 (1981). | MR | Zbl

[R9] Rees (D.).— Generalizations of reductions and mixed multiplicities, J. London Math. Soc., (2), 29, 397-414 (1984). | MR | Zbl

[R10] Rees (D.).— The general extension of a local ring and mixed multiplicities, Springer Lecture Notes in mathematics No. 1183, 339-360 (1986). | MR | Zbl

[R11] Rees (D.).— Asymptotic properties of ideals, London Math. Soc. Lecture Series, 113 (1988). | Zbl

[R12] Rees (D.).— Izumi’s theorem, Commutative Algebra (Berkeley, CA., 1987), Math. Sci. Res. Inst. Publications, 15, Springer New-York (1989). | MR | Zbl

[R-S] Rees (D.), Sharp (R. Y.).— On a Theorem of B. Teissier on multiplicities of ideals in local rings, J. London Math. Soc., (2), 18, 449-463 (1978). | MR | Zbl

[Sa] Samuel (P.), Some asymptotic properties of powers of ideals, Annals of Math., (2), 56, 11-21 (1952). | MR | Zbl

[T1] Teissier (B.).— Cycles évanescents, sections planes, et conditions de Whitney, Singularités à Cargèse, Astérisque No. 7-8, S.M.F., 285-362 (1973). | MR | Zbl

[T2] Teissier (B.).— Jacobian Newton polyhedra and equisingularity, Proceedings R.I.M.S. Conference on singularities, Kyoto, April 1978. (Publ. R.I.M.S. 1978) et traduction dans : Séminaire sur les singularités, Publ. Math. Université Paris VII no.7, (1980), 193-211. Fichier .pdf disponible sur http://people.math.jussieu.fr/~teissier/articles-Teissier.html | MR

[T3] Teissier (B.).— Variétés polaires I ; invariants polaires des singularités d’hypersurfaces, Inventiones Math. 40, 267-292 (1977). | EuDML | MR | Zbl

[T4] Teissier (B.).— Variétés polaires II ; multiplicités polaires, sections planes, et conditions de Whitney, Proc. Conf. Algebraic Geometry, La Rábida, Springer Lecture Notes in Math., no. 961, 314-491. | MR | Zbl

[T5] Teissier (B.).— Appendice : la courbe monomiale et ses déformations, in : Oscar Zariski, « Le problème des modules pour les branches planes », Publ. Ecole Polytechnique, Paris 1975, reprinted by Hermann ed., Paris, 1986, English translation by Ben Lichtin in The moduli problem for plane branches, University Lecture Series, Vol. 39, A.M.S., (2006). | MR

[T6] Teissier (B.).— The Hunting of invariants in the Geometry of discriminants, in : Real and complex singularities, Oslo 1976, Per Holm editeur, Sijthoff & Noordhoff, p. 565-677 ( 1977). | MR | Zbl

[T7] Teissier (B.).— Résolution simultanée II, in Séminaire sur les Singularités des Surfaces, Lecture Notes in Mathematics, No. 777. Springer, Berlin, 1980. 82-146. Fichier .pdf disponible sur http://people.math.jussieu.fr/~teissier/articles-Teissier.html | Numdam | Zbl

[V] Wolmer (V.).— Integral closure, Rees algebras, multiplicities, algorithms, Springer Monographs in Mathematics. Springer-Verlag, Berlin (2005). | MR | Zbl

[V-S] Hà Huy (V.), Tien So’n (P.).— Newton-Puiseux approximation and Łojasiewicz exponents, Kodai Math. J. 26, no. 1, 1-15 (2003). | MR | Zbl

[Z] Zariski (O.).— Le problème des modules pour les branches planes, Publ. Ecole Polytechnique, Paris 1975, reprinted by Hermann ed., Paris, 1986, English translation by Ben Lichtin in The moduli problem for plane branches, University Lecture Series, Vol. 39, A.M.S., (2006). | MR | Zbl