Relations among analytic functions. I

Bierstone, Edward ; Milman, P. D.

Annales de l'Institut Fourier, Tome 37 (1987), p. 187-239 / Harvested from Numdam

Résumé

Ni les ensembles analytiques réels, ni les images d’applications analytiques réelles ou complexes sont cohérents, en général. Soit $\Phi :X\rightarrow Y$ un morphisme d’espaces analytiques, et soit $\Psi :{\cal G}\rightarrow {\cal F}$ un homomorphisme de modules cohérents au-dessus de l’homomorphisme induit d’anneaux $\Phi ^*:{\cal O}_Y\rightarrow {\cal O}_X$. On conjecture que, malgré le manque de cohérence, certains invariants discrets naturels des modules de relations formelles ${\cal R}_a=~{\rm Ker}\widehat{\Psi }_a$, $a\in X$, sont semicontinus supérieurement pour la topologie de Zariski analytique de $X$. On démontre la semicontinuité dans plusieurs cas (par exemple, dans la catégorie algébrique). La semicontinuité du “diagramme des exposants initiaux” fournit un point de vue unifié et des techniques nouvelles et explicites qui se substituent à la cohérence dans des problèmes géoémtriques sur les images d’applications (ensembles semianalytiques ou sousanalytiques) et dans des problèmes analytiques sur les singularités de fonctions différentiables (en particulier, les problèmes classiques de division et composition).

Abstract

Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let $\Phi :X\rightarrow Y$ be a morphism of real analytic spaces, and let $\Psi :{\cal G}\rightarrow {\cal F}$ be a homomorphism of coherent modules over the induced ring homomorphism $\Phi ^*:{\cal O}_Y\rightarrow {\cal O}_X$. We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations ${\cal R}_a=~{\rm Ker}\widehat{\Psi }_a$, $a\in X$, are upper semi-continuous in the analytic Zariski topology of $X$. We prove semicontinuity in many cases (e.g. in the algebraic category). Semicontinuity of the “diagram of initial exponents” provides a unified point of view and explicit new techniques which substitute for coherence in both geometric problems on the images of mappings (semianalytic and subanalytic sets) and analytic problems on the singularities of differentiable functions (in particular, the classical division and composition problems).

Détail

Publié le : 1987-01-01

MR 88g:32013a | Zbl 0611.32002

DOI : https://doi.org/10.5802/aif.1082

Bibliographie

[1] J.M. Aroca, H. Hironaka and J.L. Vicente, The theory of the maximal contact, Mem. Mat. Inst. Jorge Juan, No. 29, Consejo Superior de Investigaciones Científicas, Madrid, 1975. | MR 56 #3344 | Zbl 0366.32008

[2] M. Artin, Algebraic approximation of structures over complete local rings, Inst. Hautes Etudes Sci. Publ. Math., 36 (1969), 23-58. | Numdam | MR 42 #3087 | Zbl 0181.48802

[3] M. Artin, Algebraic spaces, Yale Math. Monographs, No. 3, Yale University Press, New Haven, 1971. | MR 53 #10795 | Zbl 0226.14001

[4] J. Becker and W.R. Zame, Applications of functional analysis to the solution of power series equations, Math. Ann., 243 (1979), 37-54. | Zbl 0413.13015

[5] E. Bierstone and P.D. Milman, Composite differentiable functions, Ann. of Math., 116 (1982), 541-558. | Zbl 0519.58003

[6] E. Bierstone and P.D. Milman, The Newton diagram of an analytic morphism, and applications to differentiable functions, Bull. Amer. Math. Soc. (N.S.), 9 (1983), 315-318. | Zbl 0548.58004

[7] E. Bierstone and G.W. Schwarz, Continuous linear division and extension of C∞ functions, Duke Math. J., 50 (1983), 233-271. | MR 86b:32010 | Zbl 0521.32008

[8] J. Briancon, Weierstrass préparé à la Hironaka, Astérisque, 7, 8 (1973), 67-73. | MR 50 #13584 | Zbl 0297.32004

[9] D.A. Buchsbaum and D. Eisenbud, Some structure theorems for finite free resolutions, Adv. in Math., 12 (1974), 84-139. | MR 49 #4995 | Zbl 0297.13014

[10] C. Chevalley, On the theory of local rings, Ann. of Math., 44 (1943), 690-708. | MR 5,171d | Zbl 0060.06908

[11] Z. Denkowska, S. Ňojasiewicz and J. Stasica, Sur le nombre des composantes connexes de la section d'un sous-analytique, Bull. Acad. Polon. Sci. Sér. Sci. Math., 30 (1982), 333-335. | Zbl 0527.32007

[12] A.M. Gabrielov, Projections of semi-analytic sets, Functional Anal. Appl., 2 (1968), 282-291 = Funkcional. Anal. i PriloŽen., 2, No. 4 (1968), 18-30. | MR 39 #7137 | Zbl 0179.08503

[13] A.M. Gabrielov, Formal relations between analytic functions, Functional Anal. Appl., 5 (1971), 318-319 = Funkeional. Anal. i PriloŽen., 5, No. 4 (1971), 64-65. | MR 46 #2073 | Zbl 0254.32009

[14] A.M. Gabrielov, Formal relations between analytic functions, Math. USSR Izvestija, 7 (1973), 1056-1088 = Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 1056-1090. | Zbl 0297.32007

[15] A. Galligo, Théorème de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier, Grenoble, 29-2 (1979), 107-184. | Numdam | MR 81e:32009 | Zbl 0412.32011

[16] G. Glaeser, Fonctions composées différentiables, Ann. of Math., 77 (1963), 193-209. | MR 26 #624 | Zbl 0106.31302

[17] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Etudes Sci. Publ. Math., 5 (1960). | Numdam | Zbl 0100.08001

[18] H. Grauert, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math., 15 (1972), 171-198. | MR 45 #2206 | Zbl 0237.32011

[19] H. Grauert and R. Remmert, Analytische Stellenalgebren, Springer, Berlin-Heidelberg-New York, 1971. | MR 47 #5290 | Zbl 0231.32001

[20] R.M. Hardt, Stratification of real analytic mappings and images, Invent. Math., 28 (1975), 193-208. | MR 51 #8453 | Zbl 0298.32003

[21] H. Hironaka, Subanalytic sets, Number Theory, Algebraic Geometry and Commutative Algebra, pp. 453-493, Kinokuniya, Tokyo, 1973. | MR 51 #13275 | Zbl 0297.32008

[22] H. Hironaka, Introduction to the theory of infinitely near singular points, Mem. Mat. Inst. Jorge Juan, No. 28, Consejo Superior de Investigaciones Científicas, Madrid, 1974. | MR 53 #3349 | Zbl 0366.32007

[23] H. Hironaka, Stratification and flatness, Real and Complex Singularities, Oslo 1976, Proc. Nineth Nordic Summer School/NAVF Sympos. Math., pp. 199-265, Sijthoff and Noordhoff, Alphen aan den Rijn, 1977. | Zbl 0424.32004

[24] D. Knutson, Algebraic spaces, Lecture Notes in Math., No. 203, Springer, Berlin-Heidelberg-New York, 1971. | MR 46 #1791 | Zbl 0221.14001

[25] M. Lejeune and B. Teissier, Contribution à l'étude des singularités du point de vue du polygone de Newton, Thèse, Université Paris VII, 1973.

[26] S. Ňojasiewicz, Ensembles semi-analytiques, Inst. Hautes Etudes Sci., Bures-sur-Yvette, 1964.

[27] B. Malgrange, Ideals of Differentiable Functions, Oxford University Press, Bombay, 1966.

[28] J. Merrien, Applications des faisceaux analytiques semi-cohérents aux fonctions différentiables, Ann. Inst. Fourier, Grenoble, 31-1 (1981), 63-82. | Numdam | MR 82g:58015 | Zbl 0462.58005

[29] P.D. Milman, The Malgrange-Mather division theorem, Topology, 16 (1977), 395-401. | MR 57 #17699 | Zbl 0397.58011

[30] P.D. Milman, Analytic and polynomial homomorphisms of analytic rings, Math. Ann., 232 (1978), 247-253. | MR 58 #11486 | Zbl 0357.32005

[31] D. Mumford, Algebraic Geometry I. Complex Projective Varieties, Springer, Berlin-Heidelberg-New York, 1976. | Zbl 0356.14002

[32] R. Narasimhan, Introduction to the theory of analytic spaces, Lecture Notes in Math., No. 25, Springer, Berlin-Heidelberg-New York, 1966. | MR 36 #428 | Zbl 0168.06003

[33] D. Popescu, General Néron desingularization and approximation. I, II (preprints, National Institute for Scientific and Technical Creation, Bucharest, 1983).

[34] G.W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology, 14 (1975), 63-68. | MR 51 #6870 | Zbl 0297.57015

[35] Y.-T. Siu, ON -Approximable and holomorphic functions on complex spaces, Duke Math. J., 36 (1969), 451-454. | MR 39 #7144 | Zbl 0181.36202

[36] M. Tamm, Subanalytic sets in the calculus of variations, Acta Math., 146 (1981), 167-199. | MR 82h:32012 | Zbl 0478.58010

[37] J.Cl. Tougeron, Idéaux de Fonctions Différentiables, Springer, Berlin-Heidelberg-New York, 1972. | Zbl 0251.58001

[38] J.Cl. Tougeron, Fonctions composées différentiables : cas algébrique, Ann. Inst. Fourier, Grenoble, 30-4 (1980), 51-74. | Numdam | MR 82e:58020 | Zbl 0427.58007

[39] J.Cl. Tougeron, Existence de bornes uniformes pour certaines familles d'idéaux de l'anneau des séries formelles k [[x]]. Applications (to appear).

[40] O. Zariski and P. Samuel, Commutative Algebra, Vol. II, Springer, New York-Heidelberg-Berlin, 1975.