Relations among analytic functions. I
Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 187-239.

Ni les ensembles analytiques réels, ni les images d’applications analytiques réelles ou complexes sont cohérents, en général. Soit Φ:XY un morphisme d’espaces analytiques, et soit Ψ:𝒢 un homomorphisme de modules cohérents au-dessus de l’homomorphisme induit d’anneaux Φ * :𝒪 Y 𝒪 X . On conjecture que, malgré le manque de cohérence, certains invariants discrets naturels des modules de relations formelles a = Ker Ψ ^ a , aX, 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).

Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let Φ:XY be a morphism of real analytic spaces, and let Ψ:𝒢 be a homomorphism of coherent modules over the induced ring homomorphism Φ * :𝒪 Y 𝒪 X . We conjecture that, despite the failure of coherence, certain natural discrete invariants of the modules of formal relations a = Ker Ψ ^ a , aX, 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).

@article{AIF_1987__37_1_187_0,
     author = {Bierstone, Edward and Milman, P. D.},
     title = {Relations among analytic functions. {I}},
     journal = {Annales de l'Institut Fourier},
     pages = {187--239},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {1},
     year = {1987},
     doi = {10.5802/aif.1082},
     mrnumber = {88g:32013a},
     zbl = {0611.32002},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1082/}
}
TY  - JOUR
AU  - Bierstone, Edward
AU  - Milman, P. D.
TI  - Relations among analytic functions. I
JO  - Annales de l'Institut Fourier
PY  - 1987
SP  - 187
EP  - 239
VL  - 37
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1082/
DO  - 10.5802/aif.1082
LA  - en
ID  - AIF_1987__37_1_187_0
ER  - 
%0 Journal Article
%A Bierstone, Edward
%A Milman, P. D.
%T Relations among analytic functions. I
%J Annales de l'Institut Fourier
%D 1987
%P 187-239
%V 37
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1082/
%R 10.5802/aif.1082
%G en
%F AIF_1987__37_1_187_0
Bierstone, Edward; Milman, P. D. Relations among analytic functions. I. Annales de l'Institut Fourier, Tome 37 (1987) no. 1, pp. 187-239. doi : 10.5802/aif.1082. http://www.numdam.org/articles/10.5802/aif.1082/

[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 | Zbl

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

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

[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

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

[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

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

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

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

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

[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

[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 | Zbl

[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 | Zbl

[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

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

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

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

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

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

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

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

[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 | Zbl

[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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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

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

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

[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.

Cité par Sources :