Relations among analytic functions. I

Edward Bierstone; P. D. Milman

Annales de l'institut Fourier (1987)

  • Volume: 37, Issue: 1, page 187-239
  • ISSN: 0373-0956

Abstract

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Neither real analytic sets nor the images of real or complex analytic mappings are, in general, coherent. Let Φ : X Y 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 , a 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).

How to cite

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Bierstone, Edward, and Milman, P. D.. "Relations among analytic functions. I." Annales de l'institut Fourier 37.1 (1987): 187-239. <http://eudml.org/doc/74743>.

@article{Bierstone1987,
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).},
author = {Bierstone, Edward, Milman, P. D.},
journal = {Annales de l'institut Fourier},
keywords = {real analytic spaces; semicontinuity; singularities of differentiable functions; division; composition; analytic function},
language = {eng},
number = {1},
pages = {187-239},
publisher = {Association des Annales de l'Institut Fourier},
title = {Relations among analytic functions. I},
url = {http://eudml.org/doc/74743},
volume = {37},
year = {1987},
}

TY - JOUR
AU - Bierstone, Edward
AU - Milman, P. D.
TI - Relations among analytic functions. I
JO - Annales de l'institut Fourier
PY - 1987
PB - Association des Annales de l'Institut Fourier
VL - 37
IS - 1
SP - 187
EP - 239
AB - 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).
LA - eng
KW - real analytic spaces; semicontinuity; singularities of differentiable functions; division; composition; analytic function
UR - http://eudml.org/doc/74743
ER -

References

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  1. [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. Zbl0366.32008MR56 #3344
  2. [2] M. ARTIN, Algebraic approximation of structures over complete local rings, Inst. Hautes Etudes Sci. Publ. Math., 36 (1969), 23-58. Zbl0181.48802MR42 #3087
  3. [3] M. ARTIN, Algebraic spaces, Yale Math. Monographs, No. 3, Yale University Press, New Haven, 1971. Zbl0226.14001MR53 #10795
  4. [4] J. BECKER and W.R. ZAME, Applications of functional analysis to the solution of power series equations, Math. Ann., 243 (1979), 37-54. Zbl0413.13015
  5. [5] E. BIERSTONE and P.D. MILMAN, Composite differentiable functions, Ann. of Math., 116 (1982), 541-558. Zbl0519.58003
  6. [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. Zbl0548.58004
  7. [7] E. BIERSTONE and G.W. SCHWARZ, Continuous linear division and extension of C∞ functions, Duke Math. J., 50 (1983), 233-271. Zbl0521.32008MR86b:32010
  8. [8] J. BRIANCON, Weierstrass préparé à la Hironaka, Astérisque, 7, 8 (1973), 67-73. Zbl0297.32004MR50 #13584
  9. [9] D.A. BUCHSBAUM and D. EISENBUD, Some structure theorems for finite free resolutions, Adv. in Math., 12 (1974), 84-139. Zbl0297.13014MR49 #4995
  10. [10] C. CHEVALLEY, On the theory of local rings, Ann. of Math., 44 (1943), 690-708. Zbl0060.06908MR5,171d
  11. [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. Zbl0527.32007
  12. [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. Zbl0179.08503MR39 #7137
  13. [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. Zbl0254.32009MR46 #2073
  14. [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. Zbl0297.32007
  15. [15] A. GALLIGO, Théorème de division et stabilité en géométrie analytique locale, Ann. Inst. Fourier, Grenoble, 29-2 (1979), 107-184. Zbl0412.32011MR81e:32009
  16. [16] G. GLAESER, Fonctions composées différentiables, Ann. of Math., 77 (1963), 193-209. Zbl0106.31302MR26 #624
  17. [17] H. GRAUERT, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Inst. Hautes Etudes Sci. Publ. Math., 5 (1960). Zbl0100.08001
  18. [18] H. GRAUERT, Über die Deformation isolierter Singularitäten analytischer Mengen, Invent. Math., 15 (1972), 171-198. Zbl0237.32011MR45 #2206
  19. [19] H. GRAUERT and R. REMMERT, Analytische Stellenalgebren, Springer, Berlin-Heidelberg-New York, 1971. Zbl0231.32001MR47 #5290
  20. [20] R.M. HARDT, Stratification of real analytic mappings and images, Invent. Math., 28 (1975), 193-208. Zbl0298.32003MR51 #8453
  21. [21] H. HIRONAKA, Subanalytic sets, Number Theory, Algebraic Geometry and Commutative Algebra, pp. 453-493, Kinokuniya, Tokyo, 1973. Zbl0297.32008MR51 #13275
  22. [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. Zbl0366.32007MR53 #3349
  23. [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. Zbl0424.32004
  24. [24] D. KNUTSON, Algebraic spaces, Lecture Notes in Math., No. 203, Springer, Berlin-Heidelberg-New York, 1971. Zbl0221.14001MR46 #1791
  25. [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. [26] S. ňOJASIEWICZ, Ensembles semi-analytiques, Inst. Hautes Etudes Sci., Bures-sur-Yvette, 1964. 
  27. [27] B. MALGRANGE, Ideals of Differentiable Functions, Oxford University Press, Bombay, 1966. 
  28. [28] J. MERRIEN, Applications des faisceaux analytiques semi-cohérents aux fonctions différentiables, Ann. Inst. Fourier, Grenoble, 31-1 (1981), 63-82. Zbl0462.58005MR82g:58015
  29. [29] P.D. MILMAN, The Malgrange-Mather division theorem, Topology, 16 (1977), 395-401. Zbl0397.58011MR57 #17699
  30. [30] P.D. MILMAN, Analytic and polynomial homomorphisms of analytic rings, Math. Ann., 232 (1978), 247-253. Zbl0357.32005MR58 #11486
  31. [31] D. MUMFORD, Algebraic Geometry I. Complex Projective Varieties, Springer, Berlin-Heidelberg-New York, 1976. Zbl0356.14002
  32. [32] R. NARASIMHAN, Introduction to the theory of analytic spaces, Lecture Notes in Math., No. 25, Springer, Berlin-Heidelberg-New York, 1966. Zbl0168.06003MR36 #428
  33. [33] D. POPESCU, General Néron desingularization and approximation. I, II (preprints, National Institute for Scientific and Technical Creation, Bucharest, 1983). 
  34. [34] G.W. SCHWARZ, Smooth functions invariant under the action of a compact Lie group, Topology, 14 (1975), 63-68. Zbl0297.57015MR51 #6870
  35. [35] Y.-T. SIU, ON -Approximable and holomorphic functions on complex spaces, Duke Math. J., 36 (1969), 451-454. Zbl0181.36202MR39 #7144
  36. [36] M. TAMM, Subanalytic sets in the calculus of variations, Acta Math., 146 (1981), 167-199. Zbl0478.58010MR82h:32012
  37. [37] J.Cl. TOUGERON, Idéaux de Fonctions Différentiables, Springer, Berlin-Heidelberg-New York, 1972. Zbl0251.58001
  38. [38] J.Cl. TOUGERON, Fonctions composées différentiables : cas algébrique, Ann. Inst. Fourier, Grenoble, 30-4 (1980), 51-74. Zbl0427.58007MR82e:58020
  39. [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. [40] O. ZARISKI and P. SAMUEL, Commutative Algebra, Vol. II, Springer, New York-Heidelberg-Berlin, 1975. 

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