×

On Gabrielov’s regularity condition for analytic mappings. (English) Zbl 0773.32009

Gabrielov introduced three kinds of analytic invariants for an analytic mapping \(\varphi:X\to Y\), where \(X\) and \(Y\) are analytic manifolds over \(K=\mathbb{R}\) or \(\mathbb{C}\): \(r_ 1(x)=\) the generic rank of \(\varphi\) near \(x\), \(r_ 3(x)=\) the Krull dimension of \({\hat{\mathcal O}}_{Y,\varphi(x)}/\text{Ker} \hat\varphi^*_ x\), \(r_ 2(x)=\) the Krull dimension of \({\mathcal O}_{Y,\varphi(x)}/\text{Ker} \varphi^*_ x\), with \(\varphi_ x^*:{\mathcal O}_{Y,\varphi(x)}\to{\mathcal O}_{X,x}\) and “\(^ \wedge\)” standing for completion. It is easy to see that \(r_ 1(x)\leq r_ 2(x)\leq r_ 3(x)\).
E. Bierstone and P. D. Milman studied these invariants [Ann. Math., II. Ser. 116, No. 3, 541-558 (1982; Zbl 0519.58003), Proc. Symp. Pure Math. 40, Part 1, 127-136 (1983; Zbl 0519.58004), Ann. Inst. Fourier 37, No. 1, 187-239 (1987; Zbl 0611.32002), and Ann. Inst. Fourier 37, No. 2, 49-77 (1987; Zbl 0611.32003)] and introduced the notion of the mapping \(\varphi\) being regular and \(x\) if \(r_ 1(x)=r_ 3(x)\). They used this notion to prove the composition hypothesis of Glaeser.
The author proves that the set \(nR\varphi=\{x\in X:\varphi\) is not regular at \(x\}\) is nowhere dense and analytic in \(X\) (Theorem 1). He also constructs a local filtration by analytic subsets \(U=X_ 0\supset X_ 1\supset\cdots\supset X_{s+1}=nR(\varphi)\cap U\) such that, for each \(j=0,\dots,s\), the sheaf of ideals \(\text{Ker} \varphi^*\) is finitely generated over \(X_ j-X_{j+1}\) (Theorem 2). Both results are difficult and useful.
Reviewer: M. Hervé (Paris)

MSC:

32B20 Semi-analytic sets, subanalytic sets, and generalizations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] J. Becker and W. R. Zame, Applications of functional analysis to the solution of power series equations , Math. Ann. 243 (1979), no. 1, 37-54. · Zbl 0413.13015
[2] E. Bierstone and P. D. Milman, Composite differentiable functions , Ann. of Math. (2) 116 (1982), no. 3, 541-558. JSTOR: · Zbl 0519.58003
[3] E. Bierstone and P. D. Milman, Algebras of composite differentiable functions , Singularities, Part 1 (Arcata, Calif., 1981), Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 127-136. · Zbl 0519.58004
[4] 1 E. Bierstone and P. D. Milman, Relations among analytic functions. I , Ann. Inst. Fourier (Grenoble) 37 (1987), no. 1, 187-239. · Zbl 0611.32002
[5] 2 E. Bierstone and P. D. Milman, Relations among analytic functions. II , Ann. Inst. Fourier (Grenoble) 37 (1987), no. 2, 49-77. · Zbl 0611.32003
[6] E. Bierstone and P. D. Milman, Local analytic invariants and splitting theorems in differential analysis , Israel J. Math. 60 (1987), no. 3, 257-280. · Zbl 0653.58007
[7] E. Bierstone and P. D. Milman, Semianalytic and subanalytic sets , Inst. Hautes Études Sci. Publ. Math. (1988), no. 67, 5-42. · Zbl 0674.32002
[8] E. Bierstone and G. W. Schwarz, Continuous linear division and extension of \(\mathcal C^\infty \) functions , Duke Math. J. 50 (1983), no. 1, 233-271. · Zbl 0521.32008
[9] Z. Denkowska, S. Łojasiewicz, and J. Stasica, Certaines propriétés élémentaires des ensembles sous-analytiques , Bull. Acad. Polish. Sci. Sér. Sci. Math. 27 (1979), no. 7-8, 529-536 (1980). · Zbl 0435.32006
[10] Z. Denkowska, S. Łojasiewicz, and J. Stasica, Sur le théorème du complémentaire pour les ensembles sous-analytiques , Bull. Acad. Polish. Sci. Sér. Sci. Math. 27 (1979), no. 7-8, 537-539 (1980). · Zbl 0457.32003
[11] A. M. Gabrièlov, Formal relations among analytic functions , Izv. Akad. Nauk USSR Ser. Math. 37 (1973), 1056-1090. · Zbl 0288.32008
[12] M. Galbiati, Stratifications et ensemble de non-cohérence d’un espace analytique réel , Invent. Math. 34 (1976), no. 2, 113-128. · Zbl 0314.32006
[13] G. Glaeser, Fonctions composées différentiables , Ann. of Math. (2) 77 (1963), 193-209. · Zbl 0106.31302
[14] H. Grauert and R. Remmert, Analytische Stellenalgebren , Springer-Verlag, Berlin, 1971. · Zbl 0231.32001
[15] R. M. Hardt, Stratification of real analytic mappings and images , Invent. Math. 28 (1975), 193-208. · Zbl 0298.32003
[16] R. M. Hardt, Triangulation of subanalytic sets and proper light subanalytic maps , Invent. Math. 38 (1976/77), no. 3, 207-217. · Zbl 0338.32006
[17] M. Hervé, Several complex variables. Local theory , Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1963. · Zbl 0113.29003
[18] H. Hironaka, Subanalytic sets , Number theory, algebraic geometry and commutative algebra, in honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, pp. 453-493. · Zbl 0297.32008
[19] H. Hironaka, Triangulations of algebraic sets , Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975, pp. 165-185. · Zbl 0332.14001
[20] S. Izumi, Gabrielov’s rank condition is equivalent to an inequality of reduced orders , Math. Ann. 276 (1986), no. 1, 81-89. · Zbl 0612.32013
[21] S. Izumi, The rank condition and convergence of formal functions , Duke Math. J. 59 (1989), no. 1, 241-264. · Zbl 0688.32008
[22] S. Łojasiewicz, Ensembles semi-analytiques , Inst. Hautes Études Sci. (1964), Bures-sur-Yvette.
[23] S. Łojasiewicz, Sur la séparation régulière , Geometry seminars, 1985 (Italian) (Bologna, 1985), Univ. Stud. Bologna, Bologna, 1986, pp. 119-121. · Zbl 0612.32008
[24] S. Łojasiewicz, Stratifications et triangulations sous-analytiques , Geometry Seminars, 1986 (Italian) (Bologna, 1986), Univ. Stud. Bologna, Bologna, 1988, pp. 83-97. · Zbl 0673.58006
[25] J. Merrien, Faisceaux analytiques semi-cohérents , Ann. Inst. Fourier (Grenoble) 30 (1980), no. 4, 165-219. · Zbl 0425.32011
[26] P. D. Milman, Analytic and polynomial homomorphisms of analytic rings , Math. Ann. 232 (1978), no. 3, 247-253. · Zbl 0357.32005
[27] P. D. Milman, Complex analytic and formal solutions of real analytic equations in \(\mathbf C\spn\) , Math. Ann. 233 (1978), no. 1, 1-7. · Zbl 0364.32003
[28] R. Narasimhan, Introduction to the theory of analytic spaces , Lecture Notes in Mathematics, vol. 25, Springer-Verlag, Berlin, 1966. · Zbl 0168.06003
[29] W. F. Osgood, Lehrbuch der Funktionentheorie, Band II.1 , Teubner, Leipzig, 1929. · JFM 54.0326.10
[30] W. Pawłucki, Points de Nash des ensembles sous-analytiques , Mem. Amer. Math. Soc. 84 (1990), no. 425, vi+76. · Zbl 0694.32002
[31] W. Pawłucki, Sur les points de Nash des ensembles sous-analytiques , Bull. Polish Acad. Sci. Math. 34 (1986), no. 9-10, 541-545 (1987). · Zbl 0615.32003
[32] W. Pawłucki, On relations among analytic functions and geometry of subanalytic sets , Bull. Polish Acad. Sci. Math. 37 (1989), no. 1-6, 117-125 (1990). · Zbl 0769.32003
[33] J.-B. Poly, Problème des bords: maximale complexité forte , Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, années 1983/1984, Lecture Notes in Math., vol. 1198, Springer, Berlin, 1986, pp. 196-205. · Zbl 0591.32009
[34] J. Souville, Ensembles analytiques complexes à bord semi-analytique , C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 3, 111-114. · Zbl 0589.32016
[35] M. Spivakovsky, On the structure of valuations centered in a local domain , Harvard Univ., · Zbl 0716.13003
[36] J.-Cl. Tougeron, Sur les racines d’un polynôme à coefficients séries formelles , Rennes Univ., · Zbl 0698.13008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.