Exposé on a conjecture of Tougeron

Joseph Becker

Annales de l'institut Fourier (1977)

  • Volume: 27, Issue: 4, page 9-27
  • ISSN: 0373-0956

Abstract

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An algebra homomorphism of the locatized affine rings of an algebraic variety is continuous in the Krull topology of the respective local rings. It is not necessarily open or closed in the Krull topology. However, we show that the induced map on the associated analytic local rings is also open and closed in the Krull topology. To do this we prove a conjecture of Tougeron which states that if η is an analytic curve on an analytic variety V and f is a formal power series which is convergent when restricted to all curves η ' on V near η (in the Krull topology), then f is convergent when restricted to V .

How to cite

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Becker, Joseph. "Exposé on a conjecture of Tougeron." Annales de l'institut Fourier 27.4 (1977): 9-27. <http://eudml.org/doc/74342>.

@article{Becker1977,
abstract = {An algebra homomorphism of the locatized affine rings of an algebraic variety is continuous in the Krull topology of the respective local rings. It is not necessarily open or closed in the Krull topology. However, we show that the induced map on the associated analytic local rings is also open and closed in the Krull topology. To do this we prove a conjecture of Tougeron which states that if $\eta $ is an analytic curve on an analytic variety $V$ and $f$ is a formal power series which is convergent when restricted to all curves $\eta ^\{\prime \}$ on $V$ near $\eta $ (in the Krull topology), then $f$ is convergent when restricted to $V$.},
author = {Becker, Joseph},
journal = {Annales de l'institut Fourier},
language = {eng},
number = {4},
pages = {9-27},
publisher = {Association des Annales de l'Institut Fourier},
title = {Exposé on a conjecture of Tougeron},
url = {http://eudml.org/doc/74342},
volume = {27},
year = {1977},
}

TY - JOUR
AU - Becker, Joseph
TI - Exposé on a conjecture of Tougeron
JO - Annales de l'institut Fourier
PY - 1977
PB - Association des Annales de l'Institut Fourier
VL - 27
IS - 4
SP - 9
EP - 27
AB - An algebra homomorphism of the locatized affine rings of an algebraic variety is continuous in the Krull topology of the respective local rings. It is not necessarily open or closed in the Krull topology. However, we show that the induced map on the associated analytic local rings is also open and closed in the Krull topology. To do this we prove a conjecture of Tougeron which states that if $\eta $ is an analytic curve on an analytic variety $V$ and $f$ is a formal power series which is convergent when restricted to all curves $\eta ^{\prime }$ on $V$ near $\eta $ (in the Krull topology), then $f$ is convergent when restricted to $V$.
LA - eng
UR - http://eudml.org/doc/74342
ER -

References

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  1. [1] S. ABHYANKAR, Resolution of singularities of embedded algebraic surfaces, Academic Press, 1966. Zbl0147.20504MR36 #164
  2. [2] S. ABHYANKAR and M. VANDER PUT, Homomorphism of analytic local rings, Creile's J., 242 (1970), 26-60. Zbl0193.00501
  3. [3] M. ARTIN, On solutions to analytic equations, Invent. Math., 5 (1968), 277-291. Zbl0172.05301MR38 #344
  4. [4] A. M. GABRIELOV, The formal relations between analytic functions, Funck, Analiz. Appl., 5 (1971), 64-65. Zbl0254.32009MR46 #2073
  5. [5] A. M. GABRIELOV, Formal relations between analytic functions, Izv. Akad. Nauk. SSR, 37 (1973), 1056-1088. Zbl0297.32007
  6. [6] H. HIRONAKA, Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. Math., 79 (1964), 109-326. Zbl0122.38603MR33 #7333
  7. [7] M. NAGATA, Local Rings, Interscience Publishers, 1962. Zbl0123.03402MR27 #5790
  8. [8] J. C. TOUGERON, Courbes analytiques sur un germe d'espace analytique et applications, Ann. Inst. Fourier, Grenoble, 26, 2 (1976), 117-131. Zbl0318.32005MR54 #3011

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