# Exposé on a conjecture of Tougeron

Annales de l'institut Fourier (1977)

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

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topBecker, 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

top- [1] S. ABHYANKAR, Resolution of singularities of embedded algebraic surfaces, Academic Press, 1966. Zbl0147.20504MR36 #164
- [2] S. ABHYANKAR and M. VANDER PUT, Homomorphism of analytic local rings, Creile's J., 242 (1970), 26-60. Zbl0193.00501
- [3] M. ARTIN, On solutions to analytic equations, Invent. Math., 5 (1968), 277-291. Zbl0172.05301MR38 #344
- [4] A. M. GABRIELOV, The formal relations between analytic functions, Funck, Analiz. Appl., 5 (1971), 64-65. Zbl0254.32009MR46 #2073
- [5] A. M. GABRIELOV, Formal relations between analytic functions, Izv. Akad. Nauk. SSR, 37 (1973), 1056-1088. Zbl0297.32007
- [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] M. NAGATA, Local Rings, Interscience Publishers, 1962. Zbl0123.03402MR27 #5790
- [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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