Exposé on a conjecture of Tougeron
Annales de l'Institut Fourier, Volume 27 (1977) no. 4, p. 9-27

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.

Une algèbre d’homomorphismes d’anneaux localisés affines d’une variété algébrique est continue par rapport à la topologie de Krull. Elle peut être ni ouverte ni fermée. Cependant, on montre que l’image induite sur l’anneau local analytique associé est également ouverte et fermée par rapport à la topologie de Krull. Afin de démontrer ceci, on prouve la conjecture suivante de Tougeron : si η est une courbe analytique sur un ensemble analytique V et si f est une série formelle de puissances dont la restriction à toute courbe η sur V dans un voisinage de η (topologie de Krull) est convergente, alors la restriction de f à V est convergente.

@article{AIF_1977__27_4_9_0,
     author = {Becker, Joseph},
     title = {Expos\'e on a conjecture of Tougeron},
     journal = {Annales de l'Institut Fourier},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {27},
     number = {4},
     year = {1977},
     pages = {9-27},
     doi = {10.5802/aif.670},
     zbl = {0337.14002},
     mrnumber = {58 \#10904},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1977__27_4_9_0}
}
Becker, Joseph. Exposé on a conjecture of Tougeron. Annales de l'Institut Fourier, Volume 27 (1977) no. 4, pp. 9-27. doi : 10.5802/aif.670. http://www.numdam.org/item/AIF_1977__27_4_9_0/

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