The class group of a one-dimensional affinoid space

Marius Van Der Put

Annales de l'institut Fourier (1980)

  • Volume: 30, Issue: 4, page 155-164
  • ISSN: 0373-0956

Abstract

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A curve X over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space Y is trivial if and only if Y is a subspace of P 1 . As a consequence, X has locally a trivial class group if and only if the stable reduction of X has only rational components.

How to cite

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Put, Marius Van Der. "The class group of a one-dimensional affinoid space." Annales de l'institut Fourier 30.4 (1980): 155-164. <http://eudml.org/doc/74469>.

@article{Put1980,
abstract = {A curve $X$ over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space $Y$ is trivial if and only if $Y$ is a subspace of $\{\bf P\}^1$. As a consequence, $X$ has locally a trivial class group if and only if the stable reduction of $X$ has only rational components.},
author = {Put, Marius Van Der},
journal = {Annales de l'institut Fourier},
keywords = {curve over a non-archimedean valued field; class group of a one-dimensional connected regular affinoid space; trivial class group},
language = {eng},
number = {4},
pages = {155-164},
publisher = {Association des Annales de l'Institut Fourier},
title = {The class group of a one-dimensional affinoid space},
url = {http://eudml.org/doc/74469},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Put, Marius Van Der
TI - The class group of a one-dimensional affinoid space
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 4
SP - 155
EP - 164
AB - A curve $X$ over a non-archimedean valued field is with respect to its analytic structure a finite union of affinoid spaces. The main result states that the class group of a one dimensional, connected, regular affinoid space $Y$ is trivial if and only if $Y$ is a subspace of ${\bf P}^1$. As a consequence, $X$ has locally a trivial class group if and only if the stable reduction of $X$ has only rational components.
LA - eng
KW - curve over a non-archimedean valued field; class group of a one-dimensional connected regular affinoid space; trivial class group
UR - http://eudml.org/doc/74469
ER -

Citations in EuDML Documents

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  1. Q. Liu, Ouverts analytiques d'une courbe algébrique en géométrie rigide
  2. Jean Fresnel, Marius Van Der Put, Localisation formelle et groupe de Picard
  3. Qing Liu, Ouverts analytiques d'une courbe algébrique en géométrie rigide
  4. M. Van der Put, Cohomology on affinoid spaces
  5. Vladimir G. Berkovich, Étale cohomology for non-Archimedean analytic spaces
  6. Marius Van der Put, De Rham cohomology of affinoid spaces
  7. Jean Fresnel, Marius van der Put, Uniformisation des variétés abéliennes

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