Universal covering spaces and fundamental groups in algebraic geometry as schemes

Ravi Vakil[1]; Kirsten Wickelgren[2]

  • [1] Department of Mathematics, Stanford University Stanford CA USA 94305
  • [2] Dept. of Mathematics, Harvard University One Oxford St. Cambridge MA USA 02138

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 2, page 489-526
  • ISSN: 1246-7405

Abstract

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In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.

How to cite

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Vakil, Ravi, and Wickelgren, Kirsten. "Universal covering spaces and fundamental groups in algebraic geometry as schemes." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 489-526. <http://eudml.org/doc/219852>.

@article{Vakil2011,
abstract = {In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.},
affiliation = {Department of Mathematics, Stanford University Stanford CA USA 94305; Dept. of Mathematics, Harvard University One Oxford St. Cambridge MA USA 02138},
author = {Vakil, Ravi, Wickelgren, Kirsten},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {fundamental group; group scheme; universal cover},
language = {eng},
month = {6},
number = {2},
pages = {489-526},
publisher = {Société Arithmétique de Bordeaux},
title = {Universal covering spaces and fundamental groups in algebraic geometry as schemes},
url = {http://eudml.org/doc/219852},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Vakil, Ravi
AU - Wickelgren, Kirsten
TI - Universal covering spaces and fundamental groups in algebraic geometry as schemes
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/6//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 2
SP - 489
EP - 526
AB - In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the étale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.
LA - eng
KW - fundamental group; group scheme; universal cover
UR - http://eudml.org/doc/219852
ER -

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