Universal covering spaces and fundamental groups in algebraic geometry as schemes
Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, p. 489-526

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.

En topologie, les notions de groupe fondamental et de revêtement universel sont lieés l’une à l’autre. En suivant l’exemple topologique, on construit un schéma en groupes fondamentaux d’un revêtement universel, qui sont tous les deux des schémas. Une fibre géométrique du schéma en groupes fondamentaux est homéomorphe au groupe fondamental étale. Ces constructions s’appliquent à tout schéma quasi-compact et quasi-séparé. Avec des méthodes et des hypothèses différentes, ce schéma en groupes fondamentaux a déjà été construit par Deligne.

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     author = {Vakil, Ravi and Wickelgren, Kirsten},
     title = {Universal covering spaces and fundamental groups in algebraic geometry as schemes},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {2},
     year = {2011},
     pages = {489-526},
     doi = {10.5802/jtnb.774},
     mrnumber = {2817942},
     zbl = {1228.14019},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2011__23_2_489_0}
}
Vakil, Ravi; Wickelgren, Kirsten. Universal covering spaces and fundamental groups in algebraic geometry as schemes. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 489-526. doi : 10.5802/jtnb.774. http://www.numdam.org/item/JTNB_2011__23_2_489_0/

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