Nonabelian ${{H}^{1}}$ and the Étale Van Kampen Theorem

Misamore, Michael D.

Canadian journal of mathematics, Tome 63 (2011) no. 6, pp. 1388-1415

Résumé

Generalized étale homotopy pro-groups $\pi _{1}^{\acute{e}t}(C, x)$ associated with pointed, connected, small Grothendieck sites $(C, x)$ are defined, and their relationship to Galois theory and the theory of pointed torsors for discrete groups is explained.Applications include new rigorous proofs of some folklore results around $\pi _{1}^{\acute{e}t}(\acute{e}t(X) x)$ , a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale van Kampen theorem that gives a simple statement about a pushout square of pro-groups that works for covering families that do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups $\text{ }\!\!\pi\!\!\text{ }_{1}^{\text{Gal}}$ immediately follows.

Détail

DOI : 10.4153/CJM-2011-030-x

Mots-clés : 18G30, 14F35, étale homotopy theory, simplicial sheaves

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