1 results
Nonabelian
${{H}^{1}}$ and the Étale Van Kampen Theorem
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- Journal:
- Canadian Journal of Mathematics / Volume 63 / Issue 6 / 01 December 2011
- Published online by Cambridge University Press:
- 20 November 2018, pp. 1388-1415
- Print publication:
- 01 December 2011
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- Article
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- You have access
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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.
