Pointed torsors and Galois groups

Pointed torsors and Galois groups

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Rick Jardine, University of Western Ontario
Fine Hall 314

Suppose that $H$ is an algebraic group which is defined over a field $k$, and let $L$ be the algebraic closure of $k$. The canonical stalk for the etale topology on $k$ induces a simplicial set map from the classifying space $B(H-tors)$ of the groupoid of $H$-torsors (aka. principal $H$-bundles) to the space $BH(L)$. The homotopy fibres of this map are groupoids of pointed torsors. These fibres are analyzed with cocycle techniques, and their path components are categorical representations of the absolute Galois groupoid (suitably defined) in $H$. Analogous results hold for finite etale sites: pointed torsors in that context are classified by continuous morphisms defined on a Grothendieck fundamental groupoid.