Notes on the Proof of Weil Ii
نویسنده
چکیده
1.1. Fundamental groups. Let FET/X be the category of schemes finite étale over X. Choose x̄ a geometric point of X. The fiber functor Fx̄ : FET → SET is defined as Fx̄(Y ) = HomX(x̄, Y ). Then the étale fundamental group is defined as π1(X, x̄) = AutFx̄ π1(X, x̄) is a profinite group. If f : X → Y is a morphism, then it induces f∗ : π1(X, x̄) → π1(Y, f(x̄)). In the following, we will often omit the ”base” point x̄ and then f∗ is well-defined up to inner automorphisms. If X is the spectrum of a field K, then π1(K, K̄) is just the Galois group Gal(K̄/K). If X is geometrically connected, one has the exact sequence
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