Étale Cohomology Seminar Lecture 2
نویسنده
چکیده
Proposition 1.1. (a) Any open immersion is étale. (b) The composite of two étale morphisms is étale. (c) Any base change of an étale morphism is étale. (d) If φ ◦ ψ and φ are étale, then so is ψ. Proposition 1.2. Let f : X → Y be an étale morphism. (a) For all x ∈ X, OX,x and OY,f(x) have the same Krull dimension. (b) The morphism f is quasi-finite. (c) The morphism f is open. (d) If Y is reduced, then so is X. (e) If Y is normal, then so is X. (f) If Y is regular, then so is X.
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Étale Cohomology Seminar Lecture 4
In general, the sheaf criterion on the étale topology may be difficult to verify directly, as a scheme will in general have many étale covers. It is clear that a necessary condition for a presheaf F to be a sheaf on Xet is that it be a sheaf with respect to Zariski covers (i.e., its restriction to Xzar is a sheaf), and that it be a sheaf with respect to one-piece étale covers (V → U) such that ...
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