Proposition 59.70.3. Let j : U \to X be an étale morphism of schemes. Let \mathcal{F} in \textit{Ab}(U_{\acute{e}tale}). If \overline{x} : \mathop{\mathrm{Spec}}(k) \to X is a geometric point of X, then
In particular, j_! is an exact functor.
Proposition 59.70.3. Let j : U \to X be an étale morphism of schemes. Let \mathcal{F} in \textit{Ab}(U_{\acute{e}tale}). If \overline{x} : \mathop{\mathrm{Spec}}(k) \to X is a geometric point of X, then
In particular, j_! is an exact functor.
Proof. Exactness of j_! is very general, see Modules on Sites, Lemma 18.19.3. Of course it does also follow from the description of stalks. The formula for the stalk follows from Modules on Sites, Lemma 18.38.1 and the description of points of the small étale site in terms of geometric points, see Lemma 59.29.12.
For later use we note that the isomorphism
constructed in Modules on Sites, Lemma 18.38.1 sends (V, \overline{v}, \varphi , s) to the class of s in the stalk of \mathcal{F} at \overline{u} = \varphi (\overline{v}). \square
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Comment #3543 by Dario Weißmann on
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