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The Stacks project

Lemma 63.3.2. Let j : U \to X be a separated étale morphism. Let \mathcal{F} be an abelian sheaf on U_{\acute{e}tale}. The image of the injective map j_!\mathcal{F} \to j_*\mathcal{F} of Étale Cohomology, Lemma 59.70.6 is the subsheaf of Lemma 63.3.1.

Proof. The construction of j_!\mathcal{F} \to j_*\mathcal{F} in the proof of Étale Cohomology, Lemma 59.70.6 is via the construction of a map j_{p!}\mathcal{F} \to j_*\mathcal{F} of presheaves whose image is clearly contained in the subsheaf of Lemma 63.3.1. Hence since j_!\mathcal{F} is the sheafification of j_{p!}\mathcal{F} we conclude the image of j_!\mathcal{F} \to j_*\mathcal{F} is contained in this subsheaf. Conversely, let s \in j_*\mathcal{F}(V) have support Z proper over V. Then Z \to V is finite with closed image Z' \subset V, see More on Morphisms, Lemma 37.44.1. The restriction of s to V \setminus Z' is zero and the zero section is contained in the image of j_!\mathcal{F} \to j_*\mathcal{F}. On the other hand, if v \in Z', then we can find an étale neighbourhood (V', v') \to (V, v) such that we have a decomposition U_{V'} = W \amalg U'_1 \amalg \ldots \amalg U'_ n into open and closed subschemes with U'_ i \to V' an isomorphism and with T_{V'} \subset U'_1 \amalg \ldots \amalg U'_ n, see Étale Morphisms, Lemma 41.18.2. Inverting the isomorphisms U'_ i \to V' we obtain n morphisms \varphi '_ i : V' \to U and sections s'_ i over V' by pulling back s. Then the section \sum (\varphi '_ i, s'_ i) of j_{p!}\mathcal{F} over V', see formula for j_{p!}\mathcal{F}(V') in proof of Étale Cohomology, Lemma 59.70.6, maps to the restriction of s to V' by construction. We conclude that s is étale locally in the image of j_!\mathcal{F} \to j_*\mathcal{F} and the proof is complete. \square


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