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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