The Stacks project

Lemma 59.46.3. Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{G}$ be a sheaf of sets on $Z_{\acute{e}tale}$. Let $\overline{x}$ be a geometric point of $X$. Then

\[ (i_{small, *}\mathcal{G})_{\overline{x}} = \left\{ \begin{matrix} * & \text{if} & \overline{x} \not\in Z \\ \mathcal{G}_{\overline{x}} & \text{if} & \overline{x} \in Z \end{matrix} \right. \]

where $*$ denotes a singleton set.

Proof. Note that $i_{small, *}\mathcal{G}|_{U_{\acute{e}tale}} = *$ is the final object in the category of étale sheaves on $U$, i.e., the sheaf which associates a singleton set to each scheme étale over $U$. This explains the value of $(i_{small, *}\mathcal{G})_{\overline{x}}$ if $\overline{x} \not\in Z$.

Next, suppose that $\overline{x} \in Z$. Note that

\[ (i_{small, *}\mathcal{G})_{\overline{x}} = \mathop{\mathrm{colim}}\nolimits _{(U, \overline{u})} \mathcal{G}(U_ Z) \]

and on the other hand

\[ \mathcal{G}_{\overline{x}} = \mathop{\mathrm{colim}}\nolimits _{(V, \overline{v})} \mathcal{G}(V). \]

Let $\mathcal{C}_1 = \{ (U, \overline{u})\} ^{opp}$ be the opposite of the category of étale neighbourhoods of $\overline{x}$ in $X$, and let $\mathcal{C}_2 = \{ (V, \overline{v})\} ^{opp}$ be the opposite of the category of étale neighbourhoods of $\overline{x}$ in $Z$. The canonical map

\[ \mathcal{G}_{\overline{x}} \longrightarrow (i_{small, *}\mathcal{G})_{\overline{x}} \]

corresponds to the functor $F : \mathcal{C}_1 \to \mathcal{C}_2$, $F(U, \overline{u}) = (U_ Z, \overline{x})$. Now Lemmas 59.46.2 and 59.46.1 imply that $\mathcal{C}_1$ is cofinal in $\mathcal{C}_2$, see Categories, Definition 4.17.1. Hence it follows that the displayed arrow is an isomorphism, see Categories, Lemma 4.17.2. $\square$

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