Before stating and proving Proposition 59.46.4 in its correct generality we briefly state and prove it for closed immersions. Namely, some of the preceding arguments are quite a bit easier to follow in the case of a closed immersion and so we repeat them here in their simplified form.
In the rest of this section $i : Z \to X$ is a closed immersion. The functor
will be denoted $U \mapsto U_ Z$ as indicated. Since being a closed immersion is preserved under arbitrary base change the scheme $U_ Z$ is a closed subscheme of $U$.
Lemma 59.46.1. Let $i : Z \to X$ be a closed immersion of schemes. Let $U, U'$ be schemes étale over $X$. Let $h : U_ Z \to U'_ Z$ be a morphism over $Z$. Then there exists a diagram such that $a_ Z : W_ Z \to U_ Z$ is an isomorphism and $h = b_ Z \circ (a_ Z)^{-1}$.
\[ \xymatrix{ U & W \ar[l]_ a \ar[r]^ b & U' } \]
Proof. Consider the scheme $M = U \times _ X U'$. The graph $\Gamma _ h \subset M_ Z$ of $h$ is open. This is true for example as $\Gamma _ h$ is the image of a section of the étale morphism $\text{pr}_{1, Z} : M_ Z \to U_ Z$, see Étale Morphisms, Proposition 41.6.1. Hence there exists an open subscheme $W \subset M$ whose intersection with the closed subset $M_ Z$ is $\Gamma _ h$. Set $a = \text{pr}_1|_ W$ and $b = \text{pr}_2|_ W$. $\square$
Lemma 59.46.2. Let $i : Z \to X$ be a closed immersion of schemes. Let $V \to Z$ be an étale morphism of schemes. There exist étale morphisms $U_ i \to X$ and morphisms $U_{i, Z} \to V$ such that $\{ U_{i, Z} \to V\} $ is a Zariski covering of $V$.
Proof. Since we only have to find a Zariski covering of $V$ consisting of schemes of the form $U_ Z$ with $U$ étale over $X$, we may Zariski localize on $X$ and $V$. Hence we may assume $X$ and $V$ affine. In the affine case this is Algebra, Lemma 10.143.10. $\square$
If $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$ is a geometric point of $X$, then either $\overline{x}$ factors (uniquely) through the closed subscheme $Z$, or $Z_{\overline{x}} = \emptyset $. If $\overline{x}$ factors through $Z$ we say that $\overline{x}$ is a geometric point of $Z$ (because it is) and we use the notation “$\overline{x} \in Z$” to indicate this.
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 where $*$ denotes a singleton set.
\[ (i_{small, *}\mathcal{G})_{\overline{x}} = \left\{ \begin{matrix} *
& \text{if}
& \overline{x} \not\in Z
\\ \mathcal{G}_{\overline{x}}
& \text{if}
& \overline{x} \in Z
\end{matrix} \right. \]
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
and on the other hand
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
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. $\square$
Proposition 59.46.4. Let $i : Z \to X$ be a closed immersion of schemes.
The functor
is fully faithful and its essential image is those sheaves of sets $\mathcal{F}$ on $X_{\acute{e}tale}$ whose restriction to $X \setminus Z$ is isomorphic to $*$, and
the functor
is fully faithful and its essential image is those abelian sheaves on $X_{\acute{e}tale}$ whose support is contained in $Z$.
\[ i_{small, *} : \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \]
\[ i_{small, *} : \textit{Ab}(Z_{\acute{e}tale}) \longrightarrow \textit{Ab}(X_{\acute{e}tale}) \]
In both cases $i_{small}^{-1}$ is a left inverse to the functor $i_{small, *}$.
Proof. Let's discuss the case of sheaves of sets. For any sheaf $\mathcal{G}$ on $Z$ the morphism $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism by Lemma 59.46.3 (and Theorem 59.29.10). This implies formally that $i_{small, *}$ is fully faithful, see Sites, Lemma 7.41.1. It is clear that $i_{small, *}\mathcal{G}|_{U_{\acute{e}tale}} \cong *$ where $U = X \setminus Z$. Conversely, suppose that $\mathcal{F}$ is a sheaf of sets on $X$ such that $\mathcal{F}|_{U_{\acute{e}tale}} \cong *$. Consider the adjunction mapping
Combining Lemmas 59.46.3 and 59.36.2 we see that it is an isomorphism. This finishes the proof of (1). The proof of (2) is identical. $\square$
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