Lemma 59.99.1. Let $S$ be a scheme. Let $T$ be an object of $(\mathit{Sch}/S)_{\acute{e}tale}$.
If $\mathcal{I}$ is injective in $\textit{Ab}((\mathit{Sch}/S)_{\acute{e}tale})$, then
$i_ f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(T_{\acute{e}tale})$,
$\mathcal{I}|_{S_{\acute{e}tale}}$ is injective in $\textit{Ab}(S_{\acute{e}tale})$,
If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}((\mathit{Sch}/S)_{\acute{e}tale})$, then
$i_ f^{-1}\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(T_{\acute{e}tale})$,
$\mathcal{I}^\bullet |_{S_{\acute{e}tale}}$ is a K-injective complex in $\textit{Ab}(S_{\acute{e}tale})$,
The corresponding statements for modules do not hold.
Proof.
Parts (1)(b) and (2)(b) follow formally from the fact that the restriction functor $\pi _{S, *} = i_ S^{-1}$ is a right adjoint of the exact functor $\pi _ S^{-1}$, see Homology, Lemma 12.29.1 and Derived Categories, Lemma 13.31.9.
Parts (1)(a) and (2)(a) can be seen in two ways. First proof: We can use that $i_ f^{-1}$ is a right adjoint of the exact functor $i_{f, !}$. This functor is constructed in Topologies, Lemma 34.4.13 for sheaves of sets and for abelian sheaves in Modules on Sites, Lemma 18.16.2. It is shown in Modules on Sites, Lemma 18.16.3 that it is exact. Second proof. We can use that $i_ f = i_ T \circ f_{big}$ as is shown in Topologies, Lemma 34.4.17. Since $f_{big}$ is a localization, we see that pullback by it preserves injectives and K-injectives, see Cohomology on Sites, Lemmas 21.7.1 and 21.20.1. Then we apply the already proved parts (1)(b) and (2)(b) to the functor $i_ T^{-1}$ to conclude.
Let $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$ and consider the map $2 : \mathcal{O}_ S \to \mathcal{O}_ S$. This is an injective map of $\mathcal{O}_ S$-modules on $S_{\acute{e}tale}$. However, the pullback $\pi _ S^*(2) : \mathcal{O} \to \mathcal{O}$ is not injective as we see by evaluating on $\mathop{\mathrm{Spec}}(\mathbf{F}_2)$. Now choose an injection $\alpha : \mathcal{O} \to \mathcal{I}$ into an injective $\mathcal{O}$-module $\mathcal{I}$ on $(\mathit{Sch}/S)_{\acute{e}tale}$. Then consider the diagram
\[ \xymatrix{ \mathcal{O}_ S \ar[d]_2 \ar[rr]_{\alpha |_{S_{\acute{e}tale}}} & & \mathcal{I}|_{S_{\acute{e}tale}} \\ \mathcal{O}_ S \ar@{..>}[rru] } \]
Then the dotted arrow cannot exist in the category of $\mathcal{O}_ S$-modules because it would mean (by adjunction) that the injective map $\alpha $ factors through the noninjective map $\pi _ S^*(2)$ which cannot be the case. Thus $\mathcal{I}|_{S_{\acute{e}tale}}$ is not an injective $\mathcal{O}_ S$-module.
$\square$
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