The Stacks project

Lemma 61.17.1. Let $S$ be a scheme. Let $T$ be an object of $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$.

  1. If $\mathcal{I}$ is injective in $\textit{Ab}((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$, then

    1. $i_ f^{-1}\mathcal{I}$ is injective in $\textit{Ab}(T_{pro\text{-}\acute{e}tale})$,

    2. $\mathcal{I}|_{S_{pro\text{-}\acute{e}tale}}$ is injective in $\textit{Ab}(S_{pro\text{-}\acute{e}tale})$,

  2. If $\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}((\mathit{Sch}/S)_{pro\text{-}\acute{e}tale})$, then

    1. $i_ f^{-1}\mathcal{I}^\bullet $ is a K-injective complex in $\textit{Ab}(T_{pro\text{-}\acute{e}tale})$,

    2. $\mathcal{I}^\bullet |_{S_{pro\text{-}\acute{e}tale}}$ is a K-injective complex in $\textit{Ab}(S_{pro\text{-}\acute{e}tale})$,

Proof. Proof of (1)(a) and (2)(a): $i_ f^{-1}$ is a right adjoint of an exact functor $i_{f, !}$. Namely, recall that $i_ f$ corresponds to a cocontinuous functor $u : T_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ which is continuous and commutes with fibre products and equalizers, see Lemma 61.12.12 and its proof. Hence we obtain $i_{f, !}$ by Modules on Sites, Lemma 18.16.2. It is shown in Modules on Sites, Lemma 18.16.3 that it is exact. Then we conclude (1)(a) and (2)(a) hold by Homology, Lemma 12.29.1 and Derived Categories, Lemma 13.31.9.

Parts (1)(b) and (2)(b) are special cases of (1)(a) and (2)(a) as $i_ S = i_{\text{id}_ S}$. $\square$

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