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 $i_ S = i_{\text{id}_ S}$. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0F67. Beware of the difference between the letter 'O' and the digit '0'.