Loading web-font TeX/Math/Italic

The Stacks project

Lemma 35.8.5. Let S be a scheme. Denote

\begin{matrix} \text{id}_{\tau , Zar} & : & (\mathit{Sch}/S)_\tau \to S_{Zar}, & \tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} \\ \text{id}_{\tau , {\acute{e}tale}} & : & (\mathit{Sch}/S)_\tau \to S_{\acute{e}tale}, & \tau \in \{ {\acute{e}tale}, smooth, syntomic, fppf\} \\ \text{id}_{small, {\acute{e}tale}, Zar} & : & S_{\acute{e}tale}\to S_{Zar}, \end{matrix}

the morphisms of ringed sites of Remark 35.8.4. Let \mathcal{F} be a sheaf of \mathcal{O}_ S-modules which we view a sheaf of \mathcal{O}-modules on S_{Zar}. Then

  1. (\text{id}_{\tau , Zar})^*\mathcal{F} is the \tau -sheafification of the Zariski sheaf

    (f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F})

    on (\mathit{Sch}/S)_\tau , and

  2. (\text{id}_{small, {\acute{e}tale}, Zar})^*\mathcal{F} is the étale sheafification of the Zariski sheaf

    (f : T \to S) \longmapsto \Gamma (T, f^*\mathcal{F})

    on S_{\acute{e}tale}.

Let \mathcal{G} be a sheaf of \mathcal{O}-modules on S_{\acute{e}tale}. Then

  1. (\text{id}_{\tau , {\acute{e}tale}})^*\mathcal{G} is the \tau -sheafification of the étale sheaf

    (f : T \to S) \longmapsto \Gamma (T, f_{small}^*\mathcal{G})

    where f_{small} : T_{\acute{e}tale}\to S_{\acute{e}tale} is the morphism of ringed small étale sites of Remark 35.8.4.

Proof. Proof of (1). We first note that the result is true when \tau = Zar because in that case we have the morphism of topoi i_ f : \mathop{\mathit{Sh}}\nolimits (T_{Zar}) \to \mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{Zar}) such that \text{id}_{\tau , Zar} \circ i_ f = f_{small} as morphisms T_{Zar} \to S_{Zar}, see Topologies, Lemmas 34.3.13 and 34.3.17. Since pullback is transitive (see Modules on Sites, Lemma 18.13.3) we see that i_ f^*(\text{id}_{\tau , Zar})^*\mathcal{F} = f_{small}^*\mathcal{F} as desired. Hence, by the remark preceding this lemma we see that (\text{id}_{\tau , Zar})^*\mathcal{F} is the \tau -sheafification of the presheaf T \mapsto \Gamma (T, f^*\mathcal{F}).

The proof of (3) is exactly the same as the proof of (1), except that it uses Topologies, Lemmas 34.4.13 and 34.4.17. We omit the proof of (2). \square


Comments (2)

Comment #2959 by Ko Aoki on

Typo in the proof: "" should be replaced by "".

There are also:

  • 3 comment(s) on Section 35.8: Quasi-coherent sheaves and topologies, I

Your email address will not be published.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.