The Stacks project

Lemma 95.9.3. Let $\mathcal{F}$ be an ├ętale sheaf on $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$.

  1. If $\varphi : x \to y$ and $\psi : y \to z$ are morphisms of $\mathcal{X}$ lying over $a : U \to V$ and $b : V \to W$, then the composition

    \[ a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_{\acute{e}tale}})) \xrightarrow {a_{small}^{-1}c_\psi } a_{small}^{-1}(\mathcal{F}|_{V_{\acute{e}tale}}) \xrightarrow {c_\varphi } \mathcal{F}|_{U_{\acute{e}tale}} \]

    is equal to $c_{\psi \circ \varphi }$ via the identification

    \[ (b \circ a)_{small}^{-1}(\mathcal{F}|_{W_{\acute{e}tale}}) = a_{small}^{-1}(b_{small}^{-1} (\mathcal{F}|_{W_{\acute{e}tale}})). \]
  2. If $\varphi : x \to y$ lies over an ├ętale morphism of schemes $a : U \to V$, then (95.9.2.2) is an isomorphism.

  3. Suppose $f : \mathcal{Y} \to \mathcal{X}$ is a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ and $y$ is an object of $\mathcal{Y}$ lying over the scheme $U$ with image $x = f(y)$. Then there is a canonical identification $f^{-1}\mathcal{F}|_{U_{\acute{e}tale}} = \mathcal{F}|_{U_{\acute{e}tale}}$.

  4. Moreover, given $\psi : y' \to y$ in $\mathcal{Y}$ lying over $a : U' \to U$ the comparison map $c_\psi : a_{small}^{-1}(f^{-1}\mathcal{F}|_{U_{\acute{e}tale}}) \to f^{-1}\mathcal{F}|_{U'_{\acute{e}tale}}$ is equal to the comparison map $c_{f(\psi )} : a_{small}^{-1}\mathcal{F}|_{U_{\acute{e}tale}} \to \mathcal{F}|_{U'_{\acute{e}tale}}$ via the identifications in (3).

Proof. The verification of these properties is omitted. $\square$


Comments (0)


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Your email address will not be published.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 075D. Beware of the difference between the letter 'O' and the digit '0'.