The Stacks project

Lemma 59.42.1. Let $f : X \to Y$ be a morphism of schemes. Assume (A).

  1. $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ reflects injections and surjections,

  2. $f_{small}^{-1}f_{small, *}\mathcal{F} \to \mathcal{F}$ is surjective for any abelian sheaf $\mathcal{F}$ on $X_{\acute{e}tale}$,

  3. $f_{small, *} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(Y_{\acute{e}tale})$ is faithful.

Proof. Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Let $U$ be an object of $X_{\acute{e}tale}$. By assumption we can find a covering $\{ W_ i \to U\} $ in $X_{\acute{e}tale}$ such that each $W_ i$ is an open and closed subscheme of $X \times _ Y V_ i$ for some object $V_ i$ of $Y_{\acute{e}tale}$. The sheaf condition shows that

\[ \mathcal{F}(U) \subset \prod \mathcal{F}(W_ i) \]

and that $\mathcal{F}(W_ i)$ is a direct summand of $\mathcal{F}(X \times _ Y V_ i) = f_{small, *}\mathcal{F}(V_ i)$. Hence it is clear that $f_{small, *}$ reflects injections.

Next, suppose that $a : \mathcal{G} \to \mathcal{F}$ is a map of abelian sheaves such that $f_{small, *}a$ is surjective. Let $s \in \mathcal{F}(U)$ with $U$ as above. With $W_ i$, $V_ i$ as above we see that it suffices to show that $s|_{W_ i}$ is étale locally the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(W_ i)$ is a direct summand of $\mathcal{F}(X \times _ Y V_ i)$ it suffices to show that for any $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$ any element $s \in \mathcal{F}(X \times _ Y V)$ is étale locally on $X \times _ Y V$ the image of a section of $\mathcal{G}$ under $a$. Since $\mathcal{F}(X \times _ Y V) = f_{small, *}\mathcal{F}(V)$ we see by assumption that there exists a covering $\{ V_ j \to V\} $ such that $s$ is the image of $s_ j \in f_{small, *}\mathcal{G}(V_ j) = \mathcal{G}(X \times _ Y V_ j)$. This proves $f_{small, *}$ reflects surjections.

Parts (2), (3) follow formally from part (1), see Modules on Sites, Lemma 18.15.1. $\square$


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Your email address will not be published.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 04DK. Beware of the difference between the letter 'O' and the digit '0'.