Lemma 59.104.1. Let $f : X \to Y$ be a morphism of schemes which has a section. Then the functor
sending $\mathcal{G}$ in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ to the canonical descent datum is an equivalence of categories.
Lemma 59.104.1. Let $f : X \to Y$ be a morphism of schemes which has a section. Then the functor
sending $\mathcal{G}$ in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ to the canonical descent datum is an equivalence of categories.
Proof. This is formal and depends only on functoriality of the pullback functors. We omit the details. Hint: If $s : Y \to X$ is a section, then a quasi-inverse is the functor sending $(\mathcal{F}, \varphi )$ to $s_{small}^{-1}\mathcal{F}$. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)