Lemma 59.104.5. Let $f : X \to Y$ be a morphism of schemes which is surjective, flat, locally of finite presentation. The functor

is an equivalence of categories.

Lemma 59.104.5. Let $f : X \to Y$ be a morphism of schemes which is surjective, flat, locally of finite presentation. The functor

\[ \mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}) \longrightarrow \text{descent data for étale sheaves wrt }\{ X \to Y\} \]

is an equivalence of categories.

**Proof.**
Exactly as in the proof of Lemma 59.104.2 we claim a quasi-inverse is given by the functor sending $(\mathcal{F}, \varphi )$ to

\[ \mathcal{G} = \text{Equalizer}\left( \xymatrix{ f_*\mathcal{F} \ar@<1ex>[r] \ar@<-1ex>[r] & f_{1, *}\mathcal{F}_1 } \right) \]

and in order to prove this it suffices to show that $f^{-1}\mathcal{G} \to \mathcal{F}$ is an isomorphism. This we may check locally, hence we may and do assume $Y$ is affine. Then we can find a finite surjective morphism $Z \to Y$ such that there exists an open covering $Z = \bigcup W_ i$ such that $W_ i \to Y$ factors through $X$. See More on Morphisms, Lemma 37.48.6. Applying Lemma 59.104.4 we see that it suffices to prove the lemma after replacing $Y$ by $Z$ and $Z \times _ Y Z$ and $f$ by its base change. Thus we may assume $f$ has sections Zariski locally. Of course, using that the problem is local on $Y$ we reduce to the case where we have a section which is Lemma 59.104.1. $\square$

Your email address will not be published.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).

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.

## Comments (0)