Lemma 59.104.8 (0GF5)—The Stacks project

Processing math: 100%

Table of contents
Part 3: Topics in Scheme Theory
Chapter 59: Étale Cohomology
Section 59.104: Descending étale sheaves
Lemma 59.104.8 (cite)

previous lemma

numberstags

Lemma 59.104.8. Let $S$ be a scheme. Then the category fibred in groupoids

$p : \mathcal{S} \longrightarrow (\mathit{Sch}/S)_ h$

whose fibre category over $U$ is the category $\mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale})$ of sheaves on the small étale site of $U$ is a stack in groupoids.

Proof. To prove the lemma we will check conditions (1), (2), and (3) of More on Flatness, Lemma 38.37.13.

Condition (1) holds because we have glueing for sheaves (and Zariski coverings are étale coverings). See Sites, Lemma 7.26.4.

To see condition (2), suppose that $f : X \to Y$ is a surjective, flat, proper morphism of finite presentation over $S$ with $Y$ affine. Then we have descent for $\{ X \to Y\}$ by either Lemma 59.104.5 or Lemma 59.104.3.

Condition (3) follows immediately from the more general Lemma 59.104.7. $\square$

Comments (0)

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).

Comments (0)

Post a comment