The Stacks project

Lemma 96.16.1. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\} $. Let $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ be an object lying over the scheme $U$. Let $\mathcal{F}$ be an object of $\textit{Ab}(\mathcal{X}_\tau )$ or $\textit{Mod}(\mathcal{X}_\tau , \mathcal{O}_\mathcal {X})$. Then

\[ H^ p_\tau (x, \mathcal{F}) = H^ p((\mathit{Sch}/U)_\tau , x^{-1}\mathcal{F}) \]

and if $\tau = {\acute{e}tale}$, then we also have

\[ H^ p_{\acute{e}tale}(x, \mathcal{F}) = H^ p(U_{\acute{e}tale}, \mathcal{F}|_{U_{\acute{e}tale}}). \]

Proof. The first statement follows from Cohomology on Sites, Lemma 21.7.1 and the equivalence of Lemma 96.9.4. The second statement follows from the first combined with Étale Cohomology, Lemma 59.20.3. $\square$

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Your email address will not be published.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


View Lemma 96.16.1 as pdf