The Stacks project

Lemma 95.26.3. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. The comparision morphism $\epsilon : \mathcal{X}_{affine, {\acute{e}tale}} \to \mathcal{X}_{affine}$ satisfies the assumptions and conclusions of Cohomology on Sites, Lemma 21.43.12.

Proof. Assumption (1) holds by definition of $\mathcal{X}_{affine}$. For condition (2) we use that for $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{X})$ lying over the affine scheme $U = p(x)$ we have an equivalence $\mathcal{X}_{affine, {\acute{e}tale}}/x = (\textit{Aff}/U)_{\acute{e}tale}$ compatible with structure sheaves; see discussion in Section 95.9. Thus it suffices to show: given an affine scheme $U = \mathop{\mathrm{Spec}}(R)$ and a complex of $R$-modules $M^\bullet $ the total cohomology of the complex of modules on $(\textit{Aff}/U)_{\acute{e}tale}$ associated to $M^\bullet $ is quasi-isomorphic to $M^\bullet $. This follows from a combination of: Derived Categories of Schemes, Lemma 36.3.5 (total cohomology of complexes of modules over affines in the Zariski topology), Derived Categories of Spaces, Remark 74.6.3 (agreement between total cohomology in small Zariski and étale topologies for quasi-coherent complexes of modules), and Étale Cohomology, Lemma 59.99.3 (to see that the étale cohomology of a complex of modules on the big étale site of a scheme may be computed after restricting to the small étale site). $\square$


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Your email address will not be published.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 0H0J. Beware of the difference between the letter 'O' and the digit '0'.