The Stacks project

Lemma 95.26.6. Let $S$ be a scheme. Let $\mathcal{X} \to (\mathit{Sch}/S)_{fppf}$ be a category fibred in groupoids. The comparision morphism $\epsilon : \mathcal{X}_{affine, fppf} \to \mathcal{X}_{affine}$ satisfies the assumptions and conclusions of Cohomology on Sites, Lemma 21.43.12.

Proof. The proof is exactly the same as the proof of Lemma 95.26.3. 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 $(\textit{Aff}/U)_{fppf}$ associated to $M^\bullet $ is quasi-isomorphic to $M^\bullet $. This is √Čtale Cohomology, Lemma 59.101.3. $\square$

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Your email address will not be published.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 0H0X. Beware of the difference between the letter 'O' and the digit '0'.