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). $\square$

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