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. $\square$


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