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. $\square$


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