The Stacks project

Lemma 59.20.3. Let $\tau \in \{ {\acute{e}tale}, Zariski\} $. If $\mathcal{F}$ is an abelian sheaf defined on $(\mathit{Sch}/S)_\tau $, then the cohomology groups of $\mathcal{F}$ over $S$ agree with the cohomology groups of $\mathcal{F}|_{S_\tau }$ over $S$.

Proof. By Topologies, Lemmas 34.3.14 and 34.4.14 the functors $S_\tau \to (\mathit{Sch}/S)_\tau $ satisfy the hypotheses of Sites, Lemma 7.21.8. Hence our lemma follows from Cohomology on Sites, Lemma 21.7.2. $\square$

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