The Stacks project

Lemma 59.20.6. Let $\tau \in \{ fppf, syntomic, smooth, {\acute{e}tale}, Zariski\} $. Let $S$ be a scheme. Let $(\mathit{Sch}/S)_\tau $ and $(\mathit{Sch}'/S)_\tau $ be two big $\tau $-sites of $S$, and assume that the first is contained in the second. In this case

  1. for any abelian sheaf $\mathcal{F}'$ defined on $(\mathit{Sch}'/S)_\tau $ and any object $U$ of $(\mathit{Sch}/S)_\tau $ we have

    \[ H^ p_\tau (U, \mathcal{F}'|_{(\mathit{Sch}/S)_\tau }) = H^ p_\tau (U, \mathcal{F}') \]

    In words: the cohomology of $\mathcal{F}'$ over $U$ computed in the bigger site agrees with the cohomology of $\mathcal{F}'$ restricted to the smaller site over $U$.

  2. for any abelian sheaf $\mathcal{F}$ on $(\mathit{Sch}/S)_\tau $ there is an abelian sheaf $\mathcal{F}'$ on $(\mathit{Sch}/S)_\tau '$ whose restriction to $(\mathit{Sch}/S)_\tau $ is isomorphic to $\mathcal{F}$.

Proof. By Topologies, Lemma 34.12.2 the inclusion functor $(\mathit{Sch}/S)_\tau \to (\mathit{Sch}'/S)_\tau $ satisfies the assumptions of Sites, Lemma 7.21.8. This implies (2) and (1) follows from Cohomology on Sites, Lemma 21.7.2. $\square$

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