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 $\mathcal{F}$.