The Stacks project

Lemma 61.31.4. Let $S$ be a scheme. Let $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ and $(\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$ be two big pro-étale sites of $S$ as in Definition 61.12.8. Assume that the first is contained in the second. In this case

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

   $$H^ p(U, \mathcal{F}'|_{(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}}) = H^ p(U, \mathcal{F}')$$

   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 $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ there is an abelian sheaf $\mathcal{F}'$ on $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}'$ whose restriction to $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}$ is isomorphic to $\mathcal{F}$.