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 $\mathcal{F}$.

Proof. By Lemma 61.31.3 the inclusion functor $(\mathit{Sch}/S)_{pro\text{-}\acute{e}tale}\to (\mathit{Sch}'/S)_{pro\text{-}\acute{e}tale}$ 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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