The Stacks project

Theorem 59.51.3. Let $X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i$ be a limit of a directed system of schemes with affine transition morphisms $f_{i'i} : X_{i'} \to X_ i$. We assume that $X_ i$ is quasi-compact and quasi-separated for all $i \in I$. Let $(\mathcal{F}_ i, \varphi _{i'i})$ be a system of abelian sheaves on $(X_ i, f_{i'i})$. Denote $f_ i : X \to X_ i$ the projection and set $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i$. Then

\[ \mathop{\mathrm{colim}}\nolimits _{i\in I} H_{\acute{e}tale}^ p(X_ i, \mathcal{F}_ i) = H_{\acute{e}tale}^ p(X, \mathcal{F}). \]

for all $p \geq 0$.

Proof. By Topologies, Lemma 34.4.12 we can compute the cohomology of $\mathcal{F}$ on $X_{affine, {\acute{e}tale}}$. Thus the result by a combination of Lemma 59.51.2 and Cohomology on Sites, Lemma 21.16.6. $\square$

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