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The Stacks project

Lemma 59.52.3. Let S be a scheme. Let X = \mathop{\mathrm{lim}}\nolimits _{i \in I} X_ i be a limit of a directed system of schemes over S 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 K \in D^+(S_{\acute{e}tale}). Then

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

for all p \in \mathbf{Z} where K|_{X_ i} and K|_ X are the pullbacks of K to X_ i and X.

Proof. We may represent K by a bounded below complex \mathcal{G}^\bullet of abelian sheaves on S_{\acute{e}tale}. Say \mathcal{G}^ n = 0 for n < a. Denote \mathcal{F}^\bullet _ i and \mathcal{F}^\bullet the pullbacks of this complex of X_ i and X. These complexes represent the objects K|_{X_ i} and K|_ X and we have \mathcal{F}^\bullet = \mathop{\mathrm{colim}}\nolimits f_ i^{-1}\mathcal{F}_ i^\bullet termwise. Hence the lemma follows from Lemma 59.52.1. \square


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