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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