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

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

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