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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