The Stacks project

Remark 61.14.4. Let $X$ be a scheme. Because $X_{pro\text{-}\acute{e}tale}$ has enough weakly contractible objects for all $K$ in $D(X_{pro\text{-}\acute{e}tale})$ we have $K = R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$ by Cohomology on Sites, Proposition 21.51.2. Since $R\Gamma $ commutes with $R\mathop{\mathrm{lim}}\nolimits $ by Injectives, Lemma 19.13.6 we see that

\[ R\Gamma (X, K) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \tau _{\geq -n}K) \]

in $D(\textit{Ab})$. This will sometimes allow us to extend results from bounded below complexes to all complexes.

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