Lemma 74.5.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Then the canonical map $E \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}E$ is an isomorphism1.

Proof. Denote $\mathcal{H}^ i = H^ i(E)$ the $i$th cohomology sheaf of $E$. Let $\mathcal{B}$ be the set of affine objects of $X_{\acute{e}tale}$. Then $H^ p(U, \mathcal{H}^ i) = 0$ for all $p > 0$, all $i \in \mathbf{Z}$, and all $U \in \mathcal{B}$ as $U$ is an affine scheme. See discussion in Cohomology of Spaces, Section 68.3 and Cohomology of Schemes, Lemma 30.2.2. Thus the lemma follows from Cohomology on Sites, Lemma 21.23.10 with $d = 0$. $\square$

[1] In particular, $E$ has a K-injective representative as in Cohomology on Sites, Lemma 21.24.1.

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