Lemma 15.91.8. Let $A$ be a ring and $I \subset A$ an ideal. If $A$ is derived complete (eg. $I$-adically complete) then any pseudo-coherent object of $D(A)$ is derived complete.

**Proof.**
(Lemma 15.91.3 explains the parenthetical statement of the lemma.) Let $K$ be a pseudo-coherent object of $D(A)$. By definition this means $K$ is represented by a bounded above complex $K^\bullet $ of finite free $A$-modules. Since $A$ is derived complete it follows that $H^ n(K)$ is derived complete for all $n$, by part (1) of Lemma 15.91.6. This in turn implies that $K$ is derived complete by part (2) of the same lemma.
$\square$

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