Lemma 15.90.8. Let $A$ be a ring and $I \subset A$ an ideal. If $A$ is $I$-adically complete then any pseudo-coherent object of $D(A)$ is derived complete.

Proof. 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 $I$-adically complete, it is derived complete (Lemma 15.90.3). It follows that $H^ n(K)$ is derived complete for all $n$, by part (1) of Lemma 15.90.6. This in turn implies that $K$ is derived complete by part (2) of the same lemma. $\square$

## Comments (2)

Comment #6207 by on

I find it weird to phrase the assumption as $A$ being $I$-adically complete. Here what we really need is $A$ being derived $I$-complete.

Comment #6208 by on

This is a good point! I will strengthen the lemma in the way you mention the next time I go through all the comments. Thanks.

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• 10 comment(s) on Section 15.90: Derived Completion

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