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. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like
$\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.