Lemma 15.91.12. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $K^\bullet$ be a complex of $A$-modules such that $f : K^\bullet \to K^\bullet$ is an isomorphism for some $f \in I$, i.e., $K^\bullet$ is a complex of $A_ f$-modules. Then the derived completion of $K^\bullet$ is zero.

Proof. Indeed, in this case the $R\mathop{\mathrm{Hom}}\nolimits _ A(K, L)$ is zero for any derived complete complex $L$, see Lemma 15.91.1. Hence $K^\wedge$ is zero by the universal property in Lemma 15.91.10. $\square$

There are also:

• 14 comment(s) on Section 15.91: Derived Completion

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).