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.
$\square$

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