Lemma 15.73.7. Let $R$ be a ring. If $K^\bullet \in D^ b(R)$ and all its cohomology modules are perfect, then $K^\bullet$ is perfect.

Proof. Follows by induction on the length of the finite complex: use Lemma 15.73.4 and the canonical truncations. $\square$

Comment #95 by Sergei on

Why is not the complex $\oplus_{i\geq 0} R[i]$ (so with trivial differential) not a counterexample to Lemma 45.7 (in More Algebra) ?

Comment #96 by Johan on

Fixed. Thanks!

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