Lemma 15.69.5. Let $R$ be a ring. Let $K \in D(R)$.

If $K$ is in $D^ b(R)$ and $H^ i(K)$ has finite injective dimension for all $i$, then $K$ has finite injective dimension.

If $K^\bullet $ represents $K$, is a bounded complex of $R$-modules, and $K^ i$ has finite injective dimension for all $i$, then $K$ has finite injective dimension.

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