The Stacks project

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

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

  2. 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.

Proof. Omitted. Hint: Apply the spectral sequences of Derived Categories, Lemma 13.21.3 to the functor $F = \mathop{\mathrm{Hom}}\nolimits _ R(N, -)$ to get a computation of $\mathop{\mathrm{Ext}}\nolimits ^ i_ A(N, K)$ and use the criterion of Lemma 15.69.2. $\square$

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