Lemma 13.20.1. Let $\mathcal{A}$ be an abelian category. Let $I \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ be an injective object. Let $I^\bullet$ be a bounded below complex of injectives in $\mathcal{A}$.

1. $I^\bullet$ computes $RF$ relative to $\text{Qis}^{+}(\mathcal{A})$ for any exact functor $F : K^{+}(\mathcal{A}) \to \mathcal{D}$ into any triangulated category $\mathcal{D}$.

2. $I$ is right acyclic for any additive functor $F : \mathcal{A} \to \mathcal{B}$ into any abelian category $\mathcal{B}$.

Proof. Part (2) is a direct consequences of part (1) and Definition 13.15.3. To prove (1) let $\alpha : I^\bullet \to K^\bullet$ be a quasi-isomorphism into a complex. By Lemma 13.18.6 we see that $\alpha$ has a left inverse. Hence the category $I^\bullet /\text{Qis}^{+}(\mathcal{A})$ is essentially constant with value $\text{id} : I^\bullet \to I^\bullet$. Thus also the ind-object

$I^\bullet /\text{Qis}^{+}(\mathcal{A}) \longrightarrow \mathcal{D}, \quad (I^\bullet \to K^\bullet ) \longmapsto F(K^\bullet )$

is essentially constant with value $F(I^\bullet )$. This proves (1), see Definitions 13.14.2 and 13.14.10. $\square$

Comment #2336 by Keenan Kidwell on

I think that, for the existence of a left inverse to $\alpha$ in $\mathcal{K}^+(\mathcal{A})$, one should cite 13.18.6 instead of 13.18.7.

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