Lemma 13.20.1 (05TH)—The Stacks project

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 $\mathcal{A}$.

$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 $\mathcal{D}$.
$I$ 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. $\square$

Comments (3)

Comment #2336 by Keenan Kidwell on December 25, 2016 at 21:55

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.

Comment #2407 by Johan on February 17, 2017 at 13:05

Thanks, Fixed here.

Comment #10176 by Quoc Ho on May 29, 2025 at 06:08

For part (1), " $I^\bullet$ computes..." should be replaced by " $F(I^\bullet)$ computes...".

Comments (3)

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