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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}.

  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


Comments (2)

Comment #2336 by Keenan Kidwell on

I think that, for the existence of a left inverse to in , one should cite 13.18.6 instead of 13.18.7.


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