The Stacks project

Lemma 13.18.8. Let $\mathcal{A}$ be an abelian category. Let $I^\bullet $ be bounded below complex consisting of injective objects. Let $L^\bullet \in K(\mathcal{A})$. Then

\[ \mathop{Mor}\nolimits _{K(\mathcal{A})}(L^\bullet , I^\bullet ) = \mathop{Mor}\nolimits _{D(\mathcal{A})}(L^\bullet , I^\bullet ). \]

Proof. Let $a$ be an element of the right hand side. We may represent $a = \gamma \alpha ^{-1}$ where $\alpha : K^\bullet \to L^\bullet $ is a quasi-isomorphism and $\gamma : K^\bullet \to I^\bullet $ is a map of complexes. By Lemma 13.18.6 we can find a morphism $\beta : L^\bullet \to I^\bullet $ such that $\beta \circ \alpha $ is homotopic to $\gamma $. This proves that the map is surjective. Let $b$ be an element of the left hand side which maps to zero in the right hand side. Then $b$ is the homotopy class of a morphism $\beta : L^\bullet \to I^\bullet $ such that there exists a quasi-isomorphism $\alpha : K^\bullet \to L^\bullet $ with $\beta \circ \alpha $ homotopic to zero. Then Lemma 13.18.7 shows that $\beta $ is homotopic to zero also, i.e., $b = 0$. $\square$


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