The Stacks project

Lemma 13.34.5. Let $\mathcal{A}$ be an abelian category having enough injectives and exact countable products. Then for every complex there is a quasi-isomorphism to a K-injective complex.

Proof. By Lemma 13.34.4 it suffices to show that $K \to R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K$ is an isomorphism for all $K$ in $D(\mathcal{A})$. Consider the defining distinguished triangle

\[ R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K \to \prod \tau _{\geq -n}K \to \prod \tau _{\geq -n}K \to (R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K)[1] \]

By Lemma 13.34.2 we have

\[ H^ p(\prod \tau _{\geq -n}K) = \prod \nolimits _{p \geq -n} H^ p(K) \]

It follows in a straightforward manner from the long exact cohomology sequence of the displayed distinguished triangle that $H^ p(R\mathop{\mathrm{lim}}\nolimits \tau _{\geq -n}K) = H^ p(K)$. $\square$

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