Lemma 13.18.3. Let $\mathcal{A}$ be an abelian category. Assume $\mathcal{A}$ has enough injectives.

Any object of $\mathcal{A}$ has an injective resolution.

If $H^ n(K^\bullet ) = 0$ for all $n \ll 0$ then $K^\bullet $ has an injective resolution.

If $K^\bullet $ is a complex with $K^ n = 0$ for $n < a$, then there exists an injective resolution $\alpha : K^\bullet \to I^\bullet $ with $I^ n = 0$ for $n < a$ such that each $\alpha ^ n : K^ n \to I^ n$ is injective.

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