Definition 13.18.1. Let $\mathcal{A}$ be an abelian category. Let $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. An injective resolution of $A$ is a complex $I^\bullet$ together with a map $A \to I^0$ such that:

1. We have $I^ n = 0$ for $n < 0$.

2. Each $I^ n$ is an injective object of $\mathcal{A}$.

3. The map $A \to I^0$ is an isomorphism onto $\mathop{\mathrm{Ker}}(d^0)$.

4. We have $H^ i(I^\bullet ) = 0$ for $i > 0$.

Hence $A[0] \to I^\bullet$ is a quasi-isomorphism. In other words the complex

$\ldots \to 0 \to A \to I^0 \to I^1 \to \ldots$

is acyclic. Let $K^\bullet$ be a complex in $\mathcal{A}$. An injective resolution of $K^\bullet$ is a complex $I^\bullet$ together with a map $\alpha : K^\bullet \to I^\bullet$ of complexes such that

1. We have $I^ n = 0$ for $n \ll 0$, i.e., $I^\bullet$ is bounded below.

2. Each $I^ n$ is an injective object of $\mathcal{A}$.

3. The map $\alpha : K^\bullet \to I^\bullet$ is a quasi-isomorphism.

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