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:

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

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

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

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

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

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

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

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

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