Definition 13.19.1. Let $\mathcal{A}$ be an abelian category. Let $A \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. An projective resolution of $A$ is a complex $P^\bullet$ together with a map $P^0 \to A$ such that:

1. We have $P^ n = 0$ for $n > 0$.

2. Each $P^ n$ is an projective object of $\mathcal{A}$.

3. The map $P^0 \to A$ induces an isomorphism $\mathop{\mathrm{Coker}}(d^{-1}) \to A$.

4. We have $H^ i(P^\bullet ) = 0$ for $i < 0$.

Hence $P^\bullet \to A$ is a quasi-isomorphism. In other words the complex

$\ldots \to P^{-1} \to P^0 \to A \to 0 \to \ldots$

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

1. We have $P^ n = 0$ for $n \gg 0$, i.e., $P^\bullet$ is bounded above.

2. Each $P^ n$ is an projective object of $\mathcal{A}$.

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

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