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:

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

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

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

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

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

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

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

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

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

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