Definition:Projective Resolution

Definition
Let $\AA$ be an abelian category.

Let $X$ be an object in $\AA$.

A projective resolution of $X$ is a chain complex $P := \family {d_i : P_i \to P_{i - 1} }_{i \mathop \in \Z}$ in $\AA$, such that: together with an augmentation map $\varepsilon : P_0 \to X$, such that the chain complex
 * $\forall i < 0 : P_i = 0$
 * $P_i$ is projective for all $i \geq 0$
 * $\begin{xy}

\xymatrix{ \dots \ar[r] & P_2 \ar[r]^{d_2} & P_1 \ar[r]^{d_1} & P_0 \ar[r]^{\varepsilon} & X \ar[r] & 0 \ar[r] & 0 \ar[r] & \dots } \end{xy}$ is exact.

Also see

 * Projective Resolution Exists Iff Enough Projectives