\begin{definition}[Definition:Projective Resolution]
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:
* $\forall i < 0 : P_i = 0$
* $P_i$ is projective for all $i \geq 0$
together with an '''augmentation map''' $\varepsilon : P_0 \to X$, such that the chain complex
:$\begin{xy}
\xymatrix{
\dots \ar[r] & P_2 \ar[r]^{d_2} & P_1 \ar[r]^{d_1} & P_0 \ar[r]^{\varepsilon} & X \ar[r] & 0 \ar[r] & 0 \ar[r] & \dots
}
\end{xy}$
is exact.
\end{definition}