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:

• $\forall i < 0 : P_i = 0$
• $P_i$ 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.

Also see

• Projective Resolution Exists Iff Enough Projectives

Sources

• 1994: Charles Weibel: An Introduction to Homological Algebra: $\text{Definition 2.2.4}$.