Definition:Exact Chain Complex

Definition

Let $\AA$ be an abelian category.

Let $\family {d_i : C_i \to C_{i - 1} }_{i \mathop \in \Z}$ be a chain complex in $\AA$.

Then $\family {d_i : C_i \to C_{i - 1} }_{i \mathop \in \Z}$ is exact if and only if it is exact at $C_i$ for all $i \mathop \in \Z$.

Also see

• Definition:Exactness of Chain Complex at Object

Sources

• 1994: Charles Weibel: An Introduction to Homological Algebra: $\S 1.1$.