Definition:Chain Complex

Definition

Let $\AA$ be an abelian category.

A chain complex in $\AA$ is a family of objects $\family {C_i}_{i \mathop \in \Z}$ of $\AA$ and a family of morphisms $\family {d_i : C_i \to C_{i - 1} }_{i \mathop \in \Z}$, such that for all $i \in \Z$, the composition $d_{i - 1} \circ d_i$ is the zero morphism $0 : C_i \to C_{i - 2}$.

Visualization

A chain complex can be visualized as a diagram:

$\cdots \longrightarrow C_{i + 1} \stackrel {d_{i + 1} } \longrightarrow C_i \stackrel {d_i} \longrightarrow C_{i - 1} \stackrel {d_{i - 1} } \longrightarrow C_{i - 2} \longrightarrow \cdots$

Also see

• Definition:Differential Complex
• Definition:Null Sequence (Homological Algebra)

Sources

• 1994: Charles Weibel: An Introduction to Homological Algebra: Definition $1.1.1.$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): chain complex
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): chain complex