Definition:Homology of Chain Complex

Definition

Let $\AA$ be an abelian category.

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

The $j$-th homology (object) $\map {H_j} C$ of $C$ is defined as the cokernel of the canonical map $\Img {d_{i + 1} } \to \ker \paren {d_i}$.

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The canonical map is induced by Homomorphisms Theorem for Categories with Zero Object since by definition $d_{j} \circ d_{j + 1} = 0$.

Also see

• Chain Complex is Exact iff Zero Homology

Sources

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