# Definition:Exactness of Chain Complex at Object

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## 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 at $C_j$ if and only if the canonical map $\Img {d_{i + 1} } \to \ker \paren {d_i}$ is an isomorphism.

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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

## Sources

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