Definition:Exactness of Chain Complex at Object

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.

This page has been identified as a candidate for refactoring of medium complexity. In particular: put the following in its own page as it is a result Until this has been finished, please leave {{Refactor}} in the code. New contributors: Refactoring is a task which is expected to be undertaken by experienced editors only. Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Refactor}} from the code.

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

• Definition:Exact Chain Complex

Sources

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