Zig-zag lemma

In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes. The result is valid in every abelian category.

In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let ${\displaystyle ({\mathcal {A}},\partial _{\bullet }),({\mathcal {B}},\partial _{\bullet }')}$ and ${\displaystyle ({\mathcal {C}},\partial _{\bullet }'')}$ be chain complexes that fit into the following short exact sequence:

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