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:

${\displaystyle 0\longrightarrow {\mathcal {A}}\mathrel {\stackrel {\alpha }{\longrightarrow }} {\mathcal {B}}\mathrel {\stackrel {\beta }{\longrightarrow }} {\mathcal {C}}\longrightarrow 0}$ 

Such a sequence is shorthand for the following commutative diagram:

where the rows are exact sequences and each column is a chain complex.

The zig-zag lemma asserts that there is a collection of boundary maps

${\displaystyle \delta _{n}:H_{n}({\mathcal {C}})\longrightarrow H_{n-1}({\mathcal {A}}),}$ 

that makes the following sequence exact:

The maps ${\displaystyle \alpha _{*}^{}}$  and ${\displaystyle \beta _{*}^{}}$  are the usual maps induced by homology. The boundary maps ${\displaystyle \delta _{n}^{}}$  are explained below. The name of the lemma arises from the "zig-zag" behavior of the maps in the sequence. A variant version of the zig-zag lemma is commonly known as the "snake lemma" (it extracts the essence of the proof of the zig-zag lemma given below).

The maps ${\displaystyle \delta _{n}^{}}$  are defined using a standard diagram chasing argument. Let ${\displaystyle c\in C_{n}}$  represent a class in ${\displaystyle H_{n}({\mathcal {C}})}$ , so ${\displaystyle \partial _{n}''(c)=0}$ . Exactness of the row implies that ${\displaystyle \beta _{n}^{}}$  is surjective, so there must be some ${\displaystyle b\in B_{n}}$  with ${\displaystyle \beta _{n}^{}(b)=c}$ . By commutativity of the diagram,

${\displaystyle \beta _{n-1}\partial _{n}'(b)=\partial _{n}''\beta _{n}(b)=\partial _{n}''(c)=0.}$ 

By exactness,

${\displaystyle \partial _{n}'(b)\in \ker \beta _{n-1}=\mathrm {im} \;\alpha _{n-1}.}$ 

Thus, since ${\displaystyle \alpha _{n-1}^{}}$  is injective, there is a unique element ${\displaystyle a\in A_{n-1}}$  such that ${\displaystyle \alpha _{n-1}(a)=\partial _{n}'(b)}$ . This is a cycle, since ${\displaystyle \alpha _{n-2}^{}}$  is injective and

${\displaystyle \alpha _{n-2}\partial _{n-1}(a)=\partial _{n-1}'\alpha _{n-1}(a)=\partial _{n-1}'\partial _{n}'(b)=0,}$ 

since ${\displaystyle \partial ^{2}=0}$ . That is, ${\displaystyle \partial _{n-1}(a)\in \ker \alpha _{n-2}=\{0\}}$ . This means ${\displaystyle a}$  is a cycle, so it represents a class in ${\displaystyle H_{n-1}({\mathcal {A}})}$ . We can now define

${\displaystyle \delta _{}^{}[c]=[a].}$ 

With the boundary maps defined, one can show that they are well-defined (that is, independent of the choices of c and b). The proof uses diagram chasing arguments similar to that above. Such arguments are also used to show that the sequence in homology is exact at each group.

• Mayer–Vietoris sequence

• Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0.
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556
• Munkres, James R. (1993). Elements of Algebraic Topology. New York: Westview Press. ISBN 0-201-62728-0.