Zig-zag lemma

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.