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. It can be regarded as a generalization of the Mayer–Vietoris sequence.
Statement In an abelian category (such as the category of abelian groups or the category of vector spaces over a given field), let ( A , ∂ ∙ ) , ( B , ∂ ∙ ′ ) {\displaystyle ({\mathcal {A}},\partial _{\bullet }),({\mathcal {B}},\partial _{\bullet }')} and ( C , ∂ ∙ ″ ) {\displaystyle ({\mathcal {C}},\partial _{\bullet }'')} be chain complexes that fit into the following short exact sequence:
0 ⟶ A ⟶ α B ⟶ β C ⟶ 0 {\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
δ n : H n ( C ) ⟶ H n − 1 ( A ) , {\displaystyle \delta _{n}:H_{n}({\mathcal {C}})\longrightarrow H_{n-1}({\mathcal {A}}),}
that makes the following sequence exact:
The maps α ∗
{\displaystyle \alpha _{*}^{}} and β ∗
{\displaystyle \beta _{*}^{}} are the usual maps induced by homology. The boundary maps δ n
{\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).
Construction of the boundary maps The maps δ n
{\displaystyle \delta _{n}^{}} are defined using a standard diagram chasing argument. Let c ∈ C n {\displaystyle c\in C_{n}} represent a class in H n ( C ) {\displaystyle H_{n}({\mathcal {C}})} , so ∂ n ″ ( c ) = 0 {\displaystyle \partial _{n}''(c)=0} . Exactness of the row implies that β n
{\displaystyle \beta _{n}^{}} is surjective, so there must be some b ∈ B n {\displaystyle b\in B_{n}} with β n
( b ) = c {\displaystyle \beta _{n}^{}(b)=c} . By commutativity of the diagram,
β n − 1 ∂ n ′ ( b ) = ∂ n ″ β n ( b ) = ∂ n ″ ( c ) = 0. {\displaystyle \beta _{n-1}\partial _{n}'(b)=\partial _{n}''\beta _{n}(b)=\partial _{n}''(c)=0.}
By exactness,
∂ n ′ ( b ) ∈ ker β n − 1 = i m α n − 1 . {\displaystyle \partial _{n}'(b)\in \ker \beta _{n-1}=\mathrm {im} \;\alpha _{n-1}.}
Thus, since α n − 1
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