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. It can be regarded as a generalization of the Mayer–Vietoris sequence.
Statement
In an abelian category, let and be chain complexes that fit into the following short exact sequence:Such a sequence is shorthand for the following commutative diagram:
commutative diagram representation of a short exact sequence of chain complexes
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
that makes the following sequence exact:
long exact sequence in homology, given by the Zig-Zag Lemma
The maps and are the usual maps induced by homology. The boundary maps 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".
Construction of the boundary maps
The maps are defined using a standard diagram chasing argument. Let represent a class in, so. Exactness of the row implies that is surjective, so there must be some with. By commutativity of the diagram,By exactness,
Thus, since is injective, there is a unique element such that. This is a cycle, since is injective and
since. That is,. This means is a cycle, so it represents a class in. We can now define
With the boundary maps defined, one can show that they are well-defined. 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.