In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, particularly
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, the zig-zag lemma asserts the existence of a particular
long exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
in the
homology group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...
s of certain
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
es. The result is valid in every
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ab ...
.
Statement
In an abelian category (such as the category of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
s or the category of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s over a given
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
), let
and
be chain complexes that fit into the following
short exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
:
:
Such a sequence is shorthand for the following
commutative diagram
350px, The commutative diagram used in the proof of the five lemma.
In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
:
commutative diagram representation of a short exact sequence of chain complexes
where the rows are
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context o ...
s and each column is a
chain complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemati ...
.
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
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance ...
" (it extracts the essence of the proof of the zig-zag lemma given below).
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 (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.
See also
*
Mayer–Vietoris sequence
In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their homology and cohomology groups. The result is due to ...
References
*
*
*{{cite book , first = James R. , last = Munkres , authorlink = James Munkres , year = 1993 , title = Elements of Algebraic Topology , publisher = Westview Press , location = New York , isbn = 0-201-62728-0
Homological algebra
Lemmas in category theory