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The snake lemma is a tool used 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 ...
, to construct
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 ...
s. The snake lemma 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 ...
and is a crucial tool in homological algebra and its applications, for instance in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
. Homomorphisms constructed with its help are generally called ''connecting homomorphisms''.


Statement

In an
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 ...
(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 ...
), consider a
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 ...
: : 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 0 is the
zero object In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism . The dual notion is that of a terminal object (also called terminal element): ...
. Then there is an exact sequence relating the
kernels Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
and
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the name: ...
s of ''a'', ''b'', and ''c'': :\ker a ~~ \ker b ~~ \ker c ~\overset~ \operatornamea ~~ \operatornameb ~~ \operatornamec where ''d'' is a homomorphism, known as the ''connecting homomorphism''. Furthermore, if the morphism ''f'' is a
monomorphism In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism ...
, then so is the morphism \ker a ~~ \ker b, and if ''g is an
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 \circ f = g_2 \circ f \ ...
, then so is \operatorname b ~~ \operatorname c. The cokernels here are: \operatornamea = A'/\operatornamea, \operatornameb = B'/\operatornameb, \operatornamec = C'/\operatornamec.


Explanation of the name

To see where the snake lemma gets its name, expand the diagram above as follows: : and then note that the exact sequence that is the conclusion of the lemma can be drawn on this expanded diagram in the reversed "S" shape of a slithering
snake Snakes are elongated, Limbless vertebrate, limbless, carnivore, carnivorous reptiles of the suborder Serpentes . Like all other Squamata, squamates, snakes are ectothermic, amniote vertebrates covered in overlapping Scale (zoology), scales. Ma ...
.


Construction of the maps

The maps between the kernels and the maps between the cokernels are induced in a natural manner by the given (horizontal) maps because of the diagram's commutativity. The exactness of the two induced sequences follows in a straightforward way from the exactness of the rows of the original diagram. The important statement of the lemma is that a ''connecting homomorphism'' ''d'' exists which completes the exact sequence. In the case of abelian groups or
modules Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
over some
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
, the map ''d'' can be constructed as follows: Pick an element ''x'' in ker ''c'' and view it as an element of ''C''; since ''g'' is
surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
, there exists ''y'' in ''B'' with ''g''(''y'') = ''x''. Because of the commutativity of the diagram, we have ''g(''b''(''y'')) = ''c''(''g''(''y'')) = ''c''(''x'') = 0 (since ''x'' is in the kernel of ''c''), and therefore ''b''(''y'') is in the kernel of ''g' ''. Since the bottom row is exact, we find an element ''z'' in ''A' '' with ''f'' '(''z'') = ''b''(''y''). ''z'' is unique by injectivity of ''f'' '. We then define ''d''(''x'') = ''z'' + ''im''(''a''). Now one has to check that ''d'' is well-defined (i.e., ''d''(''x'') only depends on ''x'' and not on the choice of ''y''), that it is a homomorphism, and that the resulting long sequence is indeed exact. One may routinely verify the exactness by
diagram chasing 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 th ...
(see the proof of Lemma 9.1 in ). Once that is done, the theorem is proven for abelian groups or modules over a ring. For the general case, the argument may be rephrased in terms of properties of arrows and cancellation instead of elements. Alternatively, one may invoke
Mitchell's embedding theorem Mitchell's embedding theorem, also known as the Freyd–Mitchell theorem or the full embedding theorem, is a result about abelian categories; it essentially states that these categories, while rather abstractly defined, are in fact concrete catego ...
.


Naturality

In the applications, one often needs to show that long exact sequences are "natural" (in the sense of
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s). This follows from the naturality of the sequence produced by the snake lemma. If : is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form :


Example

Let k be field, V be a k-vector space. V is k /math>-module by t:V \to V being a k-linear transformation, so we can tensor V and k over k /math>. : V \otimes_ k = V \otimes_ (k (t)) = V/tV = \operatorname(t) . Given a short exact sequence of k-vector spaces 0 \to M \to N \to P \to 0, we can induce an exact sequence M \otimes_ k \to N \otimes_ k \to P \otimes_ k \to 0 by right exactness of tensor product. But the sequence 0 \to M \otimes_ k \to N \otimes_ k \to P \otimes_ k \to 0 is not exact in general. Hence, a natural question arises. Why is this sequence not exact? According to the diagram above, we can induce an exact sequence \ker(t_M) \to \ker(t_N) \to \ker(t_P) \to M \otimes_ k \to N \otimes_ k \to P \otimes_ k \to 0 by applying the snake lemma. Thus, the snake lemma reflects the tensor product's failure to be exact.


In the category of groups

While many results of homological algebra, such as the
five lemma In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also w ...
or the
nine lemma right In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all co ...
, hold for abelian categories as well as in the category of groups, the snake lemma does not. Indeed, arbitrary cokernels do not exist. However, one can replace cokernels by (left) cosets A'/\operatorname a, B'/\operatorname b, and C'/\operatorname c. Then the connecting homomorphism can still be defined, and one can write down a sequence as in the statement of the snake lemma. This will always be a chain complex, but it may fail to be exact. Exactness can be asserted, however, when the vertical sequences in the diagram are exact, that is, when the images of ''a'', ''b'', and ''c'' are
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
s.


Counterexample

Consider the
alternating group In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or Basic prop ...
A_5: this contains a subgroup isomorphic to the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
S_3, which in turn can be written as a semidirect product of
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
s: S_3\simeq C_3\rtimes C_2. This gives rise to the following diagram with exact rows: :\begin & 1 & \to & C_3 & \to & C_3 & \to 1\\ & \downarrow && \downarrow && \downarrow \\ 1 \to & 1 & \to & S_3 & \to & A_5 \end Note that the middle column is not exact: C_2 is not a normal subgroup in the semidirect product. Since A_5 is
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by Johnn ...
, the right vertical arrow has trivial cokernel. Meanwhile the quotient group S_3/C_3 is isomorphic to C_2. The sequence in the statement of the snake lemma is therefore :1 \longrightarrow 1 \longrightarrow 1 \longrightarrow 1 \longrightarrow C_2 \longrightarrow 1, which indeed fails to be exact.


In popular culture

The proof of the snake lemma is taught by
Jill Clayburgh Jill Clayburgh (April 30, 1944 – November 5, 2010) was an American actress known for her work in theater, television, and cinema. She received the Cannes Film Festival Award for Best Actress and was nominated for the Academy Award for Best Act ...
's character at the very beginning of the 1980 film '' It's My Turn''.


See also

*
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. Statement In an abel ...


References

* * *


External links

*{{MathWorld, title=Snake Lemma, urlname=SnakeLemma
Snake Lemma
at
PlanetMath PlanetMath is a free, collaborative, mathematics online encyclopedia. The emphasis is on rigour, openness, pedagogy, real-time content, interlinked content, and also community of about 24,000 people with various maths interests. Intended to be c ...

Proof of the Snake Lemma
in the fil
It's My Turn
Homological algebra Lemmas in category theory