Snake lemma

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, particularly homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

, the snake lemma, a statement valid in every abelian category

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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

 and its applications, for instance in algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...

. Homomorphisms constructed with its help are generally called connecting homomorphisms.

In an abelian category

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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

 (such as the category of abelian group

Abelian group

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 their order . Abelian groups generalize the arithmetic of addition of integers...

s or the category of vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s over a given field), consider a commutative diagram

Commutative diagram

In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition...

:

image:SnakeLemma01.png

where the rows are exact sequence

Exact sequence

In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the...

s and 0 is the zero object.
Then there is an exact sequence relating the kernels

Kernel (category theory)

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra...

 and cokernel

Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

s of a, b, and c:

image:SnakeLemma02.png

Furthermore, if the morphism f is a monomorphism

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ....

, then so is the morphism ker a → ker b, and if g is an epimorphism

Epimorphism

In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

, then so is coker b → coker c.

To see where the snake lemma gets its name, expand the diagram above as follows:

image:SnakeLemma03.png

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.

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.

In mathematics

Mathematics

Mathematics is the science and study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions....

, particularly homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

, the snake lemma, a statement valid in every abelian category

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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra

Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...

 and its applications, for instance in algebraic topology

Algebraic topology

Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism...

. Homomorphisms constructed with its help are generally called connecting homomorphisms.

Statement

In an abelian category

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 prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

 (such as the category of abelian group

Abelian group

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 their order . Abelian groups generalize the arithmetic of addition of integers...

s or the category of vector space

Vector space

A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s over a given field), consider a commutative diagram

Commutative diagram

In mathematics, and especially in category theory a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition...

:

image:SnakeLemma01.png

where the rows are exact sequence

Exact sequence

In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the...

s and 0 is the zero object.
Then there is an exact sequence relating the kernels

Kernel (category theory)

In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms and the kernels of module homomorphisms and certain other kernels from algebra...

 and cokernel

Cokernel

In mathematics, the cokernel of a linear mapping of vector spaces f : X → Y is the quotient space Y/im of the codomain of f by the image of f....

s of a, b, and c:

image:SnakeLemma02.png

Furthermore, if the morphism f is a monomorphism

Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation ....

, then so is the morphism ker a → ker b, and if g is an epimorphism

Epimorphism

In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

, then so is coker b → coker c.

Explanation of the name

To see where the snake lemma gets its name, expand the diagram above as follows:

image:SnakeLemma03.png

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.

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

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the "scalars" may lie in an arbitrary ring...

 over some ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations , where each operation combines two elements to form a third element...

, 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, 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). 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 choices of y and z), that it is a homomorphism, and that the resulting long sequence is indeed exact.

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

Mitchell's embedding theorem, also known as the Freyd-Mitchell theorem, is a mathematical result about abelian categories; it states that these categories, while rather abstractly defined, are all quite concrete categories of modules...

.

Naturality

In the applications, one often needs to show that long exact sequences are "natural" (in the sense of natural transformation

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 of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

s). This follows from the naturality of the sequence produced by the snake lemma.

If

commutative diagram with exact rows

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

commutative diagram with exact rows

In popular culture

• The statement of the theorem can be seen written in a blackboard behind Dustin Hoffman
  Dustin Hoffman
  Dustin Lee Hoffman is an American actor who has had an active career in film, television, and theatre since 1960. He first drew critical praise for the 1966 Off-Broadway play Eh? for which he won a Theatre World Award and a Drama Desk Award. This was soon followed by his breakout movie role as Ben...
  
   at the very beginning of the 1967 film The Graduate
  The Graduate
  The Graduate is a American comedy-drama film directed by Mike Nichols, based on the 1963 novel The Graduate by Charles Webb, who wrote it shortly after graduating from Williams College. The screenplay is by Calder Willingham and Buck Henry, who makes a cameo appearance as the hotel clerk...
  
  .
• The proof of the snake lemma is being taught by Jill Clayburgh
  Jill Clayburgh
  -Early life:Clayburgh was born in New York City to Julia , a theatrical production secretary for David Merrick, and Albert Henry "Bill" Clayburgh, a manufacturing executive. Clayburgh's family was Jewish and upper class; she was raised in a "fashionable" neighborhood on Manhattan's Upper East Side,...
  
   at the very beginning of the 1980 film It's My Turn
  It's My Turn
  "It's My Turn" is a 1980 song, which was also developed into a film of the same name. The song, written by Carole Bayer Sager and Michael Masser for Diana Ross, was released as a single and became a top ten hit on the Billboard Hot 100, peaking at number nine...
  
  .

Literature

• M. F. Atiyah; I. G. Macdonald
  Ian G. Macdonald
  Ian Grant Macdonald is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebraic combinatorics ....
  
  : Introduction to Commutative Algebra. Oxford 1969, Addison-Wesley Publishing Company, Inc. ISBN 0-201-00361-9.
• P. Hilton; U. Stammbach: A course in homological algebra. 2. Auflage, Springer Verlag, Graduate Texts in Mathematics, 1997, ISBN 0-387-94823-6.