In mathematics, particularly homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology. Homomorphisms constructed with its help are generally called connecting homomorphisms. Euclid, detail from The School of Athens by Raphael. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... 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 nice properties. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

Statement

In an Abelian category (such as the category of Abelian groups or the category of vector spaces over a given field), consider a commutative diagram: In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

first snake lemma diagram File links The following pages link to this file: Snake lemma Categories: GFDL images ...

where the rows are exact sequences and 0 is the zero object. Then there is an exact sequence relating the kernels and cokernels of a, b, and c: In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C there... 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. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ...

second snake lemma diagram File links The following pages link to this file: Snake lemma Categories: GFDL images ...

Furthermore, if the morphism f is a monomorphism, then so is the morphism ker a → ker b, and if g' is an epimorphism, then so is coker b → coker c. In the context of abstract algebra or universal algebra, a monomorphism is simply an injective homomorphism. ... In the context of abstract algebra or universal algebra, an epimorphism is simply a homomorphism onto or surjective homomorphism. ...

Explanation of the name

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

Third snake lemma diagram---see Talk:Snake lemma for a question. ...

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 over some ring, 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. In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ...

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. In mathematics, Mitchells embedding theorem is an important 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 transformations). This follows from the naturality of the sequence produced by the snake lemma. In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

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 Image File history File links Snake_lemma_nat. ...

Image File history File links Snake_lemma_nat2. ...

Trivia

The snake lemma was proved by Jill Clayburgh in the 1980 film It's My Turn. Jill Clayburgh (born April 30, 1944) is an American actress of stage, motion pictures, and television. ...