Exact sequence

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An exact sequence is a concept 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, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

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Definition

In the context of group theory, a sequence

of groups and group homomorphisms is called exact if the image (or range) of each homomorphism is equal to the kernel of the next:

Note that the sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms. More generally, the notion of an exact sequence makes sense in any abelian category (i.e. any category with kernels and cokernels).

Short exact sequence

The most common type of exact sequence is the short exact sequence. This is an exact sequence of the form

where ƒ is a monomorphism and g is an epimorphism. In this case, A is essentially a subobject of B, and the corresponding quotient is isomorphic to C:

(being f(A) = im(f) ).

A short exact sequence of abelian groups may also be written as an exact sequence with five terms:

where 0 represents the zero object, such as the trivial group or a zero-dimensional vector space. The placement of the 0's forces ƒ to be a monomorphism and g to be an epimorphism (see below).