Exact sequence

An exact sequence is a concept in mathematics, especially in ring and module theory, homological algebra, 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.

In the context of group theory, a sequence

of groups and group homomorphisms is called exact if the image 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 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 category with kernels and cokernels.

To make sense of the definition, it is helpful to consider what it means in relatively simple cases where the sequence is finite and begins or ends with the trivial group. Traditionally, this, along with the single identity element, is denoted 0 (additive notation), when the groups are abelian, and is denoted 1 (multiplicative notation), otherwise.

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