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Concept
Exact sequence
Summary
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an ) such that the of one morphism equals the kernel of the next.
In the context of group theory, a sequence
of groups and group homomorphisms is said to be exact at if . The sequence is called exact if it is exact at each for all , i.e., if the image of each homomorphism is equal to the kernel of the next.
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 with s and cokernels, and more specially in abelian categories, where it is widely used.
To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with the trivial group. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).
Consider the sequence 0 → A → B. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (from A to B) has kernel {0}; that is, if and only if that map is a monomorphism (injective, or one-to-one).
Consider the dual sequence B → C → 0. The kernel of the rightmost map is C. Therefore the sequence is exact if and only if the image of the leftmost map (from B to C) is all of C; that is, if and only if that map is an epimorphism (surjective, or onto).
Therefore, the sequence 0 → X → Y → 0 is exact if and only if the map from X to Y is both a monomorphism and epimorphism (that is, a bimorphism), and so usually an isomorphism from X to Y (this always holds in exact categories like Set).
Short exact sequences are exact sequences of the form
As established above, for any such short exact sequence, f is a monomorphism and g is an epimorphism.
Official source
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