Гомоморфизмдер тізбегі және ядроның алдыңғы бейнене теңдігі

Sequence of homomorphisms such that each kernel equals the preceding image

An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.

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).

In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about subobjects and factor objects. The extension problem is essentially the question "Given the end terms A and C of a short exact sequence, what possibilities exist for the middle term B?" In the category of groups, this is equivalent to the question, what groups B have A as a normal subgroup and C as the corresponding factor group? This problem is important in the classification of groups. See also Outer automorphism group. Notice that in an exact sequence, the composition fi+1 ∘ fi maps Ai to 0 in Ai+2, so every exact sequence is a chain complex. Furthermore, only fi images of elements of Ai are mapped to 0 by fi+1, so the homology of this chain complex is trivial. More succinctly:
Exact sequences are precisely those chain complexes which are acyclic. Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact. If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig zag lemma. It comes up in algebraic topology in the study of relative homology; the Mayer–Vietoris sequence is another example. Long exact sequences induced by short exact sequences are also characteristic of derived functors. Exact functors are functors that transform exact sequences into exact sequences.