To be precise, fix an Abelian category (such as the category of Abelian groups or the category of vector spaces over a given field) or some other category with kernels and cokernels[?] (such as the category of all groups). Choose an index set[?] of consecutive[?] integers. Then for each integer i in the index set, let A_{i} be an object in the category and let f_{i} be a morphism from A_{i} to A_{i+1}. This defines a sequence of objects and morphisms.
The sequence is exact at A_{i} if the image of f_{i−1} is equal to the kernel of f_{i}:
A long exact sequence is a sequence indexed by the entire set of all integers. Exact sequences indexed by the natural numbers or by a finite set are also quite common.
To make sense of the definition, it is helpful to consider what it means in a relatively simple case where the sequence is finite and begins and ends with 0. Consider the sequence
Any exact sequence of this form is called a short exact sequence.
By the fact that the pimage of 0 is simply the 0 of A, exactness dictates that the kernel of q is 0; in other words, q is a monomorphism. More generally, if 0 → A → B is part of any exact sequence, then the morphism from A to B is monic.
Conversely, since the kernel of s is all of C (what other options do we have?), by exactness the image of r is C, so r is an epimorphism. Again, if B → C → 0 is part of any exact sequence, then the morphism from B to C is epic.
A consequence of these last two facts is that if 0 → X → Y → 0 is exact, then X and Y must be isomorphic.
As a more important consequence, since C is isomorphic to B/(im A) by the first isomorphism theorem, if A is a subset of B (or if we choose to identify A with its image), then the existence of the short exact sequence above tells us that C = B/A.
The splitting lemma states that if we have a morphism t: B → A such that t ^{o} q is the identity on A or a morphism u: C → B such that r ^{o} u is the identity on C, then B is a twisted direct sum[?] of A and C. (For groups, a twisted direct sum is a semidirect product; in an Abelian category, every twisted direct sum is an ordinary direct sum.) In this case, we say that the short exact sequene splits.
Applications of exact sequences
Notice that in an exact sequence, the composition f_{i+1} ^{o} f_{i} maps A_{i} to 0 in A_{i+2}, so the sequence of objects and morphisms is a chain complex. Furthermore, only f_{i}images of elements of A_{i} are mapped to 0 by f_{i+1}, so the homology of this chain complex is trivial. Conversely, given any chain complex, its homology can 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 derived from this a long exact sequence by repeated application of the snake lemma. This comes up in algebraic topology in the study of relative homology[?]. The MayerVietoris sequence[?] is another example.
The extension problem of group theory is essentially the question, given A and C in a short exact sequence, of what B can be. It is important in the classification of groups[?].
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