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Exact sequence

In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

Definition

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 Ai be an object in the category and let fi be a morphism from Ai to Ai+1. This defines a sequence of objects and morphisms.

The sequence is exact at Ai if the image of fi−1 is equal to the kernel of fi:

im fi−1 = ker fi.
The sequence is exact, period, if it is exact at each object.

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.

Short exact sequences

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 p-image 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 → AB 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 BC → 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 → XY → 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: BA such that t o q is the identity on A or a morphism u: CB 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 fi+1 o fi maps Ai to 0 in Ai+2, so the sequence of objects and morphisms 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. 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 Mayer-Vietoris 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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