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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 kernelss 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.

must include more examples....