Table of contents |

2 Short exact sequences 3 Applications of exact sequences |

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 *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}:

- im
*f*_{i−1}= ker*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.

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 → *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*.

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 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....*