An exact sequence may be defined in a semi-abelian category, and more generally in a homological category. It is a sequential diagram in which the cokernel of each morphism is equal to the kernel of the next morphism.
Definition in additive categories
Let be an additive category (often assumed to be an abelian category, for instance Mod for some ring).
An exact sequence in is a chain complex in with vanishing chain homology in each degree:
A short exact sequence is an exact sequence, def. 1 of the form
One usually writes this just “” or even just “”.
Explicitly, a sequence of morphisms
is short exact, def. 2, precisely if
is a monomorphism,
is an epimorphism,
and the image of equals the kernel of (equivalently, the coimage of equals the cokernel of ).
The third condition is the definition of exactness at . So we need to show that the first two conditions are equivalent to exactness at and at .
This is easy to see by looking at elements when Mod, for some ring (and the general case can be reduced to this one using one of the embedding theorems):
The sequence being exact at
means, since the image of is just the element , that the kernel of consists of just this element. But since is a group homomorphism, this means equivalently that is an injection.
Dually, the sequence being exact at
means, since the kernel of is all of , that also the image of is all of , hence equivalently that is a surjection.
In this case, may be decomposed as the biproduct (with and the usual biproduct inclusion and projection); this sense in which is ‘split’ into and is the origin of the general terms ‘split (mono/epi)morphism’.
Definition in pointed sets
It is also helpful to consider a similar notion in the case of a pointed set.
In the category of pointed sets, a sequence
is said to be exact at if .
For concrete pointed categories (ie. a category with a faithful functor ), a sequence is exact if the image under is exact.
In the case of (abelian) categories like and , the two notions of exactness coincide if we pick the point of each group/module to be . Such a general notion is useful in cases such as the long exact sequence of homotopy groups where the homotopy “groups” for small are just pointed sets without a group structure.
Computing terms in an exact sequence
A typical use of a long exact sequence, notably of the homology long exact sequence, is that it allows to determine some of its entries in terms of others.
The characterization of short exact sequences in prop. 1 is one example for this: whenever in a long exact sequence one entry vanishes as in or , it follows that the next morphism out of or into the vanishing entry is a monomorphism or epimorphism, respectively.
If part of an exact sequence looks like
then is an isomorphism and hence
Exactness and quasi-isomorphisms
Short exact sequences and quotients
The following are some basic lemmas that show how given a short exact sequence one obtains new short exact sequences from forming quotients/cokernels (see Wise).
Let be an abelian category.
an exact sequence in and for any morphism in , also
is a short exact sequence.
We have an exact sequence of complexes of length 2
and the exact sequence to be demonstrated is degreewise the cokernel of this sequence. So the statement reduces to the fact that forming cokernels is a right exact functor.
an exact sequence and any morphism, also
Let Mod Ab. For with let be the linear map/homomorphism of abelian groups which acts by the ordinary multiplication of integers by . This is clearly an injection. The cokernel of this morphism is the projection to the quotient group, which is the cyclic group . Hence we have a short exact sequence
Classes of examples
A standard introduction is for instance in section 1.1 of
The quotient lemmas from above are discussed in
in the context of the salamander lemma and the snake lemma.