and
nonabelian homological algebra
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.
Let $\mathcal{A}$ be an additive category (often assumed to be an abelian category, for instance $\mathcal{A} = R$Mod for $R$ some ring).
An exact sequence in $\mathcal{A}$ is a chain complex $C_\bullet$ in $\mathcal{A}$ 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 β$0 \to A \to B \to C \to 0$β or even just β$A \to B \to C$β.
A general exact sequence is sometimes called a long exact sequence, to distinguish from the special case of a short exact sequence.
Explicitly, a sequence of morphisms
is short exact, def. 2, precisely if
$i$ is a monomorphism,
$p$ is an epimorphism,
and the image of $i$ equals the kernel of $p$ (equivalently, the coimage of $p$ equals the cokernel of $i$).
The third condition is the definition of exactness at $B$. So we need to show that the first two conditions are equivalent to exactness at $A$ and at $C$.
This is easy to see by looking at elements when $\mathcal{A} \simeq R$Mod, for some ring $R$ (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 $0 \to A$ is just the element $0 \in A$, that the kernel of $A \to B$ consists of just this element. But since $A \to B$ is a group homomorphism, this means equivalently that $A \to B$ is an injection.
Dually, the sequence being exact at
means, since the kernel of $C \to 0$ is all of $C$, that also the image of $B \to C$ is all of $C$, hence equivalently that $B \to C$ is a surjection.
A split exact sequence is a short exact sequence as above in which $i$ is a split monomorphism, or (equivalently) in which $p$ is a split epimorphism.
In this case, $B$ may be decomposed as the biproduct $A \oplus C$ (with $i$ and $p$ the usual biproduct inclusion and projection); this sense in which $B$ is βsplitβ into $A$ and $C$ is the origin of the general terms βsplit (mono/epi)morphismβ.
It is also helpful to consider a similar notion in the case of a pointed set.
In the category $Set_*$ of pointed sets, a sequence
is said to be exact at $(B, b)$ if $im f = g^{-1}(c)$.
For concrete pointed categories (ie. a category $\mathcal{C}$ with a faithful functor $F: \mathcal{C} \to Set_*$), a sequence is exact if the image under $F$ is exact.
In the case of (abelian) categories like $Ab$ and $R-Mod$, the two notions of exactness coincide if we pick the point of each group/module to be $0$. Such a general notion is useful in cases such as the long exact sequence of homotopy groups where the homotopy βgroupsβ for small $n$ are just pointed sets without a group structure.
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 $\cdot \to 0 \to C_n \to \cdot$ or $\cdot \to C_n \to 0 \to \cdots$, it follows that the next morphism out of or into the vanishing entry is a monomorphism or epimorphism, respectively.
In particular:
If part of an exact sequence looks like
then $\partial_n$ is an isomorphism and hence
A chain complex $C_\bullet$ is exact (is a long exact sequence), precisely if the unique chain map from the initial object, the 0-complex
is a quasi-isomorphism.
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 $\mathcal{A}$ be an abelian category.
For
an exact sequence in $\mathcal{A}$ and for $X \to B$ any morphism in $\mathcal{A}$, 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.
For
an exact sequence and $X \to A$ any morphism, also
is exact.
Let $\mathcal{A} = \mathbb{Z}$Mod $\simeq$ Ab. For $n \in \mathbb{N}$ with $n \geq 1$ let $\mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z}$ be the linear map/homomorphism of abelian groups which acts by the ordinary multiplication of integers by $n$. This is clearly an injection. The cokernel of this morphism is the projection to the quotient group, which is the cyclic group $\mathbb{Z}_n \coloneqq \mathbb{Z}/n\mathbb{Z}$. Hence we have a short exact sequence
The connecting homomorphism of the long exact sequence in homology induced from short exact sequences of the form in example 1 is called a Bockstein homomorphism.
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.