Exact sequences

Factorization into monomorphism and epimorphism

In an abelian category $A$ , every morphism $f$ has a factorization $f = m e$ with $m$ a monomorphism and $e$ an epimorphism. Moreover,

m = \ker (coker f), e = coker (\ker f) .

We can thus define the (usual) image and coimage of $f = m e : a \to b$ as

m = im (f), e = coim (f) .

The image and coimage are unique up to isomorphism, so the image of $f$ is really a subobject of $b$ , while the coimage is a quotient object of $a$ .

With images and coimages now defined, we can talk about exact sequences.

Exact sequence in an abelian category

In an abelian category $A$ , a pair of composable morphisms

a \overset{f}{\to} b \overset{g}{\to} c

is exact at $b$ when $im (f) \equiv \ker (g)$ (or equivalently, when $coker (f) \equiv coim (g)$ ).

Here the symbol $\equiv$ indicates equivalence as subobjects (which are isomorphism classes of monomorphisms to a common object).

Short exact sequence in an abelian category

In an abelian category, a diagram

0 \to a \overset{f}{\to} b \overset{g}{\to} c \to 0

is a short exact sequence when it is exact at $a$ , $b$ , and $c$ .

Equivalently, $f = \ker (g)$ and $g = coker (f)$ .

Exact functors

Definition of exact functor

A functor $T : A \to B$ between abelian categories is exact when it preserves all finite limits and colimits.

In particular, an exact functor preserves kernels and cokernels:

\ker (T (f)) = T (\ker (f)) and coker (T (f)) = T (coker (f)) .

It also preserves images and coimages, and carries exact sequences to exact sequences.

A functor is left exact when it preserves all finite limits; equivalently, when it is additive and preserves short left exact sequences. The dual notion is a functor that is right exact.