A short exact sequence of groups 
, 
, and 
is given by two maps 
and 
and is written
 |
(1)
|
Because it is an exact sequence, 
is injective, and 
is surjective. Moreover,
the group kernel of 
is the image of 
. Hence, the group 
can be considered as a (normal) subgroup of 
, and 
is isomorphic to 
.
A short exact sequence is said to split if there is a map 
such that 
is the identity on 
.
This only happens when 
is the direct product of
and 
.
The notion of a short exact sequence also makes sense for modules and sheaves. Given a module 
over a unit
ring 
,
all short exact sequences
 |
(2)
|
are split iff 
is projective,
and all short exact sequences
 |
(3)
|
is A short exact sequence of vector spaces is always split.
