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)
|
A short exact sequence of vector spaces is always split.