An exact sequence is a sequence of homomorphisms
 |
(1)
|
between a sequence of spaces 
, which satisfies
 |
(2)
|
where 
denotes the image and 
the group kernel. That is,
for 
,
iff 
for
some 
.
It follows that 
.
The notion of exact sequence makes sense when the spaces
are groups, modules, chain
complexes, or sheaves. The notation
for the maps may be suppressed and the sequence
written on a single line as
 |
(3)
|
An exact sequence may be of either finite or infinite length. The special case of length five,
 |
(4)
|
beginning and ending with zero, meaning the zero module 
,
is called a short exact sequence. An infinite
exact sequence is called a long exact sequence.
For example, the sequence where 
and 
is given by multiplying
by 2,
 |
(5)
|
is a long exact sequence because at each stage the kernel and image are
equal to the subgroup 
.
Special information is conveyed when one of the spaces 
is the zero
module. For instance, the sequence
 |
(6)
|
is exact iff the map 
is injective. Similarly,
 |
(7)
|
is exact iff the map 
is surjective.
