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A sequence

$$ 
\dots \rightarrow  A _ {n}  \rightarrow ^ { {\alpha _ n} } \ 
A _ {n+1}  \rightarrow ^ { {\alpha _ n+1} } \ 
A _ {n+2}  \rightarrow \dots
$$

of objects of an Abelian category  $  \mathfrak A $
and of morphisms  $  \alpha _ {i} $
such that

$$ 
 \mathop{\rm Ker}  \alpha _ {n+1}  = \ 
 \mathop{\rm Im}  \alpha _ {n} .
$$

An exact sequence  $  0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $
is called short, and consists of an object  $  B $, 
a subobject  $  A $
of it and the corresponding quotient object  $  C $.

====Comments====
Exact sequences often occur and are often used in (co)homological considerations. There are, e.g., the long homology exact sequence

$$ 
\dots \rightarrow  H _ {r} ( A)  \rightarrow  H _ {r} ( X)  \rightarrow  H _ {r} ( X , A )
 \rightarrow  H _ {r-1} ( A)  \rightarrow \dots
$$

of a pair  $  ( X , A ) $, 
$  A $
a subspace of  $  X $, 
and the long cohomology exact sequence

$$ 
\dots \rightarrow  H  ^ {r-1} ( A)  \rightarrow  H  ^ {r} ( X , A )  \rightarrow  H  ^ {r} ( X)  \rightarrow  H  ^ {r} ( X , A )  \rightarrow \dots .
$$

Analogous long exact sequences occur in a variety of other homology and cohomology theories. Cf. [[Homology theory|Homology theory]]; [[Cohomology|Cohomology]]; [[Cohomology sequence|Cohomology sequence]]; [[Homology sequence|Homology sequence]], and various articles on the (co)homology of various kinds of objects, such as [[Cohomology of algebras]]; [[Cohomology of groups|Cohomology of groups]]; [[Cohomology of Lie algebras|Cohomology of Lie algebras]].

An exact sequence of the form  $  0 \rightarrow A _ {1} \rightarrow A \rightarrow A _ {2} $
is sometimes called a left short exact sequence and one of the form  $  A _ {1} \rightarrow A \rightarrow A _ {2} \rightarrow 0 $
a right short exact sequence. The exact sequence of a morphism  $  \alpha :  X \rightarrow Y $
in an Abelian category is the exact sequence

$$ 
0  \rightarrow   \mathop{\rm Ker} \ 
\alpha  \rightarrow  X  \rightarrow  Y  \rightarrow   \mathop{\rm Coker}  \alpha  \rightarrow  0 .
$$