Exact sequence

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; Cohomology; Cohomology 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 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

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