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

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