Exact sequence

A sequence

of objects of an Abelian category and of morphisms such that

An exact sequence is called short, and consists of an object , a subobject of it and the corresponding quotient object .

Comments

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

of a pair , a subspace of , and the long cohomology exact sequence

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 is sometimes called a left short exact sequence and one of the form a right short exact sequence. The exact sequence of a morphism in an Abelian category is the exact sequence