Exact sequence

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In mathematics, particularly in abstract algebra and homological algebra, an exact sequence is a sequence of algebraic objects and morphisms which is used to describe or analyse algebraic structure.

In general the concept of an exact sequence makes sense when dealing with algebraic structures for which there are the concepts of "objects", "homomorphisms" between objects and of "subobjects" attached to morphisms which play the role of "kernel" and "image".

We shall expound the concept in group theory: very similar remarks apply to module theory.

A sequence will simply be a collection of group homomorphisms and groups with , indexed by some subset of the integers. A sequence will be termed exact at the term if there are maps and to the left and right of the term and the condition that the kernel of is equal to the image of holds. An exact sequence is one which is exact at every term at which the condition makes sense.

Exactness can be used to unify several concepts in group theory. For example, the assertion that the sequence

is exact asserts that f is injective. We see this by noting that the only possible map i from the trivial group has as image the trivial subgroup of consisting of the identity, and the exactness condition is thus that the kernel of is equal to this trivial subgroup, which is equivalent to the statement that is injective.

Similarly, the assertion that the sequence

is exact asserts that f is surjective. We see this by noting that the only possible map j to the trivial group has as kernel the whole of , and the exactness condition is thus that the image of is equal to this group, which is equivalent to the statement that is surjective.

Combining these, exactness of

asserts that is an isomorphism.

A short exact sequence is one of the form

It expresses the condition that is the quotient of by a subgroup isomorphic to : this may be expressed as saying that is an extension of by .