In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.

Definition

of groups and group homomorphisms is said to be exact at. The sequence is called exact, i.e., if the image of each homomorphism is equal to the kernel of the next.

The sequence of groups and homomorphisms may be either finite or infinite.

A similar definition can be made for other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms. More generally, the notion of an exact sequence makes sense in any category with kernelss and cokernels, and more specially in abelian categories, where it is widely used.

Simple cases

To understand the definition, it is helpful to consider relatively simple cases where the sequence is of group homomorphisms, is finite, and begins or ends with the trivial group. Traditionally, this, along with the single identity element, is denoted 0 (additive notation, usually when the groups are abelian), or denoted 1 (multiplicative notation).

• ; that is, if and only if that map is a monomorphism (injective, or one-to-one).

• ; that is, if and only if that map is an epimorphism (surjective, or onto).

• is both a monomorphism and epimorphism (that is, a bimorphism), and so it is an isomorphism and more generally in any exact category).

Short exact sequence

as a subobject as the corresponding factor (or quotient

is called split. It follows that if these are abelian groups is isomorphic to the direct sum

Long exact sequence

A general exact sequence is sometimes called a long exact sequence, to distinguish from the special case of a short exact sequence.

A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence

with n ≥ 2, we can split it up into the short sequences

. By construction, the sequences (2)'s (regardless of the exactness of (1)). Furthermore, (1) is a long exact sequence if and only if (2) are all short exact sequences.

See weaving lemma for details on how to re-form the long exact sequence from the short exact sequences.

Examples

Integers modulo two

is a monomorphism indicates an epimorphism of the monomorphism is the kernel of the epimorphism. Essentially "the same" sequence can also be written as

even though the two are isomorphic as groups.

to the quotient group is given by reducing integers modulo 2. This is indeed an exact sequence:

• .

• .

• .

. It is not possible for a finite group to be mapped by inclusion (that is, by a monomorphism) as a proper subgroup of itself. Instead the sequence that emerges from the first isomorphism theorem is

as these groups are not supposed to be abelian).

As a more concrete example of an exact sequence on finite groups:

is the cyclic group of order n is the dihedral group of order 2n, which is a non-abelian group.

Intersection and sum of modules

be two ideals to the element of the direct sum maps each element to .

These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence

Passing to quotient modules yields another exact sequence

Properties

The splitting lemma the following conditions are equivalent.

• is the identity on .

• is the identity on .

• such that is the direct sum.

For non-commutative groups, the splitting lemma does not apply, and one has only the equivalence between the two last conditions, with "the direct sum" replaced with "a semidirect product".

In both cases, one says that such a short exact sequence splits.

The snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case.

The five lemma gives conditions under which the middle map in a commutative diagram with exact rows of length 5 is an isomorphism; the short five lemma is a special case thereof applying to short exact sequences.

Weaving lemma

The importance of short exact sequences is underlined by the fact that every exact sequence results from "weaving together" several overlapping short exact sequences. Consider for instance the exact sequence

which implies that there exist objects Ck in the category such that

.

Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:

(This is true for a number of interesting categories, including any abelian category such as the abelian groups; but it is not true for all categories that allow exact sequences, and in particular is not true for the category of groups by the conjugate closure.) Then we obtain a commutative diagram in which all the diagonals are short exact sequences:

implies that it is a normal subgroup of the next morphism.

Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.

Applications of exact sequences

In the theory of abelian categories, short exact sequences are often used as a convenient language to talk about subobjects and factor objects.

The extension problem as the corresponding factor group? This problem is important in the classification of groups. See also Outer automorphism group.

, so every exact sequence is a chain complex, so the homology of this chain complex is trivial. More succinctly:

1. Exact sequences are precisely those chain complexes which are acyclic.

Given any chain complex, its homology can therefore be thought of as a measure of the degree to which it fails to be exact.

If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig-zag lemma. It comes up in algebraic topology in the study of relative homology; the Mayer–Vietoris sequence is another example. Long exact sequences induced by short exact sequences are also characteristic of derived functors.

Exact functors are functors that transform exact sequences into exact sequences.

de Rham complex

The term "exact" originates from exact differential forms in the context of the de Rham complex: this sequence can be written as: represent the gradient, curl, and divergence. The sequence then yields the identities from advanced calculus.

. The de Rham complex is an exact sequence of modules if and only if every closed form is exact, which explains why the word exact is an adjective of "exact sequence".

References

1. Citations

   Sources