Exact sequences

Exact sequence

In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next.

Definition

To be precise, fix an abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or some other category with kernels and cokernels (such as the category of all groups). Choose an index set of consecutive integers. Then for each integer i in the index set, let Ai be an object in the category and let fi be a morphism from Ai to Ai+1. This defines a sequence of objects and morphisms.

The sequence is exact at Ai if the image of fi−1 is equal to the kernel of fi:

im fi−1 = ker fi.
The sequence itself is exact if it is exact at each object (except at the very first and the very last object, where exactness doesn't make sense).

Example

Consider the following sequence of abelian groups:

0to Bbb{Z} xrightarrow{2cdot} Bbb{Z} to Bbb{Z}/2Bbb{Z}to 0
where 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z/2Z is given by reducing integers modulo 2. This is indeed an exact sequence:

  • the image of the map 0→Z is {0}, and the kernel of multiplication by 2 is also {0}, so the sequence is exact at the first Z.
  • the image of multiplication by 2 is 2Z, and the kernel of reducing modulo 2 is also 2Z, so the sequence is exact at the second Z.
  • the image of reducing modulo 2 is all of Z/2Z, and the kernel of the zero map is also all of Z/2Z, so the sequence is exact at the position Z/2Z

Another example, from differential geometry, especially relevant for work on the Maxwell equations:

Bbb{H}_1 xrightarrow{mbox{grad}} Bbb{H}_mbox{curl} xrightarrow{mbox{curl}} Bbb{H}_mbox{div} xrightarrow{mbox{div}} Bbb{L}_2,
based on the fact that
mbox{curl},(mbox{grad},f ) = nabla times (nabla f) = 0
mbox{div},(mbox{curl},vec v ) = nabla cdot nabla times vec{v} = 0
on properly defined Hilbert spaces.

Special cases

To make sense of the definition, it is helpful to consider what it means in relatively simple cases where the sequence is finite and begins or ends with 0.

  • The sequence 0 → AB is exact at A if and only if the map from A to B has kernel {0}, i.e. if and only if that map is a monomorphism.
  • Dually, the sequence BC → 0 is exact at C if and only if the image of the map from B to C is all of C, i.e. if and only if that map is an epimorphism.
  • A consequence of these last two facts is that the sequence 0 → XY → 0 is exact if and only if the map from X to Y is a bimorphism.

When dealing with exact sequences of groups, it is common to write 1 instead of 0 for the trivial group with a single element.

Important are short exact sequences, which are exact sequences of the form

: By the above, we know that for any such short exact sequence, f is a monomorphism and g is an epimorphism. Furthermore, the image of f is equal to the kernel of g. It is helpful to think of A as a subobject of B with f being the embedding of A into B, and of C as the corresponding factor object B/A, with the map g being the natural projection from B to B/A (whose kernel is exactly A).

Facts

The splitting lemma states that if the above short exact sequence admits a morphism t: BA such that t o f is the identity on A or a morphism u: CB such that g o u is the identity on C, then B is a twisted direct sum of A and C. (For groups, a twisted direct sum is a semidirect product; in an abelian category, every twisted direct sum is an ordinary direct sum.) In this case, we say that the 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.

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

A_1to A_2to A_3to A_4to A_5to A_6 ,!

and define

C_k = ker (A_kto A_{k+1})= operatorname{im} (A_{k-1}to A_k) = operatorname{coker} (A_{k-2}to A_{k-1}).
Then we obtain a commutative diagram in which all the diagonals are short exact sequences:
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 sub- and factor objects.

The extension problem is essentially the question, given the end terms A and C of a short exact sequence, what possibilities exist for the middle term B? In the category of groups, this is equivalent to the question, what groups B have A as a normal subgroup and C as the corresponding factor group? This problem is important in the classification of groups. See also Outer automorphism group.

Notice that in an exact sequence, the composition fi+1 o fi maps Ai to 0 in Ai+2, so every exact sequence is a chain complex. Furthermore, only fi-images of elements of Ai are mapped to 0 by fi+1, so the homology of this chain complex is trivial. More succinctly:

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 (i.e. an exact sequence indexed by the natural numbers) by repeated application of the snake lemma. This is explained in the article on homology. 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.

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