An exact sequence is a sequence of morphisms between objects (for example, groups
) such that the image
of one morphism equals the kernel of the next.
In the context of group theory
, a sequence
of groups and group homomorphism
s is called exact 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 structure
s. For example, one could have an exact sequence of vector space
s and linear map
s, or of modules and module homomorphism
s. More generally, the notion of an exact sequence makes sense in any category
s and cokernel
To understand the definition, it is helpful to consider relatively simple cases where the sequence 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).
* Consider the sequence 0 → ''A'' → ''B''. The image of the leftmost map is 0. Therefore the sequence is exact if and only if the rightmost map (from ''A'' to ''B'') has kernel ; that is, if and only if that map is a monomorphism
(injective, or one-to-one).
* Consider the dual sequence ''B'' → ''C'' → 0. The kernel of the rightmost map is C. Therefore the sequence is exact if and only if the image of the leftmost map (from ''B'' to ''C'') is all of ''C''; that is, if and only if that map is an epimorphism
(surjective, or onto).
* Therefore, the sequence 0 → ''X'' → ''Y'' → 0 is exact if and only if the map from ''X'' to ''Y'' is both a monomorphism and epimorphism (that is, a bimorphism
), and thus, in many cases, an isomorphism
from ''X'' to ''Y''.
Short exact sequence
Important are short exact sequences, which are exact sequences of the form
As established above, 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'' embedding ''A'' into ''B'', and of ''C'' as the corresponding factor object (or quotient
), ''B''/''A'', with ''g'' inducing an isomorphism
The short exact sequence
is called split if there exists a homomorphism ''h'' : ''C'' → ''B'' such that the composition ''g'' ∘ ''h'' is the identity map on ''C''. It follows that if these are abelian groups, ''B'' is isomorphic to the direct sum
of ''A'' and ''C'' (see Splitting lemma
Long exact sequence
A long exact sequence is an exact sequence consisting of more than three nonzero terms, often an infinite exact sequence.
A long exact sequence
is equivalent to a sequence of short exact sequences
Integers modulo two
Consider the following sequence of abelian group
The first homomorphism maps each element ''i'' in the set of integers Z to the element 2''i'' in Z. The second homomorphism maps each element ''i'' in Z to an element ''j'' in the quotient group; that is, ''j'' = ''i'' mod 2. Here the hook arrow
indicates that the map 2× from Z to Z is a monomorphism, and the two-headed arrow
indicates an epimorphism (the map mod 2). This is an exact sequence because the image 2Z of the monomorphism is the kernel of the epimorphism. Essentially "the same" sequence can also be written as
In this case the monomorphism is 2''n'' ↦ 2''n'' and although it looks like an identity function, it is not onto (that is, not an epimorphism) because the odd numbers don't belong to 2Z. The image of 2Z through this monomorphism is however exactly the same subset of Z as the image of Z through ''n'' ↦ 2''n'' used in the previous sequence. This latter sequence does differ in the concrete nature of its first object from the previous one as 2Z is not the same set as Z even though the two are isomorphic as groups.
The first sequence may also be written without using special symbols for monomorphism and epimorphism:
Here 0 denotes the trivial group, 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 , and the kernel of multiplication by 2 is also , 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 Z/2Z, and the kernel of the zero map is also Z/2Z, so the sequence is exact at the position Z/2Z.
The first and third sequences are somewhat of a special case owing to the infinite nature of Z. 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
As a more concrete example of an exact sequence on finite groups:
is the cyclic group
of order ''n'' and
is the dihedral group
of order 2''n'', which is a non-abelian group.
Intersection and sum of modules
Let and be two ideal
s of a ring .
is an exact sequence of -modules, where the module homomorphism
maps each element of
to the element of the direct sum
, and the homomorphsim
maps each element of
These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence
Passing to quotient module
s yield another exact sequence
Grad, curl and div in differential geometry
Another example can be derived from differential geometry
, especially relevant for work on the Maxwell equations
Consider the Hilbert space
of scalar-valued square-integrable functions on three dimensions
. Taking the gradient
of a function
moves us to a subset of
, the space of vector valued, still square-integrable functions on the same domain
-- specifically, the set of such functions that represent conservative vector fields. (The generalized Stokes' theorem
has preserved integrability.)
First, note the curl
of all such fields is zero -- since
for all such . However, this only proves that the image of the gradient
is a subset of the kernel of the curl. To prove that they are in fact the same set, prove the converse: that if the curl of a vector field
is 0, then
is the gradient of some scalar function. This follows almost immediately from Stokes' theorem
(see the proof at conservative force
.) The image of the gradient
is then precisely the kernel of the curl, and so we can then take the curl to be our next morphism, taking us again to a (different) subset of
Similarly, we note that
so the image of the curl is a subset of the kernel of the divergence
. The converse is somewhat involved:
Having thus proved that the image of the curl is precisely the kernel of the divergence, this morphism in turn takes us back to the space we started from
. Since definitionally we have landed on a space of integrable functions, any such function can (at least formally) be integrated in order to produce a vector field which divergence is that function -- so the image of the divergence is the entirety of
, and we can complete our sequence:
Equivalently, we could have reasoned in reverse: in a simply connected
space, a curl-free vector field (a field in the kernel of the curl) can always be written as a gradient of a scalar function
(and thus is in the image of the gradient). Similarly, a divergence
less field can be written as a curl of another field.
(Reasoning in this direction thus makes use of the fact that 3-dimensional space is topologically trivial.)
This short exact sequence also permits a much shorter proof of the validity of the Helmholtz decomposition
that does not rely on brute-force vector calculus. Consider the subsequence
Since the divergence of the gradient is the Laplacian
, and since the Hilbert space of square-integrable functions can be spanned by the eigenfunctions of the Laplacian, we already see that some inverse mapping
must exist. To explicitly construct such an inverse, we can start from the definition of the vector Laplacian
Since we are trying to construct an identity mapping by composing some function with the gradient, we know that in our case
. Then if we take the divergence of both sides
we see that if a function is an eigenfunction of the vector Laplacian, its divergence must be an eigenfunction of the scalar Laplacian with the same eigenvalue. Then we can build our inverse function
simply by breaking any function in
into the vector-Laplacian eigenbasis, scaling each by the inverse of their eigenvalue, and taking the divergence; the action of
is thus clearly the identity. Thus by the splitting lemma
or equivalently, any square-integrable vector field on
can be broken into the sum of a gradient and a curl -- which is what we set out to prove.
The splitting lemma
states that if the short exact sequence
admits a morphism such that is the identity on or
a morphism such that is the identity on , then is a direct sum
of and (for non-commutative groups, this is a semidirect product
). 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.
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 group
s; but it is not true for all categories that allow exact sequences, and in particular is not true for the category of groups
, in which coker(''f'') : ''G'' → ''H'' is not ''H''/im(''f'') but
, the quotient of ''H'' by the conjugate closure
of im(''f'').) Then we obtain a commutative diagram in which all the diagonals are short exact sequences:
:Image:long short exact sequences.png
The only portion of this diagram that depends on the cokernel condition is the object
and the final pair of morphisms
. If there exists any object
is exact, then the exactness of
is ensured. Again taking the example of the category of groups, the fact that im(''f'') is the kernel of some homomorphism on ''H'' implies that it is a normal subgroup
, which coincides with its conjugate closure; thus coker(''f'') is isomorphic to the image ''H''/im(''f'') 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 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 ''f''''i''+1
to 0 in ''A''''i''+2
, so every exact sequence is a chain complex
. Furthermore, only ''f''''i''
-images of elements of ''A''''i''
are mapped to 0 by ''f''''i''+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 (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 functor
s are functor
s that transform exact sequences into exact sequences.