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.
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 kernels and cokernels, and more specially in abelian categories, where it is widely used.
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).
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
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:
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
In this case the monomorphism is 2n ↦ 2n 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 ↦ 2n 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:
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 is
These homomorphisms are restrictions of similarly defined homomorphisms that form the short exact sequence
so the image of the curl is a subset of the kernel of the divergence. The converse is somewhat involved:
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 divergenceless 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
admits a morphism t : B → A such that t ∘ f is the identity on A or a morphism u: C → B such that g ∘ u is the identity on C, then B is a direct sum of A and C (for non-commutative groups, this is a semidirect product). One says that such a short exact sequence splits.
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
Suppose in addition that the cokernel of each morphism exists, and is isomorphic to the image of the next morphism in the sequence:
Conversely, given any list of overlapping short exact sequences, their middle terms form an exact sequence in the same manner.
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 ∘ 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:
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.