Exact Sequence - Example

Example

Consider the following sequence of abelian groups:

The first operation forms an element in the set of integers, Z, using multiplication by 2 on an element from Z j = 2i. The second operation forms an element in the quotient space, 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.

This sequence may also be written without using special symbols for monomorphism and epimorphism:

Here 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:

,

based on the fact that on properly defined Hilbert spaces,

in addition, curl-free vector fields can always be written as a gradient of a scalar function (as soon as the space is assumed to be simply connected, see Note 1 below), and that a divergenceless field can be written as a curl of another field.

Note 1: this example makes use of the fact that 3-dimensional space is topologically trivial.

Note 2: and are the domains for the curl and div operators respectively.