I’ve heard it said that the concept of modules was invented to put ideals of rings on the same footing as quotient rings. Both the inclusion of an ideal into a ring and the quotient map are homomorphisms of -modules. In fact, these two homomorphisms have a special relationship: the elements of that are sent to zero in are exactly those in the image of .
In general we say that a composable chain of two homomorphisms is exact (or exact at ) if the elements of sent to in (those in the kernel of ) are exactly those in the image of .
In the special case that , this just says that the only element of sent to in is , i.e. is injective. Similarly, the chain is exact if and only if is surjective: everything in is sent to zero, so the image of must be all of .
So given a ring and an ideal , we actually have two more examples of exactness: not only is exact, but so are and . This collection of information is expressed by saying that the sequence
is an exact sequence. In general, a finite or infinite sequence of morphisms is called an exact sequence if it is exact at each , i.e. each consecutive pair is exact. For example, the operations (gradient), (curl), and (divergence) on vector fields on fit into an exact sequence
because those smooth vector fields whose curls vanish are exactly the gradients of smooth functions, and those whose divergences vanish are exactly the curls of smooth vector fields.
Incidentally, an exact sequence of the type is called a short exact sequence. A short exact sequence is so small that it conveys a very precise relationship between the two homomorphisms and , which can be summed up as
is isomorphic to a submodule of , and is isomorphic to the quotient of by that submodule.
(Incidentally, exact sequences that extend infinitely in one or both directions are called long exact sequences.)
Exact sequences also arise as a way of understanding module presentations. The -module has a single generator , so we get a surjection sending . We can extend this to an exact sequence : exactness on the right is surjectivity of , and exactness on the left says that the integers mapped to are exactly the multiples of .
For a more complicated example, say that an -module is abstractly presented as , i.e. it has three generators, , , and , subject only to the relation . The three generators give us a surjection sending to and so on, which we can interpret as an exact sequence . We can express the fact that every relation between the elements , , and is a multiple of by saying that the kernel of is generated by . In other words, the homomorphism sending to makes the sequence exact. In general, if we present a module with generators and relations, we get an exact sequence .
(In the case of , we see that we are presenting with a single generator modulo a single relation.)
From the exact sequence , all the information defining is contained in the homomorphism ; if we know it, then is its cokernel, the quotient of by the image of in it. In general, if is the cokernel of a general module homomorphism , then we get an exact sequence . Similarly, if is the kernel of then we get an exact sequence . In fact, the kernel and cokernel of are the only modules (up to isomorphism) that fit into the blank spots in to give an exact sequence.
Generalizing to a general element gives another useful example: the map given by multiplication by has image , thus cokernel . The kernel of is the set of elements whose product with is zero, an ideal called the annihilator of . So we get an exact sequence
Now, the image of is the ideal , which is the kernel of , so we also get a short exact sequence as at the top of this post. Less obviously, we also find that is the cokernel of , since the map is surjective with kernel . Therefore we have another short exact sequence . The four-term exact sequence above is thus “built” out of two smaller short exact sequences.
In fact, this is a general phenomenon. If
is an exact sequence of any length, then by definition each image is the same submodule of as the following kernel , so we have short exact sequences
for each .
This lets us prove facts about general exact sequences from facts about short exact sequences. For example, if is a short exact sequence of vector spaces, then the rank-nullity theorem says that , but we can do better. If is any exact sequence of vector spaces, with the maps denoted , then , since both sums are equal to .
For modules over general rings, there isn’t a notion of dimension that behaves additively in short exact sequences , but the principle of getting information about from information about and appears again and again, making short exact sequences an exceptionally useful concept.
(Incidentally, if you know the origin of the rough “same footing” quotation at the top of this post, please tell me! I’d love to credit its originator.)