Before diving into the definition of an exact sequence (and morphisms of exact sequences), we examine a few illustrative examples.
Submodules and quotient modules
Suppose is a submodule of an -module , where is a ring (with unity). We then have a pair of -module morphisms connecting the submodule to the quotient module , namely the injective morphism into and the other a surjective morphism , and these morphisms are related by the fact that
Direct sums of modules
Suppose and are two -modules. We have seen that their direct sum and direct product are isomorphic as -modules. The former comes with injective -module morphisms ; the latter comes with surjective -module morphisms . If we let be the -module isomorphism that sends , then pre-composing each of the projections with gives surjective -module morphisms .
What does all of this have in common with the previous example? There is now a very similar connection between the module and the module by way of the module , namely a sequence of -module morphisms
where is injective, is surjective, and .
Generators and relations for a module
Suppose is an -module and is any subset of the elements of . The submodule of generated by is exactly the image of the -module morphism
where is the free -module on and is the morphism that sends each to itself (considered as an element in ). In particular, the set generates as an -module exactly when is surjective.
The kernel of is the submodule consisting of all formal sums that simplify to in the module . In other words, it consists of all -linear relations among the elements in . If we let be the inclusion morphism, then we have a sequence of morphisms
where is injective, is surjective, and . This information amounts to the classic "generators and relations" description of a module.
However, as the suspicious subscripts might indicate, there is a new feature available in this example. Unlike in the previous examples, here it is clear that we can continue this process. That's because is an -module, and so a set of generators for corresponds to another surjection
Composing with then gives a sequence of morphisms
While it's true is no longer surjective, we still do have . And we can once again continue the process, finding the kernel of , then a set of generators for that kernel, hence a surjection from another free module onto that kernel, and so on. In doing so, we are slowly building a free resolution of the module :
Moreover, at every spot in this sequence we have .
One could reasonably hope that most (maybe even all?) properties of are encoded in this sequence of morphisms. There's a catch, though. The choice of generators (at every step!) is not unique. So for a given module there could be (and almost always are) many other such sequences (of free modules with the same kernel-image relationships). How could we compare one sequence to another? We would probably want a notion of "morphism" between such sequences...