Illustrative examples

Illustrative examples

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 N is a submodule of an R-module M, where R is a ring (with unity). We then have a pair of R-module morphisms connecting the submodule N to the quotient module M/N, namely the injective morphism i:Nβ†’M into M and the other a surjective morphism Ο€:Mβ†’M/N, and these morphisms are related by the fact that ker⁑(Ο€)=im(i):

N→iM→πM/N.

Direct sums of modules

Suppose M1 and M2 are two R-modules. We have seen that their direct sum M1βŠ•M2 and direct product M1Γ—M2 are isomorphic as R-modules. The former comes with injective R-module morphisms ji:Miβ†’M1βŠ•M2; the latter comes with surjective R-module morphisms pi:M1Γ—M2β†’Mi. If we let f:M1βŠ•M2β†’βˆΌM1Γ—M2 be the R-module isomorphism that sends m1+m2↦(m1,m2), then pre-composing each of the projections pi with f gives surjective R-module morphisms Ο€i:M1βŠ•M2β†’Mi.

What does all of this have in common with the previous example? There is now a very similar connection between the module M1 and the module M2 by way of the module M1βŠ•M2, namely a sequence of R-module morphisms

M1β†’j1M1βŠ•M2β†’Ο€2M2

where j1 is injective, Ο€2 is surjective, and ker⁑(Ο€2)=im(j1).

Generators and relations for a module

Suppose M is an R-module and A1 is any subset of the elements of M. The submodule of M generated by A1 is exactly the image of the R-module morphism

F(A1)β†’Ο€1M

where F(A1) is the free R-module on A1 and Ο€1 is the morphism that sends each a∈A1 to itself (considered as an element in M). In particular, the set A1 generates M as an R-module exactly when Ο€1 is surjective.

The kernel of Ο€1 is the submodule consisting of all formal sums βˆ‘a∈A1raβ‹…a that simplify to 0M in the module M. In other words, it consists of all R-linear relations among the elements in A1. If we let i1:ker⁑(Ο€1)β†’F(A1) be the inclusion morphism, then we have a sequence of morphisms

ker⁑(Ο€1)β†’i1F(A1)β†’Ο€1M

where i1 is injective, Ο€1 is surjective, and ker⁑(Ο€1)=im(i1). 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 ker⁑(Ο€1) is an R-module, and so a set of generators for ker⁑(Ο€1) corresponds to another surjection

F(A2)β†’Ο€2β€²ker⁑(Ο€1).

Composing with i1 then gives a sequence of morphisms

F(A2)β†’Ο€2F(A1)β†’Ο€1M.

While it's true Ο€2 is no longer surjective, we still do have ker⁑(Ο€1)=im(Ο€2). And we can once again continue the process, finding the kernel of Ο€2, 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 M:

β‹―β†’Ο€n+1F(An)β†’Ο€nF(Anβˆ’1)β†’Ο€nβˆ’1β‹―β†’Ο€2F(A1)β†’Ο€1M.

Moreover, at every spot in this sequence we have ker⁑(Ο€n)=im(Ο€n+1).

One could reasonably hope that most (maybe even all?) properties of M 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 M 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...