Exact Sequences I - Illustrative Examples

#module_theory

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 $i : N \to M$ into $M$ and the other a surjective morphism $π : M \to M / N$ , and these morphisms are related by the fact that $\ker (π) = im (i) :$

N \overset{i}{\to} M \overset{π}{\to} M / N .

Direct sums of modules

Suppose $M_{1}$ and $M_{2}$ are two $R$ -modules. We have seen that their direct sum $M_{1} \oplus M_{2}$ and direct product $M_{1} \times M_{2}$ are isomorphic as $R$ -modules. (Later we'll see this as an instance of something called a biproduct.) The former comes with injective $R$ -module morphisms $j_{i} : M_{i} \to M_{1} \oplus M_{2}$ ; the latter comes with surjective $R$ -module morphisms $p_{i} : M_{1} \times M_{2} \to M_{i}$ . If we let $f : M_{1} \oplus M_{2} \tilde{\to} M_{1} \times M_{2}$ be the $R$ -module isomorphism that sends $m_{1} + m_{2} \mapsto (m_{1}, m_{2})$ , then pre-composing each of the projections $p_{i}$ with $f$ gives surjective $R$ -module morphisms $π_{i} : M_{1} \oplus M_{2} \to M_{i}$ .

What does all of this have in common with the previous example? There is now a very similar connection between the module $M_{1}$ and the module $M_{2}$ by way of the module $M_{1} \oplus M_{2}$ , namely a sequence of $R$ -module morphisms

M_{1} \overset{j_{1}}{\to} M_{1} \oplus M_{2} \overset{π_{2}}{\to} M_{2}

where $j_{1}$ is injective, $π_{2}$ is surjective, and $\ker (π_{2}) = im (j_{1})$ .

Generators and relations for a module

Suppose $M$ is an $R$ -module and $X_{1}$ is any subset of the elements of $M$ . The submodule of $M$ generated by $X_{1}$ is exactly the image of the $R$ -module morphism

F (X_{1}) \overset{π_{1}}{\to} M

where $F (X_{1})$ is the free $R$ -module on $X_{1}$ and $π$ is the $R$ -module morphism corresponding to the inclusion $i : X_{1} \to U (M)$ . In particular, the set $X_{1}$ generates $M$ as an $R$ -module exactly when $π_{1}$ is surjective.

The kernel of $π_{1}$ is the submodule consisting of all formal sums $\sum_{x \in X_{1}} r_{x} \cdot x$ that simplify to $0_{M}$ in the module $M$ . In other words, it consists of all $R$ -linear relations among the elements in $X_{1}$ . If we let $i_{1} : \ker (π_{1}) \to F (X_{1})$ be the inclusion morphism, then we have a sequence of morphisms

\ker (π_{1}) \overset{i_{1}}{\to} F (X_{1}) \overset{π_{1}}{\to} M

where $i_{1}$ is injective, $π_{1}$ is surjective, and $\ker (π_{1}) = im (i_{1})$ . 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 $\ker (π_{1})$ is an $R$ -module, and so a set of generators for $\ker (π_{1})$ corresponds to another surjection

F (X_{2}) \overset{π_{2}^{'}}{\to} \ker (π_{1}) .

Composing with $i_{1}$ then gives a sequence of morphisms

F (X_{2}) \overset{π_{2}}{\to} F (X_{1}) \overset{π_{1}}{\to} M .

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

\dots \overset{π_{n + 1}}{\to} F (X_{n}) \overset{π_{n}}{\to} F (X_{n - 1}) \overset{π_{n - 1}}{\to} \dots \overset{π_{2}}{\to} F (X_{1}) \overset{π_{1}}{\to} M .

Moreover, at every spot in this sequence we have $\ker (π_{n}) = im (π_{n + 1})$ .

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...

Suggested next note

Exact Sequences II - Exact Sequences