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 $i:N\beta \x86\x92M$ into $M$ and the other a surjective morphism $\mathrm{{\rm O}\x80}:M\beta \x86\x92M/N$, and these morphisms are related by the fact that $\mathrm{ker}\beta \x81\u2018(\mathrm{{\rm O}\x80})=\mathrm{im}(i):$

Suppose ${M}_{1}$ and ${M}_{2}$ are two $R$-modules. We have seen that their direct sum ${M}_{1}\beta \x8a\x95{M}_{2}$ and direct product ${M}_{1}\Gamma \x97{M}_{2}$ are isomorphic as $R$-modules. The former comes with injective $R$-module morphisms ${j}_{i}:{M}_{i}\beta \x86\x92{M}_{1}\beta \x8a\x95{M}_{2}$; the latter comes with surjective $R$-module morphisms ${p}_{i}:{M}_{1}\Gamma \x97{M}_{2}\beta \x86\x92{M}_{i}$. If we let $f:{M}_{1}\beta \x8a\x95{M}_{2}\stackrel{\beta \x88\u038c\phantom{\rule[-.25em]{0ex}{0ex}}}{{\textstyle \beta \x86\x92}}{M}_{1}\Gamma \x97{M}_{2}$ be the $R$-module isomorphism that sends ${m}_{1}+{m}_{2}\beta \x86\xa6({m}_{1},{m}_{2})$, then pre-composing each of the projections ${p}_{i}$ with $f$ gives surjective $R$-module morphisms ${\mathrm{{\rm O}\x80}}_{i}:{M}_{1}\beta \x8a\x95{M}_{2}\beta \x86\x92{M}_{i}$.

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 ${M}_{1}\beta \x8a\x95{M}_{2}$, namely a sequence of $R$-module morphisms

where ${j}_{1}$ is injective, ${\mathrm{{\rm O}\x80}}_{2}$ is surjective, and $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{2})=\mathrm{im}({j}_{1})$.

Generators and relations for a module

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

where $F({A}_{1})$ is the free $R$-module on ${A}_{1}$ and ${\mathrm{{\rm O}\x80}}_{1}$ is the morphism that sends each $a\beta \x88\x88{A}_{1}$ to itself (considered as an element in $M$). In particular, the set ${A}_{1}$ generates $M$ as an $R$-module exactly when ${\mathrm{{\rm O}\x80}}_{1}$ is surjective.

The kernel of ${\mathrm{{\rm O}\x80}}_{1}$ is the submodule consisting of all formal sums $\underset{a\beta \x88\x88{A}_{1}}{\beta \x88\x91}{r}_{a}\beta \x8b\x85a$ that simplify to ${0}_{M}$ in the module $M$. In other words, it consists of all $R$-linear relations among the elements in ${A}_{1}$. If we let ${i}_{1}:\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})\beta \x86\x92F({A}_{1})$ be the inclusion morphism, then we have a sequence of morphisms

where ${i}_{1}$ is injective, ${\mathrm{{\rm O}\x80}}_{1}$ is surjective, and $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})=\mathrm{im}({i}_{1})$. 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 $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})$ is an $R$-module, and so a set of generators for $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})$ corresponds to another surjection

While it's true ${\mathrm{{\rm O}\x80}}_{2}$ is no longer surjective, we still do have $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{1})=\mathrm{im}({\mathrm{{\rm O}\x80}}_{2})$. And we can once again continue the process, finding the kernel of ${\mathrm{{\rm O}\x80}}_{2}$, 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 $M$:

Moreover, at every spot in this sequence we have $\mathrm{ker}\beta \x81\u2018({\mathrm{{\rm O}\x80}}_{n})=\mathrm{im}({\mathrm{{\rm O}\x80}}_{n+1})$.

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...