Elementary motivations for free resolutions Let $M$ be a finitely generated module over a Noetherian ring $R$ which admits a finite free resolution $0 \to F_n \to \dots \to F_0 \to M \to 0$. There is no doubt that knowing such a resolution is very useful in practice since it allows us to compute many invariants of $M$, such as homology groups, the rank of $M$, Betti numbers in the local case, or the Hilbert polynomial in the graded case. More generally, we can compute any invariant which is easily determined for free modules and behaves well under exact sequences.

While this all is clear for me, I am not able to explain the usefulness of finite free resolutions at an elementary level. Of course it always contains a finite presentation $F_1 \to F_0 \to M \to 0$ which I can motivate: With this data we are able to describe all elements of $M$ and to prove any equation which holds in $M$. In short: a finite presentation allows us to do calculations in $M$. But this explanation creates the impression that a finite presentation is all we need to understand the module $M$, doesn't it? Are there elementary reasons why we should be interested in higher syzygies?

Maybe this is a related question: What were the historical reasons to consider free resolutions?

Thank you in advance!
Intuitively, $M$ is an alternating sum of free modules.
Intuitively, $M$ is an alternating sum of free modules.