Let $R$ be a Noetherian ring. Suppose that $M_1$ is a finitely generated $R$-module, so that after picking a list of $n_1$ generators, $M_1$ is isomorphic to $R^{n_1}/M_2$, where $M_2$ is another finitely generated $R$-module. Then we can pick a list of generators on $M_2$, which is where we would stop for the purposes of, say, the Smith Normal Form. But what if we continue? We can write $M_2 \cong R^{n_2}/M_3$, where $M_3$ is another finitely generated $R$-module. And we can repeat with $M_3$, yielding $R^{n_3}$, etc.
I'm interested in the sequence $R^{n_1}, R^{n_2}, \ldots$ I would like to know if we can essentially describe the original module in terms of these free modules. I've tried to make this inquiry more specific with the following three questions, but feel free to answer the more general inquiry in a different way.
- Can the sequence always be chosen to terminate? We say that the sequence terminates at $R^{n_k}$ when the kernel $M_{k+1}$ is trivial.
- When can the ranks of $R^{n_1}, R^{n_2}, \ldots$ be chosen to always decrease?
- Can we recover $M_1$ from $R^{n_1}, R^{n_2}, \ldots$?