In some notes I'm following, there is a statement
(Clearly) over a regular local ring each finitely generated module has a finite free resolution
Is the finitely generated assumption neccessary? If it is infact clear (for finitely generated modules), why?
The furthest I can get with what I know is as follows.
A regular local ring has finite (right) global dimension, so every module has finite projective dimension. Let $M$ have $\mathrm{pd}(M) = n$. Therefore, if we write down an exact sequence:
$0 \rightarrow K \rightarrow F_n \rightarrow \dots \rightarrow F_0 \rightarrow M \rightarrow 0$
with the $F_i$ free. By Schanuel's Lemma, $K$ is projective.
I would aim to produce a free resolution from this, but cannot see an obvious path. With the finitely generated assumption, the $F_i$ and $K$ would be finitely generated.