Let $R$ be a regular local ring and let $M$ be an $R$-module. Then there exists a finite projective resolution $P_\bullet\to M\to 0$. However, need there exist a finite projective resolution consisting of finitely generated projective modules? What if we require that $M$ be finitely generated?
A regular local ring $R$ is Noetherian by hypothesis. If $M$ is a finitely generated $R$-module, then there is a finite rank free module $F$ and a surjection $F\to M$ fitting into a short exact sequence $$0\to N\to F\to M\to0.$$ As $N$ is a submodule of the finitely generated free module $F$ over the Noetherian ring $R$ then $N$ is finitely generated. Iterating this, gives a resolution of $N$ by finitely generated free modules. As $R$ is regular, it has finite global dimension, and the iterated kernels are eventually projective (so free) and the resolution can be brought to an end.