 Resolution by finitely generated free modules

A finitely generated module over a Noetherian ring admits a possibly infinite free resolution whose every term is finitely generated. Surject a finite free module onto it; the kernel is finitely generated because a finite free module is Noetherian. Repeat this construction on the successive kernels. If the original module has finite projective dimension $n$, its $n$th syzygy is projective by dimension shifting of the Ext functor, giving a length-$n$ resolution with finite projective terms.