= 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.
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