I know that a Noetherian local ring $R$ has finite global dimension if and only if $R$ is regular in which case $\mathrm{gldim}{R} = \dim{R}$. Therefore, for regular rings, every module has projective dimension at most $\dim{R}$. When $R$ is not regular there must exist modules with infinite projective dimension.
My question is:
For $R$ Noetherian local but not regular, do there exist modules with finite but arbitrarily large projective dimension or are the projective dimensions of modules with $\mathrm{pd}(M) < \infty$ bounded? If so, are they bounded by $\dim{R}$?
Best Answer
I'll assume all rings are commutative. In this case, your question was answered precisely by Auslander and Buchsbaum in "Homological dimension in local rings". Link!
The answer is yes: these finite projective dimensions are bounded by the Krull dimension of $R$!