# Is the projective dimension of modules with finite projective dimension bounded?

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 $$\dim{R}$$?

The answer is yes: these finite projective dimensions are bounded by the Krull dimension of $$R$$!
• Thank you for your answer. To check, this is true even for modules that are not finitely generated, correct? When $M$ is finitely generated I know that $\mathrm{pd}(M) \le \dim{R}$ by the Auslander Buchsbaum formula. Jan 19 at 3:27
• Apologies, I was unclear -- this is for finitely generated modules. Indeed, this paper is the origin of the Auslander-Buchsbaum formula. For general modules, the remarkable answer is that the finite projective dimensions are still bounded, and in fact the maximum finite projective dimension is always exactly $\operatorname{dim}(R)$! This is a theorem of Reynaud and Gruson, from "Critères de platitude et de projectivité" (Theoreme 3.2.6) Jan 19 at 3:49