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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 $\dim{R}$?

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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 $R$!

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    $\begingroup$ 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. $\endgroup$
    – Ben C
    Jan 19 at 3:27
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    $\begingroup$ 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) $\endgroup$ Jan 19 at 3:49
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    $\begingroup$ So, in short, yes, this bound holds for all modules. $\endgroup$ Jan 19 at 3:52

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