TITLE: Does finite projective dimension localize?
QUESTION [6 upvotes]: Let $R$ be a commutative (but not necessarily Noetherian) ring with unity.  Let $M$ be an $R$-module.  Suppose that, for all $\mathfrak p \in\text {Spec}(R),$ $\text{pd}_{R_{\mathfrak p}}M_{\mathfrak p}< \infty $.  Is it the case that $\text{pd}_RM < \infty$?

REPLY [9 votes]: Let $R$ be a ring with $\operatorname{gldim}(R)=\infty$. Then there exist $R$-modules of arbitrary large projective dimension. By taking their direct sum we can find an $R$-module $M$ of infinite projective dimension. Now, if the global dimension of all localizations of $R$ is finite, that is, $R_{\mathfrak p}$ is a regular local ring for all $\mathfrak p\in\operatorname{Spec}(R)$, it follows that the projective dimension of all localizations of $M$ is finite. 
Such an $R$ is the classical example of Nagata of a noetherian ring of infinite Krull dimension.