Let R be a (non local) noetherian ring, and let $\mathrm{gldim}(R)$ be the global dimension. Recall that R is regular iff for every maximal ideal $m\subset R$ the localization $R_m$ is regular.
If $R$ has finite Krull dimension, then we have the following equivalence:
$R$ is regular iff $\mathrm{gldim}(R)=\dim(R)$.
But does this also hold for infinite dimensional rings? To be more precise i know that the implication "$\Rightarrow$" is true, but what about "$\Leftarrow$"?