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Is it possible for a (non-local) regular ring to have an infinite dimension? As far as I know, the well known characterization of regular rings in terms of finite global dimension is only for local case. Or does it also apply to non-local regular rings? Specifically, is the following statement true?

A (non-local) Noetherian ring $A$ is regular if and only if it has a finite global dimension. In this case, $\dim A=\operatorname{gl}\dim A$.

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up vote 3 down vote accepted

There are two questions. For the one in the title, the answer is yes. There are even such ones which are Jacobson rings (every prime ideal is an intersection of maximal ideals), see this paper by K. Fujita.

For the second question, the answer is also yes because $\dim A$ is the supremum of the dimensions of the localizations at maximal ideals $A_m$, and the same is true for the global dimensions (Matsumura, Commutative Algebra, p. 130, Lemma 5.III.)

Edit I didn't see the first part of the second question. So the equivalence is true if $A$ has finite dimension.

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