Do there exist semi-local Noetherian rings with infinite Krull dimension?

As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.

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Do there exist semi-local Noetherian rings with infinite Krull dimension?

As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.

user26857

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No, this can't exist: If ${\mathfrak m}_1,...,{\mathfrak m}_k$ are the finitely many maximal ideals of $R$, then each localization $R_{{\mathfrak m}_i}$ has finite Krull-dimension, and hence $\text{dim}(R) = \max\{\text{dim}(R_{{\mathfrak m}_i})\ |\ i=1,2,...,k\}<\infty.$

Hanno

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