The Stacks project

Remark 27.13.2. In [Hoobler-finite] a (locally Noetherian) scheme $X$ is called Japanese if for every $x \in X$ and every associated prime $\mathfrak p$ of $\mathcal{O}_{X, x}$ the ring $\mathcal{O}_{X, x}/\mathfrak p$ is Japanese. We do not use this definition since there exists a one dimensional noetherian domain with excellent (in particular Japanese) local rings whose normalization is not finite. See [Example 1, Hochster-loci] or [Heinzer-Levy] or [Exposé XIX, Traveaux]. On the other hand, we could circumvent this problem by calling a scheme $X$ Japanese if for every affine open $\mathop{\mathrm{Spec}}(A) \subset X$ the ring $A/\mathfrak p$ is Japanese for every associated prime $\mathfrak p$ of $A$.


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