The Stacks project

Lemma 110.21.1. There exists a Jacobson, universally catenary, Noetherian domain $B$ with maximal ideals $\mathfrak m_1, \mathfrak m_2$ such that $\dim (B_{\mathfrak m_1}) = 1$ and $\dim (B_{\mathfrak m_2}) = 2$.

Proof. The construction of $B$ is given above. We just point out that $B$ is universally catenary by Algebra, Lemma 10.105.4 and Morphisms, Lemma 29.17.5. $\square$

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