Lemma 109.38.1. There exists a local homomorphism $A \to B$ of local domains which is essentially of finite type and such that $A/\mathfrak m_ A \to B/\mathfrak m_ B$ is finite such that for every prime $\mathfrak q \not= \mathfrak m_ B$ of $B$ the ring map $A \to B/\mathfrak q$ is not the localization of a quasi-finite ring map.
Proof. See the discussion above. $\square$
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