The Stacks project

Lemma 110.37.1. There exists a local ring $A$, a finite type ring map $A \to B$ and a prime $\mathfrak q$ lying over $\mathfrak m_ A$ such that $B_{\mathfrak q}$ is flat over $A$, and for any element $g \in B$, $g \not\in \mathfrak q$ the ring $B_ g$ is neither finitely presented over $A$ nor flat over $A$.

Proof. See discussion above. $\square$

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