Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$ is open in $\operatorname{Spec}B$. (See, e.g., page 187 of Matsumara's Commutative Ring Theory.)

Is there a known example of a non finitely generated $A$-algebra $B$, where the set $U$ as defined above is not open?

Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $$U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$$ is open in $\operatorname{Spec}B$. (See, e.g., page 187 of Matsumara's Commutative Ring Theory.)

Is there a known example of a non finitely generated $A$-algebra $B$, where the set $U$ as defined above is not open?

Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$ is open in $\operatorname{Spec}B$.

Is there a known example of a non finitely generated $A$-algebra $B$, where the set $U$ as defined above is not open?

Let $A$ be a commutative Noetherian ring and $B$ a finitely generated $A$-algebra. Then the set $U\colon=\{P\in\operatorname{Spec}B\mid B_P\ \mathrm{is\ flat\ over}\ A\}$ is open in $\operatorname{Spec}B$. (See, e.g., page 187 of Matsumara's Commutative Ring Theory.)

Is there a known example of a non finitely generated $A$-algebra $B$, where the set $U$ as defined above is not open?