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Let $R$ be a reduced finitely generated algebra over $\Bbb Z$. Let $T$ be a finite set of prime ideals of $R$. Let $S = \bigcap_{p \not\in T} R \setminus p$.

1) Is it true that $A := S^{-1}R$ is finitely generated as an algebra over $\Bbb Z$?

2) Is it true that $B := \bigcap_{p \not \in T} ((R \setminus p)^{-1}R)$ is finitely generated as an algebra over $\Bbb Z$?

It seems to work if $R$ is the ring of integers of a number field, and we have $A \cong B$. I am wondering about the general case. Maybe I should add some assumptions to make this true (e.g. $R$ integral domain...).

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