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All rings are commutative with unit.

Let $\rho:A\rightarrow B$ be a ring homomorphism. Suppose $\mathfrak q$ is a prime ideal of $B$, and let $\mathfrak p=\rho^{-1}(\mathfrak q)$.

My question: Is $B_\mathfrak q$ a localization of $B\otimes_A A_\mathfrak p$, or is it equal to $B\otimes_A A_\mathfrak p$?

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I suspect they are equal, but the book I am reading claims $B_\mathfrak q$ is only a localization of $B\otimes_A A_\mathfrak p$.

Here is my reasoning: Let $S=A-\mathfrak p$. Then $B\otimes_A A_\mathfrak p \cong B\otimes_A S^{-1}A$. This last module is canonically isomorphic to $S^{-1}B$, which is just $B_\mathfrak q$.

Edit: @wxu has kindly pointed out the error in my reasoning. How do I show the claimed fact, then?

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