Remark 10.18.7. Let $R \to S$ be a ring map. Let $\mathfrak q$ be a prime ideal of $S$ lying over the prime ideal $\mathfrak p$ of $R$. According to Remark 10.18.5 the prime $\mathfrak q$ corresponds to a unique prime $\overline{\mathfrak q}$ of the fibre ring $F = S \otimes _ R \kappa (\mathfrak p)$. Then we have
\[ F_{\overline{\mathfrak q}} \cong S_\mathfrak q \otimes _{R_\mathfrak p} \kappa (\mathfrak p) \cong S_\mathfrak q/\mathfrak p S_\mathfrak q \]
Namely, there is an obvious ring map $F \to S_\mathfrak q \otimes _{R_\mathfrak p} \kappa (\mathfrak p)$ which is easily seen to be isomorphic to $F \to F_{\overline{\mathfrak q}}$. The second isomorphism follows from the fact that $\kappa (\mathfrak p)$ is the quotient of $R_\mathfrak p$ by $\mathfrak pR_\mathfrak p$.
Comments (0)
There are also: