Definition 10.112.5. Suppose that $R \to S$ is a ring map. Let $\mathfrak q \subset S$ be a prime lying over the prime $\mathfrak p$ of $R$. The *local ring of the fibre at $\mathfrak q$* is the local ring

\[ S_{\mathfrak q}/\mathfrak pS_{\mathfrak q} = (S/\mathfrak pS)_{\mathfrak q} = (S \otimes _ R \kappa (\mathfrak p))_{\mathfrak q} \]

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