Definition 10.125.1. Suppose that $R \to S$ is of finite type, and let $\mathfrak q \subset S$ be a prime lying over a prime $\mathfrak p$ of $R$. We define the *relative dimension of $S/R$ at $\mathfrak q$*, denoted $\dim _{\mathfrak q}(S/R)$, to be the dimension of $\mathop{\mathrm{Spec}}(S \otimes _ R \kappa (\mathfrak p))$ at the point corresponding to $\mathfrak q$. We let $\dim (S/R)$ be the supremum of $\dim _{\mathfrak q}(S/R)$ over all $\mathfrak q$. This is called the *relative dimension of* $S/R$.

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