The Stacks project

Proof. Let $x_1, \ldots , x_ n$ be generators of $S$ over $R$. By Lemma 10.122.4 the assumption means that $S \otimes _ R \kappa (\mathfrak p) = S_{\mathfrak p}/\mathfrak pS_{\mathfrak p}$ is a finite $\kappa (\mathfrak p)$-algebra. Thus we may find monic polynomials $P_ i \in R_{\mathfrak p}[X]$ such that $P_ i(x_ i)$ maps to zero in $S_{\mathfrak p}/\mathfrak pS_{\mathfrak p}$. Since $\mathfrak p$ is a minimal prime, $\mathfrak pR_{\mathfrak p}$ is a locally nilpotent ideal, see Lemma 10.25.1. Hence $\mathfrak pS_{\mathfrak p}$ is a locally nilpotent ideal, see Lemma 10.32.3. Thus there exist $e_ i \geq 1$ such that $P(x_ i)^{e_ i} = 0$ in $S_{\mathfrak p}$. Let $g_1 \in R$, $g_1 \not\in \mathfrak p$ be an element such that $P_ i$ has coefficients in $R[1/g_1]$ for all $i$. Next, let $g_2 \in R$, $g_2 \not\in \mathfrak p$ be an element such that $P(x_ i)^{e_ i} = 0$ in $S_{g_1g_2}$. Setting $g = g_1g_2$ we win. $\square$