The Stacks project

Proof. We use the characterization of dimension of Proposition 10.60.9. Let $x_1, \ldots , x_ d$ be elements of $\mathfrak p$ generating an ideal of definition of $R_{\mathfrak p}$ with $d = \dim (R_{\mathfrak p})$. Let $y_1, \ldots , y_ e$ be elements of $\mathfrak q$ generating an ideal of definition of $S_{\mathfrak q}/\mathfrak pS_{\mathfrak q}$ with $e = \dim (S_{\mathfrak q}/\mathfrak pS_{\mathfrak q})$. It is clear that $S_{\mathfrak q}/(x_1, \ldots , x_ d, y_1, \ldots , y_ e)$ has a nilpotent maximal ideal. Hence $x_1, \ldots , x_ d, y_1, \ldots , y_ e$ generate an ideal of definition of $S_{\mathfrak q}$. $\square$