The Stacks project

Proof. By Lemma 15.45.3 we know that $R^ h$ and $R^{sh}$ are Noetherian, hence the lemma makes sense. Let $\mathfrak m$ be the maximal ideal of $R$. Let $x_1, \ldots , x_ t \in \mathfrak m$ be a minimal system of generators of $\mathfrak m$, i.e., such that the images in $\mathfrak m/\mathfrak m^2$ form a basis over $\kappa = R/\mathfrak m$. Because $R \to R^ h$ and $R \to R^{sh}$ are faithfully flat, it follows that the images $x_1^ h, \ldots , x_ t^ h$ in $R^ h$, resp. $x_1^{sh}, \ldots , x_ t^{sh}$ in $R^{sh}$ are a minimal system of generators for $\mathfrak m^ h = \mathfrak mR^ h$, resp. $\mathfrak m^{sh} = \mathfrak mR^{sh}$. Regularity of $R$ by definition means $t = \dim (R)$ and similarly for $R^ h$ and $R^{sh}$. Hence the lemma follows from the equality of dimensions $\dim (R) = \dim (R^ h) = \dim (R^{sh})$ of Lemma 15.45.7 $\square$