normal domain implies r one

Given: The objects and hypotheses in the statement.

serre r k and s k conditions: For a commutative Noetherian ring $R$ and an integer $j\ge 0$, condition $(R_j)$ means that $R_{\mathfrak{p}}$ is regular whenever $\mathrm{ht}\mathfrak{p}\le j$. Condition $(S_j)$ means that $\mathrm{depth}{R}_{\mathfrak{p}}\ge \min \{j,\dim R_{\mathfrak{p}}\}$ for every prime $\mathfrak{p}$. A finite module $M$ satisfies $(S_j)$ if ${\mathrm{depth}}_{R_{\mathfrak{p}}}{M}_{\mathfrak{p}}\ge \min \{j,\dim {\mathrm{Supp}}_{R_{\mathfrak{p}}}{M}_{\mathfrak{p}}\}$ for every prime in its support. Outside the support the condition is vacuous, consistent with depth of the zero module being $+\mathrm{\infty }$ and the empty support having no nonnegative dimension. Thus the zero module satisfies all $(S_j)$ conditions, and the zero ring satisfies both families vacuously.

Height-one localizations of normal Noetherian domains are DVRs: Let $R$ be a Noetherian integrally closed domain, and let $\mathfrak{p}$ be a prime ideal of height $1$. Then the localisation $R_{\mathfrak{p}}$ is a discrete valuation ring.

embedding dimension and regular local ring: A nonzero Noetherian local ring $(A,\mathfrak{m},k)$ is regular when ${\dim }_k(\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2)=\dim A$.