regular local rings are normal

Given: The objects and hypotheses in the statement. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

regular local domain induction: Every regular local ring is an integral domain.

regular local ring satisfies r one: Every regular local ring satisfies $(R_1)$. Its height-zero localizations are fields, and its height-one localizations are DVRs.

regular local ring satisfies s two: Every regular local ring satisfies $(S_j)$ for every integer $j\ge 0$, in particular $(S_2)$.

serre normality criterion: For every commutative Noetherian ring $R$, including rings with zero divisors and the zero ring, $R$ is normal if and only if it satisfies $(R_1)$ and $(S_2)$.

localisation and polynomial extension of regular rings: Localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular. Regularity can equivalently be tested at maximal ideals. For every nonzero such ring, $\mathrm{gldim}R=\dim R$, allowing infinity. More generally, for a finite module over any commutative Noetherian ring, projective dimension is the supremum of its prime-local projective dimensions. Dedekind domains and their finite polynomial extensions are regular.

reduced noetherian total fractions and normal components: For a reduced commutative Noetherian ring $R$ with minimal primes ${\mathfrak{p}}_1,\dots ,{\mathfrak{p}}_s$, there is a canonical isomorphism $Q(R)\cong {\prod }_{i=1}^s\mathrm{Frac}(R\mathrm{/}{\mathfrak{p}}_i)$. The following are equivalent: $R$ is normal; $R$ is integrally closed in $Q(R)$; and $R$ is a finite product of normal domains. For $R=0$ this is the empty product.