localisations of regular local rings are regular

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

auslander buchsbaum serre regularity criterion: For a nonzero Noetherian local ring $(R,\mathfrak{m},k)$ the following are equivalent: $R$ is regular; ${\mathrm{pd}}_{R}k<\mathrm{\infty }$; $\mathrm{gldim}R<\mathrm{\infty }$; and every finite $R$-module has finite projective dimension. When these hold, $\mathrm{gldim}R={\mathrm{pd}}_{R}k=\dim R$. A nonzero finite module over regular local $R$ is maximal Cohen–Macaulay (depth $\dim R$) if and only if it is free.

Localisation of modules is exact: If $0⟶M^{\mathrm{\prime }}\stackrel{f}{\to }M\stackrel{g}{\to }{M}^{\mathrm{\prime }\mathrm{\prime }}⟶0$ is a short exact sequence of $R$-modules, then $0⟶S^{-1}{M}^{\mathrm{\prime }}\stackrel{S^{-1}f}{\to }{S}^{-1}M\stackrel{S^{-1}g}{\to }{S}^{-1}{M}^{\mathrm{\prime }\mathrm{\prime }}⟶0$ is a short exact sequence of $S^{-1}R$-modules.

Height equals local dimension: Let $R$ be a commutative ring and let $\mathfrak{p}\in \mathrm{Spec}(R)$. Then $\mathrm{ht}(\mathfrak{p})=\sup \{n\ge 0:{\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_n=\mathfrak{p}\text{ is a strict chain of prime ideals in }R\}$. The supremum is allowed to be infinite.