completion preserves regular local rings

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

completion preserves embedding dimension: For a nonzero Noetherian local ring $(R,\mathfrak{m},k)$, its maximal-adic completion $\widehat{R}$ has maximal ideal $\widehat{\mathfrak{m}}=\mathfrak{m}\widehat{R}$, residue field $k$, and a canonical isomorphism $\mathfrak{m}\mathrm{/}{\mathfrak{m}}^2\cong \widehat{\mathfrak{m}}\mathrm{/}{\widehat{\mathfrak{m}}}^2$. In particular their embedding dimensions agree.

Completion preserves dimension and Hilbert-Samuel data: Assume the Axiom of Choice. Let $(R,\mathfrak{m})$ be a Noetherian local ring, let $M\mathrm{\ne }0$ be a finitely generated $R$-module, and let $\widehat{R}$, $\widehat{M}$ denote the $\mathfrak{m}$-adic completions. 1. For every $n\ge 0$, $\widehat{M}\mathrm{/}{\mathfrak{m}}^{n+1}\widehat{M}\cong M\mathrm{/}{\mathfrak{m}}^{n+1}M\mathrm{.}$ In particular the Hilbert-Samuel functions of $M$ and $\widehat{M}$ agree. 2. The Hilbert-Samuel multiplicity of $M$ equals that of $\widehat{M}$. 3. The support dimensions of $M$ and $\widehat{M}$ are equal.