Let $(R,\mathfrak{m})$ be a noetherian local domain. Let $(A,\mathfrak{n})$ be a regular local ring contains $R$ such that $\mathfrak{n}\cap R=\mathfrak{m}$ and $A$ is essential finitely generated as an $R$-algebra. Let $(\hat{R},\hat{\mathfrak{m}})$ be the completion of $R$ with respect to $\mathfrak{m}$.

Question: is the localization of $B:=A\otimes_R\hat{R}$ at the maximal ideal $\mathfrak{p}:=A\otimes_R\hat{\mathfrak{m}}+ \mathfrak{n}\otimes_R\hat{R}$ regular?

``Answer'': Yes. Let $\hat{A}$ be the $\mathfrak{n}$-adic completion of $A$ and let $\hat{B}$ be the $\mathfrak{p}$-adic completion of $B$. Since $B$ is Noetherian, $\widehat{B_{\mathfrak{p}}}\cong\hat{B} \cong \hat{A}$ (the second isomorphism is not quite obvious). The claim follows from that a Noetherian local ring is regular if and only if its completion is regular.

  • $\begingroup$ Did you intend to write that $A$ is "essentially of finite type" as an $R$-algebra? It is very rare for a local homomorphism of local rings to be a finitely generated homomorphism. (I feel like I wrote this very comment a week or two ago -- I think @darx clarified the confusion in that case.) $\endgroup$ – Jason Starr Dec 16 '18 at 15:16
  • $\begingroup$ Yes, a good point. I edit the question. $\endgroup$ – G.-S. Zhou Dec 16 '18 at 15:49

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