# Behavior of regularity under base change

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.

• 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.) – Jason Starr Dec 16 '18 at 15:16
• Yes, a good point. I edit the question. – G.-S. Zhou Dec 16 '18 at 15:49