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Let $(R,m)$ be a local Noetherian excellent domain and $\hat R$ be the $m$-adic completion of $R.$ Let $0\neq c\in R$ and $R_c$ is regular.

How to show that ${\hat R}_c$ is also regular?

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  • $\begingroup$ Hint: try to show localization commutes with completion at a maximal ideal. $\endgroup$ – KReiser Nov 23 '17 at 0:04
  • $\begingroup$ Where does this problem come from? Note that $R_c$ is not necessarily local if $\dim R > 1$. $\endgroup$ – MooS Nov 23 '17 at 8:58
  • $\begingroup$ I have edited my question and change the ring to excellent ring. This is required in the proof of existence of test element in characteristic p method $\endgroup$ – Cusp Nov 24 '17 at 6:31

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