# localization of completion

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?

• Hint: try to show localization commutes with completion at a maximal ideal. – KReiser Nov 23 '17 at 0:04
• Where does this problem come from? Note that $R_c$ is not necessarily local if $\dim R > 1$. – MooS Nov 23 '17 at 8:58
• 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 – Cusp Nov 24 '17 at 6:31