This is probably an elementary problem but I wasn't able to figure it out. Given a regular ring $A$ and an ideal $I$ which you can assume $A/I$ is also regular. Is the completion ring $\hat{A}$ necessarily regular? This is true if $A$ is local but unfortunately I am not sure whether this can be turned into the local case. Because I'm not sure whether it is possible to figure out what all maximal ideal of $\hat{A}$ look like. If not true is it true if I assume $I$ is a minimal prime ideal generated by a single element?
Maybe the problem can be turned into the local case. I was thinking given any maximal ideal of $\hat{A}$ like $m$. Its image under projection to $A/I$ is going to be an ideal which will be inside another maximal ideal $m'$ of $A/I$. Now $m'$ induces a maximal ideal on $\hat{A}$ which contains $m$. So if $m$ was maximal they have to be the same thing. This means every maximal ideal of $\hat{A}$ is induced from a maximal ideal in $A$. If this is true now we can turn the problem to the local case.
