Let $R$ be a complete DVR with fraction field $K$, $X$ be a regular scheme flat over $R$. Let $L$ be a finite field extension of $K$ and $Q$ be the integral closure of $R$ in $L$. Denote by $Y:=X \times\_R Q$ the base change of $X$. Is $Y$ a regular scheme? If not true in general, is there any additional assumption on $R$ for which this holds true?


PS. One can assume that the characteristic of $K$ is zero.



Base change of regular schemes
No. Take $R=\mathbb{Z}\_p$ and $X$ to be the affine scheme defined by $xy=p$. This is regular, but its base change to $\mathbb{Z}\_p[\sqrt{p}]$ is not.


If $X$ is smooth over $R$, then $Y$ is smooth over $Q$ (and therefore regular in many useful situations). This is of course a much stronger assumption.


