# Base change of regular schemes [closed]

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.

• Check out formally smooth and/or geometrically regular. – Olivier Apr 22 '15 at 20:19

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.
• And if you want an equal-characteristic example, take $R=\mathbf C[[t]]$ and $X$ be the closed subscheme of $\mathbf A^2_R$ defined by $xy=t$; it is regular, but its base change to $\mathbf C[[t^{1/2}]]$ is not. – ACL Apr 22 '15 at 18:56