Let $K$ and $K'$ be number fields $K \subset K'$, and let $R$ and $R'$ be the corresponding ring of integers. Let $S = Spec\ R$ and $S' = Spec\ R'$. Suppose $X \to S$ be an arithmetic surface that is a regular scheme of dimension 2 which is projective and flat over $S$. 


Is $X' = X \times\_S S'$ also regular? If not what kind of singularities may arise? What happens more generally?



Regularity of schemes under base change
I don't think this will be true in general.


Say $K=\mathbf{Q}$ and $K'=\mathbf{Q}(\sqrt{2})$, and let $X\_0$ be $Spec(R')$. Then $X\_0$ is regular of dimension 1 and the map down to $S$ is projective and flat, but the base change to $S'$ is the spectrum of $\mathbf{Z}[\sqrt{2}][X]/(X^2-2)$ which is two smooth affine curves meeting in a nasty way at the point in characteristic 2. Now let $X$ be $X\_0\times\mathbf{P}^1$.


