An inseparable lift of a regular variety.

Question

Let $X$ be a variety over an (imperfect) field $k$, that is regular as a scheme. Let $k'/k$ be an algebraic inseparable extension (I am interested in $k'$ being the perfection or the algebraic closure of $k$; yet one can assume that the extension is finite). Then the scheme $X_{k'}$ is not necessarily reduced. Yet what could one say about the corresponding reduced scheme. Is it always regular? Are there any references for this situation?

Answer 1

Even if $X_{k'}$ is reduced, it is not necessarily regular: let $p>2$ denote the caracteristic, take $t$ in $k$ which is not a $p$th power, and consider the plane curve $X$ defined by $y^2=x^p-t$. It is geometrically reduced, and over $\overline{k}$ it has a unique singular point, namely $(t^{1/p},0)$. This corresponds to the maximal ideal $(x^p-t,y)$ in $k[x,y]$; in the affine ring of $X$ this ideal is obviously generated by $y$, so the point is regular.