Remark 108.40.3. Non-excellent regular rings whose residue fields have a finite $p$-basis can be constructed even in the function field of $\mathbb {P}^2_ k$, over a characteristic $p$ field $k = \overline{k}$. See [$\mathsection 4.1$, DS18].

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).