Is there any version of Kawamata-Viehweg vanishing theorem that holds for excellent surfaces? I will be very glad if there is one such, but if not then is it at least true for a surface over a non algebraically closed field of char $p>0$?

More particularly I have the following situation:

$X$ is a variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Let $x$ be a codimension $2$ point of $X$ and $S=\text{Spec }\mathcal{O}_{X,\ x}$. Then $S$ is a surface over the residue field $k(x)$ ( which is not algebraically closed ). I need to use Kawamata-Viehweg vanishing theorem ( at least some relative version of it ) on this $S$. I know that one version of this vanishing theorem holds for surfaces over an algebraically closed field of char $p>0$ by Hiromu Tanaka ( The $X$-methods for klt surfaces in positive characteristic ).

whatever)? If you are looking for references, I know that some Grauert-Riemenschneider-type statements are even true for excellent two dimensional rings (see for example the work of Lipman on resolution of 2 dimensional singularities and also his work on rational singularities in any characteristic). Can you be more precise in terms of what exactly you need? – Karl Schwede Apr 20 '13 at 5:18