Let $\overline K$ be an algebraic closure of a $p$-adic field
$K$. We construct a separated noetherian regular scheme $X$
(nonalgebraic) equipped with an action of
$G_K=\mathrm{Gal}(\overline{K}/K)$. We have $H^0(X, O_X) = Q_p$ and
$H_1(X, O_X) = 0$...