  Let $X$ be a smooth projective algebraic variety over $Z/p$, which has a flat
lift to a scheme $X'$ over $Z/p^2$. If the absolute Frobenius morphism $F$ on
$X$ lifts to a morphism on $X'$, then an old trick by Mazur shows that
push-down of the de Rham complex under $F$ decomposes. We show that the
quasi-isomorphism in question is split. This is then applied to toric varieties
(where a glueing argument gives lifting of Frobenius to $Z/p^2$) and we derive
natural characteristic $p$ proofs of Bott vanishing and degeneration of the
Danilov spectral sequence. For flag varieties we obtain generalizations of a
result of Paranjape and Srinivas about non-lifting of Frobenius to the Witt
vectors.
