  This paper is about sheaf cohomology for varieties (schemes) in
characteristic $p>0$. We assume the presence of a Frobenius splitting. (See
V.B. Mehta and A. Ramanathan, Frobenius splitting and cohomology vanishing for
Schubert varieties, Annals of Math. 122 (1985), 27--40). The main result is
that a non-zero higher direct image under a proper map of the ideal sheaf of a
compatibly Frobenius split subvariety can not have a support whose inverse
image is contained in that subvariety. Earlier vanishing theorems for Frobenius
split varieties were based on direct limits and Serre's vanishing theorem, but
our theorem is based on inverse limits and Grothendieck's theorem on formal
functions. The result implies a Grauert--Riemenschneider type theorem.
