On a Grauert-Riemenschneider vanishing theorem for Frobenius split varieties in characteristic p

It is known that the Grauert-Riemenschneider vanishing theorem is not valid in characteristic p ([1]). Here we show that it may be restored in the presence of a suitable Frobenius splitting. The proof uses interchanging two projective limits, one involving iterated Frobenius maps, cf. [2] and [4], the other coming from Grothendieck’s theorem on formal functions. That leads to the following general vanishing theorem which we then apply in the situation of the Grauert-Riemenschneider theorem.