Frobenius splitting and ordinarity

We examine the relationship between the notion of Frobenius splitting and ordinarity for varieties. We show that the de Rham-Witt cohomology groups Hⁱ(X,W(OX)) of a smooth projective Frobenius split variety are finitely generated over W(k). We provide counterexamples to a conjecture of Mehta that Frobenius split varieties are ordinary or even Hodge-Witt.