  We prove the vanishing modulo torsion of the higher direct images of the
sheaf of Witt vectors (and the Witt canonical sheaf) for a purely inseparable
projective alteration between normal finite quotients over a perfect field. For
this, we show that the relative Hodge-Witt cohomology admits an action of
correspondences. As an application we define Witt-rational singularities which
form a broader class than rational singularities. In particular, finite
quotients have Witt-rational singularities. In addition, we prove that the
torsion part of the Witt vector cohomology of a smooth, proper scheme is a
birational invariant.
