Hodge-Witt cohomology and Witt-rational singularities

TL;DR

This paper establishes vanishing results for Witt vector cohomology in positive characteristic, introduces Witt-rational singularities as a broader class than rational singularities, and shows torsion cohomology as a birational invariant.

Contribution

It proves vanishing of higher direct images of Witt sheaves under purely inseparable alterations and defines Witt-rational singularities, expanding the class of singularities with favorable cohomological properties.

Findings

Higher direct images of Witt sheaves vanish modulo torsion.

Finite quotients have Witt-rational singularities.

Torsion part of Witt cohomology is a birational invariant.

Abstract

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.