On the Frobenius stable part of Witt vector cohomology

TL;DR

This paper extends trace formulas relating Witt vector cohomology to rational point counts for varieties over finite fields, and proves a vanishing theorem for certain étale cohomology groups in positive characteristic.

Contribution

It generalizes the trace formula to Witt vector cohomology of finite length and establishes a vanishing result for étale cohomology on affine Cohen-Macaulay varieties.

Findings

Trace formula for Witt vector cohomology of finite length

Vanishing of compactly supported étale cohomology for affine Cohen-Macaulay varieties

Extension of cohomological techniques to non-smooth varieties

Abstract

For a proper (not necessarily smooth) variety over a finite field with q elements, Berthelot-Bloch-Esnault proved a trace formula which computes the number of rational points modulo q in terms of the Witt vector cohomology. We show the analogous formula for Witt vector cohomology of finite length. In addition, we prove a vanishing result for the compactly supported \'etale cohomology of a constant p-torsion sheaf on an affine Cohen-Macaulay variety in positive characteristic p.