Integral Monsky-Washnitzer cohomology and the overconvergent de Rham-Witt complex

TL;DR

This paper proves the well-definedness of integral Monsky-Washnitzer cohomology for nonsingular affine varieties over perfect fields of characteristic p and establishes an isomorphism with overconvergent de Rham-Witt cohomology in certain degrees.

Contribution

It rigorously confirms the integral cohomology groups are well-defined and extends known isomorphism results to all dimensions for nonsingular affine varieties.

Findings

Integral Monsky-Washnitzer cohomology groups are well-defined.

Established isomorphism with overconvergent de Rham-Witt cohomology in degrees small relative to p.

Extended previous results to all dimensions of nonsingular affine varieties.

Abstract

In their paper which introduced Monsky-Washnitzer cohomology, Monsky and Washnitzer described conditions under which the definition can be adapted to give integral cohomology groups. It seems to be well-known among experts that their construction always gives well-defined integral cohomology groups, but this fact also does not appear to be explicitly written down anywhere. In this paper, we prove that the integral Monsky-Washnitzer cohomology groups are well-defined, for any nonsingular affine variety over a perfect field of characteristic p. We then compare these cohomology groups with overconvergent de Rham-Witt cohomology. It was shown earlier that if the affine variety has small dimension relative to the characteristic of the ground field, then the cohomology groups are isomorphic. We extend this result to show that for any nonsingular affine variety, regardless of dimension, we have an isomorphism between integral Monsky-Washnitzer cohomology and overconvergent de Rham-Witt cohomology in degrees which are small relative to the characteristic.