Mathematical Research Letters

Volume 21 (2014)

Number 2

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

Pages: 281 – 288

DOI: http://dx.doi.org/10.4310/MRL.2014.v21.n2.a6

Authors

Christopher Davis (Department of Mathematics, University of California at Irvine; and Department of Mathematical Sciences, University of Copenhagen, Denmark)

David Zureick-Brown (Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia, U.S.A.)

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 these 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.

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