  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.
