Let F be a polynomial mapping from C to C with n > q. We study the De Rham cohomology of its fibres and its relative cohomology groups, by introducing a special fibre F−1(∞) ”at infinity” and its cohomology. Let us fix a weighted homogeneous degree on C[x1, ..., xn] with strictly positive weights. The fibre at infinity is the zero set of the leading terms of the coordinate functions of F . We i...

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 pro...