  Recent important and powerful frameworks for the study of differential forms
by Huber-Joerder and Huber-Kebekus-Kelly based on Voevodsky's h-topology have
greatly simplified and unified many approaches. This article builds towards the
goal of putting Illusie's de Rham-Witt complex in the same framework by
exploring the h-sheafification of the rational de Rham-Witt differentials.
Assuming resolution of singularities in positive characteristic one recovers a
complete cohomological h-descent for all terms of the complex. We also provide
unconditional h-descent for the global sections and draw the expected
conclusions. The approach is to realize that a certain right Kan extension
introduced by Huber-Kebekus-Kelly takes the sheaf of rational de Rham-Witt
forms to a qfh-sheaf. As such, we state and prove many results about
qfh-sheaves which are of independent interest.
