  T. Dupuy, E. Katz, J. Rabinoff, D. Zureick-Brown introduced the module of
total $p$-differentials for a ring over $Z/p^2Z$. We study the same
construction for a ring over $Z_{(p)}$ and prove a regularity criterion. For a
local ring, the tensor product with the residue field is constructed in a
different way by O. Gabber, L. Ramero. In another article arXiv:2006.00448, we
use the sheaf of FW-differentials to define the cotangent bundle and the
micro-support of an etale sheaf.
