  Given a scheme over a complete discrete valuation ring of mixed
characteristic with perfect residue field, the Greenberg transform produces a
new scheme over the residue field thicker than the special fiber. In this
paper, we will generalize this transform to the case of imperfect residue
field. We will then construct a certain kind of cycle class map defined on this
generalized Greenberg transform applied to the N\'eron model of a semi-abelian
variety, which takes values in the relatively perfect nearby cycle functor
defined by Kato and the second author.
