  For a smooth map between noetherian schemes, Verdier relates the top relative
differentials of the map with the twisted inverse image functor `upper shriek'.
We show that the associated traces for smooth proper maps can be rendered
concrete by showing that the resulting theory of residues satisfy the residue
formulas (R1)--(R10) in Hartshorne's "Residues and Duality". We show that the
resulting abstract transitivity map relating the twisted image functors for the
composite of two smooth maps satisfies an explicit formula involving
differential forms. We also give explicit formulas for traces of differential
forms for finite flat maps (arising from Verdier's isomorphism) between schemes
which are smooth over a common base, and use this to relate Verdier's
isomorphism to Kunz and Waldi's regular differentials. These results also give
concrete realisations of traces and residues for Lipman's fundamental class map
via the results of Lipman and Neeman relating the fundamental class to
Verdier's isomorphism.
