Grothendieck Duality and Transitivity II: Traces and Residues via Verdier's isomorphism

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.