On the fundamental class of an essentially smooth scheme-map

Let f : X Z be a separated, essentially finite-type, flat map of noetherian schemes and : X X Z X the diagonal map. The fundamental class C f (globalizing residues) is a map from the relative Hochschild functor L * * f * to the relative dualizing functor f ! . We show a compatibility between C f and the derived tensor product. The main result is that, in a suitable sense, C f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f ! to relative d-forms.