  Fix a noetherian scheme S. For any flat map f: X->Y of separated
essentially-finite-type perfect S-schemes we define a canonical
derived-category map c(f):\H(X)->f^!\H(Y), the fundamental class of f, where
\H(Z) is the (pre-)Hochschild complex of an S-scheme Z and f^! is the twisted
inverse image coming from Grothendieck duality theory. When Y=S and f is
essentially smooth of relative dimension n, this gives an isomorphism from n-th
degree relative differential forms [ =H^{-n}(\H(X)) ] to f^!O_S[-n]. The basic
results concern transitivity of c(-) vis-\`a-vis compositions X->Y->Z, and
compatibility of c(-) with flat base change. These properties imply that c(-)
orients the flat maps in the bivariant theory of part I, compatibly with
essentially \'etale base change. Furthermore, c(-) leads to a dual oriented
bivariant theory, whose homology is the classical Hochschild homology of flat
S-schemes. When Y=S, c(-) is used to define a duality map
\H(X)->RHom(\H(X),f^!O_S), an isomorphism if f is essentially smooth. These
results apply in particular to flat essentially finite type maps of noetherian
rings.
