Paper detail

Bivariance, Grothendieck duality and Hochschild homology, II: the fundamental class of a flat scheme-map

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-à-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 étale 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.