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

Abstract

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 cf:HX→f!HY, the fundamental class of f, where HZ 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 Ωfn[n]=H−n(HX)[n]⟶∼f!OS. We focus mainly on transitivity of c vis-à-vis compositions X→Y→Z, and on the compatibility of c with flat base change. These properties imply that c orients the flat maps in the bivariant theory of part I [1], 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 dX:HX→RHom(HX,f!OS), an isomorphism if f is essentially smooth. These results apply in particular to flat essentially-finite-type maps of noetherian rings.