Bivariance, Grothendieck duality and Hochschild homology, II: The fundamental class of a flat scheme-map
文摘
Fix a noetherian scheme S. For any flat map of separated essentially-finite-type perfect S-schemes we define a canonical derived-category map , the fundamental class of f, where is the (pre-)Hochschild complex of an S-scheme Z and is the twisted inverse image coming from Grothendieck duality theory. When and f is essentially smooth of relative dimension n, this gives an isomorphism . We focus mainly on transitivity of vis-脿-vis compositions , and on the compatibility of with flat base change. These properties imply that orients the flat maps in the bivariant theory of part I , compatibly with essentially 茅tale base change. Furthermore, leads to a dual oriented bivariant theory, whose homology is the classical Hochschild homology of flat S-schemes. When , is used to define a duality map , an isomorphism if f is essentially smooth. These results apply in particular to flat essentially-finite-type maps of noetherian rings.