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 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.

Minor changes. To appear in Advances in Math. 93 pages

References in corpus (4)

• Deformation quantization modules
• The relative Riemann-Roch theorem from Hochschild homology
• Platitude géométrique et classes fondamentales relatives pondérées II
• Platitude géométrique et classes fondamentales relatives pondérées I

Cited by in corpus (5)

• Vanishing of the higher direct images of the structure sheaf
• Relation between two twisted inverse image pseudofunctors in duality theory
• On the fundamental class of an essentially smooth scheme-map
• The relation between Grothendieck duality and Hochschild homology
• Grothendieck Duality theories -- abstract and concrete, I: pseudo-coherent finite maps