On the fundamental class of an essentially smooth scheme-map

Abstract

Let f: X → Z be a separated, essentially finite-type, flat map of noetherian schemes and δ: X → X×ZX the diagonal map. The fundamental class Cf (globalizing residues) is a map from the relative Hochschild functor Lδ*δ*f* to the relative dualizing functor f!. We show a compatibility between Cf and the derived tensor product. The main result is that, in a suitable sense, Cf generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f! to relative d-forms. Show more