On the fundamental class of an essentially smooth scheme-map

TL;DR

This paper generalizes Verdier's classical isomorphism for smooth schemes by introducing a fundamental class for essentially smooth scheme-maps, establishing compatibility with derived tensor products and linking duality to relative differential forms.

Contribution

It introduces a fundamental class for essentially smooth scheme-maps and proves its compatibility with derived tensor products, extending classical duality results.

Findings

C_f generalizes Verdier's isomorphism for smooth maps

Compatibility between C_f and derived tensor product is established

C_f links duality functor f^! to relative differential forms

Abstract

Let f: X -> Z be a separated essentially-finite-type flat map of noetherian schemes, and \delta: X --> X \times_Z X the diagonal map. The fundamental class C_f (globalizing residues) is a map from the relative Hochschild functor L\delta^*\delta_* f^* to the relative dualizing functor f^! A compatibility between this C_f and derived tensor product is shown. The main result is that, in a suitable sense, C_f generalizes Verdier's classical isomorphism for smooth f with fibers of dimension d, an isomorphism that binds f^! to relative d-forms.