Bivariance, Grothendieck duality and Hochschild homology

TL;DR

This paper develops a new bivariant Hochschild (co)homology theory for schemes over a noetherian base using Grothendieck duality, establishing a foundation for future work on orientations and fundamental classes.

Contribution

It introduces a novel construction of bivariant Hochschild (co)homology theories via Grothendieck duality for schemes over a fixed noetherian scheme.

Findings

Defines a bivariant theory valued in symmetric graded modules

Connects Hochschild (co)homology to Ext groups over the base scheme

Provides a foundation for future orientation and fundamental class studies

Abstract

A procedure for constructing bivariant theories by means of Grothendieck duality is developed. This produces, in particular, a bivariant theory of Hochschild (co)homology on the category of schemes that are flat, separated and essentially of finite type over a fixed noetherian scheme S. The theory takes values in the category of symmetric graded modules over the graded-commutative ring \oplus_i H^i(S,O_S). In degree i, the cohomology and homology H^0(S,O_S)-modules thereby associated to such an x: X -> S, with Hochschild complex H_x, are Ext^i(H_x, H_x) and Ext^{-i}(H_x, x^!O_S). This lays the foundation for a sequel that will treat orientations in bivariant Hochschild theory through canonical relative fundamental class maps, unifying and generalizing previously known manifestations, via differential forms, of such maps.