The relation between Grothendieck duality and Hochschild homology

TL;DR

This paper surveys the relationship between Grothendieck duality and Hochschild homology, focusing on formulas for flat maps and extending duality functors to unbounded derived categories, with new results removing boundedness restrictions.

Contribution

It provides new methods to extend duality formulas to unbounded derived categories for flat maps, improving understanding of Hochschild homology computations.

Findings

Formulas for Hochschild homology and cohomology for flat maps

Extension of duality functor $f^!$ to unbounded derived categories

Removal of boundedness restrictions in duality formulas

Abstract

The article primarily surveys work that followed from the formulas discovered by Avramov and Iyengar in 2008, which permit one to compute certain Hochschild homology and cohomology modules as expressions involving dualizing complexes. One aspect of the formulas is that (so far) they are only known for maps of finite Tor-dimension---we specialize even further, for this survey we give the formulas only for flat maps. Recall that, for general maps of schemes $f:X\to Y$, the duality functor $f^!$ has traditionally been defined and studied only on the bounded-below derived category. Alonso, Jeremias and Lipman observed that, as long as we restrict to morphisms $f$ of finite Tor-dimension, there is an extension of $f^!$ to the unbounded derived category. But this extension was so poorly understood that the paper [19], revisiting the formulas of Avramov and Iyengar, was unable to prove the obvious extensions of the formulas to the unbounded setting. While most of the paper is a survey, sections 4 and 5 are new. They show how to use the results of [23] to remove the unnatural boundedness hypotheses from the formulas in [19]. This is the application that originally motivated [23].