Relation between two twisted inverse image pseudofunctors in duality theory

TL;DR

This paper explores the relationship between two key pseudofunctors in Grothendieck duality theory, establishing a canonical map with favorable properties and applications to Hochschild (co)homology and affine schemes.

Contribution

It defines and analyzes a canonical map between the two duality pseudofunctors, revealing their connection and applications in algebraic geometry.

Findings

The canonical map behaves well with flat base change.

It becomes an isomorphism under 'compactly supported' derived functors.

Applications include reduction theorems for Hochschild (co)homology.

Abstract

Grothendieck duality theory assigns to essentially-finite-type maps f of noetherian schemes a pseudofunctor f^\times right-adjoint to Rf_*, and a pseudofunctor f^! agreeing with f^\times when f is proper, but equal to the usual inverse image f^* when f is etale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by "compactly supported" versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.