Grothendieck duality for non-proper morphisms

TL;DR

This paper extends Grothendieck duality to certain non-proper morphisms by establishing an adjunction between derived pushforward and a localized version of the exceptional inverse image functor for morphisms that are proper over specified closed subsets.

Contribution

It generalizes the classical Grothendieck duality to finite type morphisms that are proper only over closed subsets, introducing a new adjunction between localized functors.

Findings

Established an adjunction between $Rf_*$ and $R ext{Gamma}_{Z'}f^!$ for non-proper morphisms

Extended duality theory to finite type morphisms with proper restrictions over closed subsets

Provided a framework for duality in more general geometric situations

Abstract

We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism and let $Z' \subseteq X$ and $Z \subseteq Y$ be closed subsets such that $f$ restricts to a proper morphism $f'\colon Z'\to Z$ of $f$. Then the functor $Rf_*$ is left adjoint to $R\Gamma_{Z'}f^!$ when considered as functors between complexes supported on $Z'$ or $Z$.