Grothendieck duality for non-proper morphisms

Abstract

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

The results in this paper are part of the authors PhD thesis