  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 $Z$.

Algebraic Geometry