  Let $\mathbb{X}$ be a semiseparated Noetherian scheme with a dualizing
complex $D$. We lift some well-known triangulated equivalences associated with
Grothendieck duality to Quillen equivalences of model categories. In the
process we are able to show that the Gorenstein flat model structure, on the
category of quasi-coherent sheaves on $\mathbb{X}$, is Quillen equivalent to
the Gorenstein injective model structure. Also noteworthy is that we extend the
recollement of Krause to hold without the Noetherian condition. Using a set of
flat generators, it holds for any quasi-compact semiseparated scheme
$\mathbb{X}$. With this we also show that the Gorenstein injective
quasi-coherent sheaves are the fibrant objects of a cofibrantly generated
abelian model structure for any semiseparated Noetherian scheme $\mathbb{X}$.
Finally, we consider both the injective and (mock) projective approach to Tate
cohomology of quasi-coherent sheaves. They agree whenever $\mathbb{X}$ is a
semiseparated Gorenstein scheme of finite Krull dimension.

Algebraic Topology;Algebraic Geometry;Rings and Algebras