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

Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves