  Let $R$ be a ring and Ch($R$) the category of chain complexes of $R$-modules.
We put an abelian model structure on Ch($R$) whose homotopy category is
equivalent to $K(Proj)$, the homotopy category of all complexes of projectives.
However, the cofibrant objects are not complexes of projectives, but rather all
complexes of flat modules. The trivial objects are what Positselski calls
contraacyclic complexes and so the homotopy category coincides with his
contraderived category. We in fact construct this model on the category of
chain complexes of quasi-coherent sheaves on any scheme $X$ admitting a flat
generator. In this case the homotopy category recovers what Murfet calls the
mock homotopy category of projectives. In the same way we construct a model for
the (mock) projective stable derived category, and we use model category
methods to recover the recollement of Murfet. Finally, we consider
generalizations by replacing the flat cotorsion pair with other complete
hereditary cotorsion pairs in Grothendieck categories.
