Pure exact structures and the pure derived category of a scheme

Let C be closed symmetric monoidal Grothendieck category. We define the pure derived category with respect to the monoidal structure via a relative injective model category structure on the category C(C) of unbounded chain complexes in C. We use lambda-Purity techniques to get this. As application we define the stalkwise pure derived category of the category of quasi-coherent sheaves on a quasi-separated scheme. We also give a different approach by using the category of flat quasi-coherent sheaves.