Pure exact structures and the pure derived category of a scheme

Abstract

Let $\mathcal C$ 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 $\mathbf{C}(\mathcal C)$ of unbounded chain complexes in $\mathcal C$. 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.