The flat model structure on complexes of sheaves

Let $\mathbf{Ch}(\mathcal{O})$ be the category of chain complexes of $\mathcal{O}$-modules on a topological space $T$ (where $\mathcal{O}$ is a sheaf of rings on $T$). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on $\mathbf{Ch}(\mathcal{O})$. As a corollary, we have a general framework for doing homological algebra in the category $\mathbf{Sh}(\mathcal{O})$ of $\mathcal{O}$-modules. I.e., we have a natural way to define the functors $\operatorname{Ext}$ and $\operatorname{Tor}$ in $\mathbf{Sh}(\mathcal{O})$.