The flat model structure on $\mathbf{Ch}(R)$

Given a cotorsion pair $(\mathcal{A},\mathcal{B})$ in an abelian category $\mathcal{C}$ with enough $\mathcal{A}$ objects and enough $\mathcal{B}$ objects, we define two cotorsion pairs in the category $\mathbf{Ch(\mathcal{C})}$ of unbounded chain complexes. We see that these two cotorsion pairs are related in a nice way when $(\mathcal{A},\mathcal{B})$ is hereditary. We then show that both of these induced cotorsion pairs are complete when $(\mathcal{A},\mathcal{B})$ is the ``flat'' cotorsion pair of $R$-modules. This proves the flat cover conjecture for (possibly unbounded) chain complexes and also gives us a new ``flat'' model category structure on $\mathbf{Ch}(R)$. In the last section we use the theory of model categories to show that we can define $\operatorname{Ext}^n_R(M,N)$using a flat resolution of $M$ and a cotorsion coresolution of $N$.