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

Abstract:

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 $N$.