The flat model structure on 𝐂𝐡(𝐑)

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