Let (F(R), Cot(R)) be a cotorsion theory, Such that F(R) is the class of flat R-modules and Cot(R) the cotorsion modules. Why this is true that, For $ N\in Cot(R) $,
$ \text{Ext}_{F(R)}^i(M, N)\cong \text{Ext}_R^i(M, N) $.

Please detail the question! What do you mean by ${\rm Ext}^i_{F(R)}(M,N)$? Usually for constructing Ext-functors one needs an abelian category.
– George C. ModoiApr 30 '13 at 17:05

This means that we should take a flat resolution of M, Instead of a projective resolution of that M.
– MaxMay 5 '13 at 9:10

If one takes flat instead of projective resolutions then we are lead to another concepts. For example instead projective dimension we have weak dimension. To be more precise the weak dimension of a ring $R$ is the largest number $i$ such that ${\rm Tor}^i_R(M,N)=0$ (or infinite is such a number does't exist). Sure every projective resolution is also a flat resolution so the weak dimension is smaller (or equal) that the projective one, but they can differ.
– George C. ModoiMay 8 '13 at 14:36