Let us suppose that $f: Spec A \rightarrow Spec B$ is a finite surjective morphism of integral algebraic varieties over a field k (these hypothesis on the field may be unneccessary). We then know that the extension $K(B) \rightarrow K(A)$ can be factored as first going through the inseparable part, and then to the separable part of the extension. My question is the following:
Can this construction, modulo localizations be done on the level of affine schemes? I seem to have been able to do this, but am not sure.