# Purely inseparable morphisms and factorizations of a morphism of finite type.

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.

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Can you explain precisely what you mean by "Can this construction, modulo localizations be done on the level of affine schemes?" ? What would that entail. –  Alex Youcis Mar 10 at 11:10
What I mean is: Can we factor f into morphisms (localizing if neccesary) $g:Spec A \rightarrow Spec C$, $h Spec C \rightarrow Spec B$ such that $K(B) \rightarrow K(C)$ is separable and $K(C) \rightarrow K(A)$ is purely inseparable? –  user101036 Mar 10 at 22:15
I forgot to add @AlexYoucis –  user101036 Mar 10 at 22:16
What I would want to do is simply use base change by $K(C)$ on Spec B, but I am not sure if it works nicely or not (I guess not). –  user101036 Mar 10 at 22:17