I asked [this question][1] a couple of weeks ago. 

The question arose while looking for a criterion of separability for extensions of fraction fields $K(A)\to K(B)$ induced by a faithfully flat morphism $A\to B$ between algebras over a field of positive characteristic which are domains with $A$ Noetherian, and such that the induced morphism $A/m\to B/mB$ is bijective for every maximal ideal $m$ of $A$.

G.Leuschke gave me [this reference][2], whose results (concretely, the Theorem 1.8) allows one to conclude that the separability is guaranteed when $B$ is also Noetherian by the argument that follows:

 1. The next theorem is an exercise from Bourbaki's *Algebra II*: 

 **Theorem 1 (Bourbaki Alg II, V.15. Ex.11):** *Let $K$ be a field of characteristic $p>0$ and let $C$ be a $K$-algebra. Then $C$ is separable if and only if for every family of elements* $\{k_{i} \}\subset K$ *linearly free over* $K^{p}$ *and every family* $\{c_{i}\}\subset C$ (*with* $c_{i}=0$ *except for a finite number of subindices) the equality*
$$\sum_{i} k_{i}c_{i}^{p}=0$$
*implies that* $c_{i}=0$ *(for every $i$).*


 2. Now, it is not hard to see that this exercise gives the following: 

 **Theorem 2:** *Let* $A\hookrightarrow B$ *be a flat extension of algebras over a field of characteristic* $p>0$ *which are domains, and denote by* $A^{1/p}$ *(resp, by* $B^{1/p}$) *the algebra* $A$ *(resp.* $B$) *seen as an* $A$ *(resp $B$) Module via the Frobenius map. Then the field extension* $K(A)\to K(B)$ *is separable if and only every finite set* $a_{1},\dots, a_{n}\in A$ *of free elements in* $A^{1/p}$ *is free in* $B^{1/p}$. *This happens if and only if the canonical map*  
$$B \otimes_{A} \left\langle a_{1},\dots ,a_{n} \right\rangle_{B^{1/p}}\to \left\langle a_{1},\dots ,a_{n} \right\rangle_{B^{1/p}}$$
  *is injective.*


 3. Here comes Frankild et al's paper: when $A, B$ are Noetherian, $A\to B$ is faithfully flat and the induced map $A/m\to B/mB$ is bijective for every maximal ideal $m$ of $A$, an easy application of the local-global principle together with Theorem 1.8 in the paper guarantees the separability of $K(A)\to K(B)$.


I thought the criterion was extendible to the case in which $B$ is not Noetherian, but there were mistakes in my argument. Does anybody have an idea on how to prove (or refute) the corresponding affirmation in such a case? (Frankild et al's paper can give you some hints, but I don't write them down in order to avoid bias).

P.S: The argument can be extended easily, I think, to the case in which the localization of $B$ at every maximal ideal $\eta$ is $\eta$-adically separated.


  [1]: http://mathoverflow.net/questions/62206/criteria-for-preservation-of-a-module-structure-under-extension-of-scalars
  [2]: http://www.math.unl.edu/~rwiegand1/ExtVanish/paper.pdf