Hi,

I made this question 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, 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:

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 $i$).*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.*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.