I need a simple commutative algebra lemma for a paper, but can't find a reference. Maybe I don't know the right keywords. Here's the setup. $F:K$ is a field extension, $A$ is an algebra over $K$, and $M$ an $A$-module. If $M \otimes\_K F$ is a free $A \otimes\_K F$ module, must $M$ have been a free $A$-module?


I can do the particular case I need: $K = \mathbb{F}\_2$, $F$ a finite extension, $A = 1 \oplus N$ for $N$ nilpotent, $M \otimes\_K F$ free of rank one. But I suspect that something more general holds, and it would be nice to quote rather than reprove.


Thanks!



If M is not a free A-module, can tensoring with a bigger field make it free?
Yes, there are such examples even with invertible $M$ and smooth $K$-algebras of dimension 1, with $K$ any field that is not algebraically closed. 


Choose such a $K$, so we may and do also choose a nontrivial primitive finite extension $F$ of $K$. Consider the projective line over $K$ and remove a closed point $\xi$ such that $F = K(\xi)$. This is an affine open $U = {\rm{Spec}}(A)$ for a Dedekind $A$, and let $M$ be the maximal ideal of $A$ corresponding to an $K$-point. This is invertible as an $A$-module (as for any Dedekind domain) but cannot be principal. Indeed, if $a \in A$ were a generator of $M$ then its divisor on $\mathbf{P}^1\_K$ has restriction to $U$ that is a single point of degree 1 yet the condition ${\rm{deg}}\_K({\rm{div}}(a)) = 0$ forces the "negative" part of the divisor to have degree $-1$ over $K$, contradicting that this negative part is supported at the closed point $\xi$ with $K$-degree larger than 1.


Clearly $A \otimes\_K F$ is the coordinate ring of the complement in $\mathbf{P}^1\_F$ of ${\rm{Spec}}(F \otimes\_K F)$, which contains an $F$-point, so $A \otimes\_K F$ is the coordinate ring of a non-empty affine open in the open complement $\mathbf{A}^1\_F = {\rm{Spec}}(F[t])$ of an $F$-point in $\mathbf{P}^1\_F$. Thus, $A \otimes\_K F$ is the localization of the PID $F[t]$ at some nonzero element, so $A \otimes\_K F$ is a PID and hence its nonzero ideal $M \otimes\_K F$ (corresponding to a single $F$-point) is principal. 


