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!