For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module. (via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$).
Assume $A \subseteq B \subseteq C$ are commutative rings, such that $C$ is separable over $A$. Is $C$ separable over $B$?
Can anyone please help me with the proof?
Adjamagbo claims that this implies that $C$ is separable over $B$, but I am not able to prove this.
Adjamagbo's claim appears on page 92 (13) in: "On separable algebras over a UFD and the Jacobian conjecture in any characteristic", in Automorphisms of affine spaces, A. van den Essen (ed.), Kluwer Academic Publishers, 1995.