Separability of $A \subseteq C$ implies separability of $B \subseteq C$, where $A \subseteq B \subseteq C$

Question

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: $f(s_1 \otimes_R s_2)=s_1s_2$).

My question:

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.

Answer 1

There is a canonical epimorphism $C\otimes_AC\to C\otimes_BC$ and its kernel annihilates $C$. Since $C$ is a projective $C\otimes_AC$-module it follows that it is also a projective $C\otimes_BC$-module.