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.

  • $\begingroup$ Can't you project down the separability idempotent for $C/A$ onto $C \otimes_B C$ (using the projection $C \otimes_A C \to C \otimes_B C$) ? $\endgroup$ – darij grinberg Jun 5 '15 at 1:24
  • $\begingroup$ Thank you very much for the hint! Denote $f_{AC}: C \otimes_A C \to C$, and $f_{BC}: C \otimes_B C \to C$. Denote the separability idempotent for $C/A$ by $e_{AC}$, and denote its image in $C \otimes_B C$ by $\bar{e}$. How I show that $\bar{e}$ satisfies the two conditions of a separability idempotent? $\endgroup$ – user237522 Jun 5 '15 at 2:10
  • $\begingroup$ Namely, how to show that: (1) $f_{BC}(\bar{e})=1$. (2) $Kerf_{BC}\bar{e}=0$. $\endgroup$ – user237522 Jun 5 '15 at 2:18
  • $\begingroup$ I don't recall there being a kernel condition in the definition of a separability idempotent. There was a commutativity-like condition, though. $\endgroup$ – darij grinberg Jun 5 '15 at 7:16
  • $\begingroup$ I have found your commutativity condition in Lemma 2.1 in the following paper of Auslander and Buchsbaum "On ramification theory in noetherian rings" people.brandeis.edu/~buchsbau/miscpapers/024.pdf It is actually equivalent to my above condition, since the kernel is of the form $\{1\otimes c- c \otimes 1\}$. $\endgroup$ – user237522 Jun 7 '15 at 1:39

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.


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