$A \subseteq B \subseteq C$ with $A \subseteq C$ separable and $pd_{B \otimes_A B}(B) = \infty$

Question

Assume $A \subseteq B \subseteq C$ are commutative rings such that $C$ is separable over $A$, namely $C$ is a projective $C \otimes_A C$-module.

Separability of $C$ over $A$ does not imply separability of $B$ over $A$. In other words, it can happen that $pd_{C \otimes_A C}(C)=0$ and $pd_{B \otimes_A B}(B) > 0$.

Can one find rings $A \subseteq B \subseteq C$ with $pd_{C \otimes_A C}(C)=0$ and $pd_{B \otimes_A B}(B)= \infty$? (I guess yes), or if $pd_{C \otimes_A C}(C)=0$, then necessarily $pd_{B \otimes_A B}(B)< \infty$?

(Notice that separability of $C$ over $A$ implies separability of $C$ over $B$).

Thank you very much.

Answer 1

Let $A$ be a field, let $C$ be a matrix algebra over $k$, and let $B$ be any subalgebra of $C$ isomorphic to $k[x]/(x^2)$. $B$ could be the subalgebra of $C$ generated by any non-zero matrix with zero square.