Assume $A$ and $B$ are commutative algebras with $1$.
There is a nice result of Wang, [Corollary 8](http://ac.els-cdn.com/0021869380902331/1-s2.0-0021869380902331-main.pdf?_tid=3e25ad1a-3e2d-11e5-b1dc-00000aab0f6b&acdnat=1439080154_a8c7a76b2c7c860d578180ec8a516365), which says the following: "Let $B = A[z] = A[Z]/(h(Z))$. Then $B$ is a separable algebra over $A$ if and only if $h'(z)$, the formal derivative of $h$ evaluated at $z$, is a unit of $B$". 


By definition, an $A$-algebra $B$ is separable over $A$ if $B$ is a projective $B \otimes\_A B$-module via $\mu: B\otimes\_A B \to B$, $b\_1 \otimes\_A b\_2 \mapsto b\_1b\_2$.
In other words, $B$ is separable over $A$ if $pd\_{B\otimes\_A B}(B)=0$.


Hence Corollary 8 says: Let $B = A[z] = A[Z]/(h(Z))$. Then $pd\_{B\otimes\_A B}(B)=0$ if and only if the ideal of $B$ generated by $h'(z)$ is $B$. 


**My question:** Can one generalize Corollary 8 to the following: Let $B = A[z] = A[Z]/(h(Z))$. Then $pd\_{B \otimes\_A B}(B) = n$ if and only if the ideal of $B$ generated by $h'(z),h''(z), \ldots, h^{(n+1)}$ is $B$.


(Remark: Maybe one should look at $n=1$ first).


Any comment will be appreciated.



Is it possible to generalize a result of Wang?
I don't see any reason to expect the higher derivatives of $h$ to be important. For a simple counterexample to your conjecture, consider $h(Z)=Z^2$ (for, say, $A$ a field of characteristic $\neq2$). Then $h''$ is a unit, but $B$ has infinite projective dimension over $B\otimes\_A B$. Another counterexample is $h(Z)=0$; then no number of iterated derivatives of $h$ can generate a nonzero ideal but the projective dimension of $B$ is $1$.


