Projective dimension of a quotient ring

Question

Assume $A$ and $B$ are commutative algebras with $1$, $B = A[z] = A[Z]/(h(Z))$, $Z$ an indeterminate. The first comment in this question says that, if $A$ is noetherian, then $pd_{B\otimes_A B}(B) \in \{0,1,\infty\}$.

Please, two questions (when $A$ is noetherian):

(1) How one proves the claim in the comment?

(2) What additional conditions one can impose in order that $pd_{B\otimes_A B}(B) \neq \infty$? For example, if we know that $B$ is separable over $A$ then, by definition, $pd_{B\otimes_A B}(B)=0$. Another (less trivial) example is: If $B$ is smooth over $A$, then by Rodicio's Corollary 2 $fd_{B\otimes_A B}(B) < \infty$. Other ideas are welcome.

Answer 1

I will try to answer to both questions together but the second one only in a few very particular cases. I'm sorry for not having complete answers.

If $\phi :A \to B$ is flat and the rings $A$, $B$ and $B\otimes_AB$ noetherian, then $\phi$ is smooth (or regular) if and only if fd$_{B\otimes_AB}(B)<\infty$ and in this case fd$_{B\otimes_AB}(B)$ is the supremum of the transcendence degrees of the fibers of $\phi$ (that are domains since they are regular local rings in this case). You can see this result in "Majadas-Rodicio, Commutative algebras of finite Hochschild homological dimension", J. of Algebra (1992), propositions 2.3 and 2.5.

That result includes the case of a flat finite type homomorphism of noetherian rings $A \to B$. In that same paper is implicit the question of whether for a flat map (not necessarily of finite type) of noetherian rings $\phi :A \to B$, fd$_{B\otimes_AB}(B)<\infty$ is equivalent to regularity plus transcendence degrees of the fibers, and some very particular cases are solved in sections 3-5, but as far as I know, this question remains essentially open.