Let $A \to B$ be a finitely generated homomorphism between two commutative noetherian rings.
As far as I understand, in various generalizations of this situation, such a map is called smooth if $B$ is a perfect object in $D(B\otimes_A B)$. (See for example Definition 2.2 of http://arxiv.org/pdf/1006.4721v2.pdf).
Is this true for commutative noetherian rings?
That is, is a finite type map $A\to B$ is smooth if and only if $B$ has a finite projective dimension over $B\otimes_A B$? Note that one direction (that smoothness implies finite projective dimension) is true because of EGA IV Proposition 17.12.4 which says that the kernel of $B\otimes_A B\to B$ is locally generated by a regular sequence. But does the converse holds?
Edit: the answer below shows that surjections, for example, are a counterexample. What if in addition we assume flatness of the map $A\to B$, so that $B\otimes^L_A B \cong B\otimes_A B$?