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 $B\otimes^L\_A B \cong B\otimes\_A B$?



Homological characterization of smooth maps
Yes to the second question. More generally, if $f:A \to B$ is a flat homomorphism of noetherian commutative rings such that the flat dimension of $B$ over $B\otimes\_A B$ is finite, then $f$ is regular. See Rodicio: Smooth algebras and vanishing of Hochschild homology, Comm. math. Helv. 65 (1990) 474-477. When $f$ is of finite type then you get your answer, since in this case regularity and smoothness are the same thing. 


