Let $A \subseteq B$ be noetherian integral domains, $A$ regular (=every localization at maximal ideal is a regular local ring) and $B$ is a smooth $A$-algebra. For the definition of a smooth algebra, please see the first page of

Robert A Morris, Stuart Sui-Sheng Wang,

A Jacobian criterion for smoothness

Journal of Algebra, Volume 69, Issue 2, April 1981, Pages 483–486

doi:10.1016/0021-8693(81)90217-9

which says that $B$ is a smooth $A$-algebra if the following two conditions are satisfied:

(1) For each $A$-algebra $C$, and each ideal $J$ in $C$ with $J^2=0$, the canonical homomorphism $Hom_{A-alg}(B,C) \to Hom_{A-alg}(B,C/J)$ is surjective.

(2) $B$ is finitely presented as an $A$-algebra.

My question: Is it true that $B$ must be regular too? If not, what additional conditions should we assume in order that $B$ will be regular?

I really apologize if this question is trivial; it's just that only recently I have started to study regular rings/smooth extensions.