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



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> Robert A Morris, Stuart Sui-Sheng Wang, *A Jacobian criterion for smoothness*  
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>  Journal of Algebra, Volume 69, Issue 2, April 1981, Pages 483–486  
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> [doi:10.1016/0021-8693(81)90217-9](http://dx.doi.org/10.1016/0021-8693(81)90217-9)
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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.



When a smooth algebra is regular?
Yes. More generally, let $A \to B$ be a homomorphism of noetherian rings satisfying the condition (1) of your question (that is, B is formally smooth in the sense of [EGA IV.17.1.1]). Let $\mathfrak q$ be a prime ideal of $B$ and $\mathfrak p$ its contraction in $A$, $k(\mathfrak p)$, $k(\mathfrak p)$ the residue fields. I will use André-Quillen homology modules and references are to André's book. We have


$$H^1(A\_{k(\mathfrak p)},B\_{k(\mathfrak q)},k(\mathfrak q))=H^1(A,B,k(\mathfrak q))=0$$


by 5.27 and 16.17. By 5.1, 7.4, 3.21, 6.26 we have an exact sequence


$$ H^1(A\_{k(\mathfrak p)},B\_{k(\mathfrak q)},k(\mathfrak q)) \to H^2(B\_{k(\mathfrak q)},k(\mathfrak q),k(\mathfrak q))\to H^2(A\_{k(\mathfrak p)},k(\mathfrak q),k(\mathfrak q))=Hom\_{k(\mathfrak p)}(H\_2(A\_{k(\mathfrak p)},k(\mathfrak p),k(\mathfrak p)),k(\mathfrak q))=0$$


Therefore $H^2(B\_{k(\mathfrak q)},k(\mathfrak q),k(\mathfrak q))=0$ and again by 6.26 $B$ is regular.


