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Let $A$ be a commutative noetherian domain of characteristic zero, $T$ an indeterminate, $h \in A[T]$, $B= A[T]/(h)$ and assume $B$ is also a domain.

When $B$ is (formally) smooth over $A$?; namely, what should we additionally assume on $h, A, B, A \subseteq B$ in order to get a smooth $A \to B$?

Of course, a legitimate answer is: $A \subseteq B$ is flat and $fd_{B \otimes_A B}(B) < \infty$ (according to Corollary 2), but I expect a more specific answer involving $h$.

This question appears as a question in a comment here.

Edit: After reading the nice comment of Jason Starr, I wonder if in the more general case where $B=A[T]/(h_1,\ldots,h_n)$ the following is true: $A \to B$ is formally smooth iff either $h_1=\ldots=h_n=0$ or the ideal of $A[T]$ generated by $h_1,h_1',\ldots,h_n,h_n'$ is $A[T]$.

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    $\begingroup$ The extension is formally smooth if and only if either $h$ is identically zero or $h(T)$ and $h'(T)$ generate the unit ideal of $A[T]$. You should look up "Jacobian criterion". $\endgroup$
    – Jason Starr
    Commented Sep 6, 2015 at 14:47
  • $\begingroup$ Thank you very much! You can write it as an answer if you like. $\endgroup$
    – user237522
    Commented Sep 6, 2015 at 16:20
  • $\begingroup$ @JasonStarr, please just to be sure we are working with the same definitions: $B$ is a smooth $A$-algebra if the following condition is satisfied: 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. $\endgroup$
    – user237522
    Commented Sep 6, 2015 at 22:41
  • $\begingroup$ @JasonStarr, please, is the claim I have suggested in the "Edit" true? I guess after your above comment that should be clear; however, I am not able to either prove or disprove it. $\endgroup$
    – user237522
    Commented Sep 9, 2015 at 0:20
  • $\begingroup$ No, that claim is not true. You should read about the "Jacobian criterion". This is discussed in many basic algebraic geometry textbooks. $\endgroup$
    – Jason Starr
    Commented Sep 9, 2015 at 1:15

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