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 $A[T]$.

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 $A[T]$.

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.

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 $A[T]$.