Lemma 15.31.6. Let $A' \to B'$ be a ring map. Let $I \subset A'$ be an ideal. Set $A = A'/I$ and $B = B'/IB'$. Let $f'_1, \ldots , f'_ r \in B'$. Assume
$A' \to B'$ is flat and of finite presentation (for example smooth),
$I$ is locally nilpotent,
the images $f_1, \ldots , f_ r \in B$ form a quasi-regular sequence,
$B/(f_1, \ldots , f_ r)$ is smooth over $A$.
Then $B'/(f'_1, \ldots , f'_ r)$ is smooth over $A'$.
Comment #3854 by Zhiyu Zhang on
Comment #3939 by Johan on
Comment #4919 by Matthieu Romagny on
Comment #4920 by Johan on