Lemma 15.30.5. 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 $A'$.

## Comments (2)

Comment #3854 by Zhiyu Zhang on

Comment #3939 by Johan on