Following question arised from following former thread of mine:
I think that I have nearly found the solution but I'm quite unsure if this works and how generally. Therefore I would be glad if someboby could look through the following argument and examine if it works:
The problem is the following:
Assume $B$ a ring with an ideal $I$ such that $I^2=0$. Denote $B_0:=B/I$. Let $g_1, ..., g_r \in B_0[T_1,..., T_n]$ are polynomials and denote by $f_i $ arbitrary lift of $g_i$ in $B_0[T_1,..., T_n]$.
Assume that $g: B_0 \to B_0[T_1,...,T_n]/(g_1,...,g_r)$ is smooth (especially flat).
The question is if the induced morphism
$f: B \to B[T_1,...,T_n]/(f_1,...,f_r)$ induced by lifts $f_i$ of $g_i$ from above stays also smooth (especially flat)?
My idea was to use Jacobian smoothness condition: the (4) from https://en.wikipedia.org/wiki/Smooth_morphism#Equivalent_definitions
Since $g$ smooth $B_0[T_1,...,T_n]/(g_1,...,g_r)$ is generated by $r \times r$-minors of matrix $(\frac{\partial g_i}{\partial T_j})$.
So the question is if always $B[T_1,...,T_n]/(f_1,...,f_r)$ is generated by $r \times r$-minors of matrix $(\frac{\partial f_i}{\partial T_j})$.
This would of course mean that all such lifts preserves smoothness. I'm not sure if the story is here so simple.
For example I don't see the point where the assumption $I^2$ play role.