Let $A$ be a smooth algebra over $k$ a finite field.
Say $B$ is a $p$-adically complete smooth algebra over the Witt ring $W(k)$, lifting $A$.
Assume $B$ is of the form $W(k)\langle t_1,\ldots, t_n\rangle / (f_1,\ldots, f_m)$.
Can the $f_j$'s be arranged so that their coefficients are all $p$-adic units, without affecting smoothness of $B$?
Here is what I would like to stay invariant:
the mod $p$ fiber
smoothness of the $B$ obtained by modifying the $f_j$'s
I guess I can equivalently ask if, for any smooth $k$-algebra $A$, one can find a $B$ of the form up there, with all $f_j$'s with $p$-adic units as coefficients