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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

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  • $\begingroup$ It is impossible to answer without clarification of you are asking for. Do you actually mean "without affecting $B$"? Do you want to keep $n$ and $m$ to be the same? $\endgroup$
    – js21
    May 22, 2018 at 14:55
  • $\begingroup$ @js21 Thanks. I added a clarification $\endgroup$
    – user124171
    May 22, 2018 at 14:58
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    $\begingroup$ A trivial observation which does not answer your question, but could be sufficient for some applications you may have in mind. (1) If $A$ is a complete intersection, i.e. $A=k[t_1, \ldots, t_n]/(f_1, \ldots, f_m)$ with ${\rm dim}(A) = n-m$, then any lifts of the $f_i$ define a smooth lifting $B$. So you can e.g. take Teichmuller lifts of the coefficients, and you're done. (2) Every smooth algebra $A/k$ is locally a complete intersection. $\endgroup$ May 22, 2018 at 16:27

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