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Suppose that $A$ is a complete neotherian local ring and that we're given a surjective homomorphism

$f: A[[x_{1}, \ldots, x_{n}]] \rightarrow A[[t_{1}, \ldots, t_{m}]]$.

Can we always find a different system of coordinates $y_{i}$ for $A[[x_{1}, \ldots, x_{n}]]$ such that in these new coordinates we have $f(y_{i}) = t_{i}$ for $i \leq m$ and $f(y_{i}) = 0$ for $i > m$?

This is certainly true if $A$ is a field, in which case any lift of $t_{1}, \ldots, t_{m}$ can be completed to a regular sequence by looking at what happens to the tangent space, but I have no intuition outside this case.

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    $\begingroup$ Homomorphism of what? of $A$-algebras? $\endgroup$
    – YCor
    Aug 1, 2019 at 21:34
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    $\begingroup$ Maybe I'm too naive, but can't we just first find the coordinates over the residue field of $A$ and then pick any lifting of them in $A$? The liftings will still form a system of coordinates since their kähler differentiels generate $\Omega^1$, by Nakayama's lemma. $\endgroup$
    – Wille Liu
    Sep 5, 2019 at 10:29

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