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Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$.

My QUESTIONS concerns the analysis of the structure of $\mathcal{O}_{X,x}$:

When $O_{X,x}^{\wedge}$ has the shape $ k[[x_1,x_2,..., x_m]]/I$?

When - more specifically - it has the shape $ k[[x_1,x_2,..., x_{n+1}]]/(f)$ for $f \in k[[x_1,x_2,..., x_{n+1}]]$ non zero divisor?

Are there any reconmendable reference with thread the structure of such local complete rings rigoriously?

My CONSIDERATIONS /starting point:

The main tool to treat this problem is of course the Cohen strucure theorem. It provides especially that $$O_{X,x}^{\wedge} \cong \Lambda [[x_1, \ldots , x_ n]]/I$$ where $\Lambda$ is the "mysterious" Cohen ring.

Which criterions are neccessary & sufficient to settle that here $\Lambda=k$ (references?)?

Another point is under which conditions we obtain a more "concretely" form $$ O_{X,x}^{\wedge}= k[[x_1,x_2,..., x_{n+1}]]/(f)$$

?

This question arises from following former thread of mine: Intuition behind RDP (Rational Double Points)

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    $\begingroup$ The residue field of $\mathcal{O}_{X, x}$ is $k(x)$, while the residue field of $k[[x_1, \ldots, x_n]]/I$ is $k$. So a necessary (and in fact sufficient) condition is that $x\in X(k)$. $\endgroup$
    – Piotr Achinger
    Commented May 11, 2019 at 22:05
  • $\begingroup$ @PiotrAchinger: Do you have a recommandable reference / sketch of the proof for the "sufficient" part? $\endgroup$
    – user267839
    Commented May 11, 2019 at 22:09
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    $\begingroup$ Pick generators $t_1, \ldots, t_n \in \mathfrak{m}_x$. The induced homomorphism $k[[T_1, \ldots, T_n]] \to \mathfrak{O}^\wedge_{X, x}$ sending $T_i$ to $t_i$ is then easily seen to be surjective. $\endgroup$
    – Piotr Achinger
    Commented May 11, 2019 at 22:19
  • $\begingroup$ @PiotrAchinger: so $O_{X,x}^{\wedge}= k(x)[[x_1,x_2,..., x_{n+1}]]/I$ holds always when the maximal ideal is finitely generated? Are there any conditions on $X$ which garantee that the completions have moreover the shape $O_{X,x}^{\wedge}= k[[x_1,x_2,..., x_{n+1}]]/(f)$? $\endgroup$
    – user267839
    Commented May 11, 2019 at 22:27
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    $\begingroup$ The condition is $\ \dim \mathfrak{m}_x/\mathfrak{m}_x^2\leq n+1$. $\endgroup$
    – abx
    Commented May 12, 2019 at 4:28

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