The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Lemma 10.154.10. Let $(R, \mathfrak m)$ be a Noetherian complete local ring. Assume $R$ is regular.

  1. If $R$ contains either $\mathbf{F}_ p$ or $\mathbf{Q}$, then $R$ is isomorphic to a power series ring over its residue field.

  2. If $k$ is a field and $k \to R$ is a ring map inducing an isomorphism $k \to R/\mathfrak m$, then $R$ is isomorphic as a $k$-algebra to a power series ring over $k$.

Proof. In case (1), by the Cohen structure theorem (Theorem 10.154.8) there exists a coefficient ring which must be a field mapping isomorphically to the residue field. Thus it suffices to prove (2). In case (2) we pick $f_1, \ldots , f_ d \in \mathfrak m$ which map to a basis of $\mathfrak m/\mathfrak m^2$ and we consider the continuous $k$-algebra map $k[[x_1, \ldots , x_ d]] \to R$ sending $x_ i$ to $f_ i$. As both source and target are $(x_1, \ldots , x_ d)$-adically complete, this map is surjective by Lemma 10.95.1. On the other hand, it has to be injective because otherwise the dimension of $R$ would be $< d$ by Lemma 10.59.12. $\square$


Comments (2)

Comment #1853 by Matthieu Romagny on

Typo in (2) : isomrphism --> isomorphism

Comment #1892 by on

This already got fixed in commit 368a679d67605624062d35132d96d4bc5b32c73d which you can find here.


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