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*}

Theorem 10.154.8 (Cohen structure theorem). Let $(R, \mathfrak m)$ be a complete local ring.

  1. $R$ has a coefficient ring (see Definition 10.154.4),

  2. if $\mathfrak m$ is a finitely generated ideal, then $R$ is isomorphic to a quotient

    \[ \Lambda [[x_1, \ldots , x_ n]]/I \]

    where $\Lambda $ is either a field or a Cohen ring.

Proof. Let us prove a coefficient ring exists. First we prove this in case the characteristic of the residue field $\kappa $ is zero. Namely, in this case we will prove by induction on $n > 0$ that there exists a section

\[ \varphi _ n : \kappa \longrightarrow R/\mathfrak m^ n \]

to the canonical map $R/\mathfrak m^ n \to \kappa = R/\mathfrak m$. This is trivial for $n = 1$. If $n > 1$, let $\varphi _{n - 1}$ be given. The field extension $\mathbf{Q} \subset \kappa $ is formally smooth by Proposition 10.152.9. Hence we can find the dotted arrow in the following diagram

\[ \xymatrix{ R/\mathfrak m^{n - 1} & R/\mathfrak m^ n \ar[l] \\ \kappa \ar[u]^{\varphi _{n - 1}} \ar@{..>}[ru] & \mathbf{Q} \ar[l] \ar[u] } \]

This proves the induction step. Putting these maps together

\[ \mathop{\mathrm{lim}}\nolimits _ n\ \varphi _ n : \kappa \longrightarrow R = \mathop{\mathrm{lim}}\nolimits _ n\ R/\mathfrak m^ n \]

gives a map whose image is the desired coefficient ring.

Next, we prove the existence of a coefficient ring in the case where the characteristic of the residue field $\kappa $ is $p > 0$. Namely, choose a Cohen ring $\Lambda $ with $\kappa = \Lambda /p\Lambda $, see Lemma 10.154.6. In this case we will prove by induction on $n > 0$ that there exists a map

\[ \varphi _ n : \Lambda /p^ n\Lambda \longrightarrow R/\mathfrak m^ n \]

whose composition with the reduction map $R/\mathfrak m^ n \to \kappa $ produces the given isomorphism $\Lambda /p\Lambda = \kappa $. This is trivial for $n = 1$. If $n > 1$, let $\varphi _{n - 1}$ be given. The ring map $\mathbf{Z}/p^ n\mathbf{Z} \to \Lambda /p^ n\Lambda $ is formally smooth by Lemma 10.154.7. Hence we can find the dotted arrow in the following diagram

\[ \xymatrix{ R/\mathfrak m^{n - 1} & R/\mathfrak m^ n \ar[l] \\ \Lambda /p^ n\Lambda \ar[u]^{\varphi _{n - 1}} \ar@{..>}[ru] & \mathbf{Z}/p^ n\mathbf{Z} \ar[l] \ar[u] } \]

This proves the induction step. Putting these maps together

\[ \mathop{\mathrm{lim}}\nolimits _ n\ \varphi _ n : \Lambda = \mathop{\mathrm{lim}}\nolimits _ n\ \Lambda /p^ n\Lambda \longrightarrow R = \mathop{\mathrm{lim}}\nolimits _ n\ R/\mathfrak m^ n \]

gives a map whose image is the desired coefficient ring.

The final statement of the theorem follows readily. Namely, if $y_1, \ldots , y_ n$ are generators of the ideal $\mathfrak m$, then we can use the map $\Lambda \to R$ just constructed to get a map

\[ \Lambda [[x_1, \ldots , x_ n]] \longrightarrow R, \quad x_ i \longmapsto y_ i. \]

Since both sides are $(x_1, \ldots , x_ n)$-adically complete this map is surjective by Lemma 10.95.1 as it is surjective modulo $(x_1, \ldots , x_ n)$ by construction. $\square$


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