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*}

10.154 The Cohen structure theorem

Here is a fundamental notion in commutative algebra.

Definition 10.154.1. Let $(R, \mathfrak m)$ be a local ring. We say $R$ is a complete local ring if the canonical map

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

to the completion of $R$ with respect to $\mathfrak m$ is an isomorphism1.

Note that an Artinian local ring $R$ is a complete local ring because $\mathfrak m_ R^ n = 0$ for some $n > 0$. In this section we mostly focus on Noetherian complete local rings.

Lemma 10.154.2. Let $R$ be a Noetherian complete local ring. Any quotient of $R$ is also a Noetherian complete local ring. Given a finite ring map $R \to S$, then $S$ is a product of Noetherian complete local rings.

Proof. The ring $S$ is Noetherian by Lemma 10.30.1. As an $R$-module $S$ is complete by Lemma 10.96.1. Hence $S$ is the product of the completions at its maximal ideals by Lemma 10.96.8. $\square$

Lemma 10.154.3. Let $(R, \mathfrak m)$ be a complete local ring. If $\mathfrak m$ is a finitely generated ideal then $R$ is Noetherian.

Proof. See Lemma 10.96.5. $\square$

Definition 10.154.4. Let $(R, \mathfrak m)$ be a complete local ring. A subring $\Lambda \subset R$ is called a coefficient ring if the following conditions hold:

  1. $\Lambda $ is a complete local ring with maximal ideal $\Lambda \cap \mathfrak m$,

  2. the residue field of $\Lambda $ maps isomorphically to the residue field of $R$, and

  3. $\Lambda \cap \mathfrak m = p\Lambda $, where $p$ is the characteristic of the residue field of $R$.

Let us make some remarks on this definition. We split the discussion into the following cases:

  1. The local ring $R$ contains a field. This happens if either $\mathbf{Q} \subset R$, or $pR = 0$ where $p$ is the characteristic of $R/\mathfrak m$. In this case a coefficient ring $\Lambda $ is a field contained in $R$ which maps isomorphically to $R/\mathfrak m$.

  2. The characteristic of $R/\mathfrak m$ is $p > 0$ but no power of $p$ is zero in $R$. In this case $\Lambda $ is a complete discrete valuation ring with uniformizer $p$ and residue field $R/\mathfrak m$.

  3. The characteristic of $R/\mathfrak m$ is $p > 0$, and for some $n > 1$ we have $p^{n - 1} \not= 0$, $p^ n = 0$ in $R$. In this case $\Lambda $ is an Artinian local ring whose maximal ideal is generated by $p$ and which has residue field $R/\mathfrak m$.

The complete discrete valuation rings with uniformizer $p$ above play a special role and we baptize them as follows.

Definition 10.154.5. A Cohen ring is a complete discrete valuation ring with uniformizer $p$ a prime number.

Lemma 10.154.6. Let $p$ be a prime number. Let $k$ be a field of characteristic $p$. There exists a Cohen ring $\Lambda $ with $\Lambda /p\Lambda \cong k$.

Proof. First note that the $p$-adic integers $\mathbf{Z}_ p$ form a Cohen ring for $\mathbf{F}_ p$. Let $k$ be an arbitrary field of characteristic $p$. Let $\mathbf{Z}_ p \to R$ be a flat local ring map such that $\mathfrak m_ R = pR$ and $R/pR = k$, see Lemma 10.153.1. Then clearly $R$ is a discrete valuation ring. Hence its completion is a Cohen ring for $k$. $\square$

Lemma 10.154.7. Let $p > 0$ be a prime. Let $\Lambda $ be a Cohen ring with residue field of characteristic $p$. For every $n \geq 1$ the ring map

\[ \mathbf{Z}/p^ n\mathbf{Z} \to \Lambda /p^ n\Lambda \]

is formally smooth.

Proof. If $n = 1$, this follows from Proposition 10.152.9. For general $n$ we argue by induction on $n$. Namely, if $\mathbf{Z}/p^ n\mathbf{Z} \to \Lambda /p^ n\Lambda $ is formally smooth, then we can apply Lemma 10.136.12 to the ring map $\mathbf{Z}/p^{n + 1}\mathbf{Z} \to \Lambda /p^{n + 1}\Lambda $ and the ideal $I = (p^ n) \subset \mathbf{Z}/p^{n + 1}\mathbf{Z}$. $\square$

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. $\square$

Remark 10.154.9. If $k$ is a field then the power series ring $k[[X_1, \ldots , X_ d]]$ is a Noetherian complete local regular ring of dimension $d$. If $\Lambda $ is a Cohen ring then $\Lambda [[X_1, \ldots , X_ d]]$ is a complete local Noetherian regular ring of dimension $d + 1$. Hence the Cohen structure theorem implies that any Noetherian complete local ring is a quotient of a regular local ring. In particular we see that a Noetherian complete local ring is universally catenary, see Lemma 10.104.9 and Lemma 10.105.3.

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. $\square$

Lemma 10.154.11. Let $(R, \mathfrak m)$ be a Noetherian complete local domain. Then there exists a $R_0 \subset R$ with the following properties

  1. $R_0$ is a regular complete local ring,

  2. $R_0 \subset R$ is finite and induces an isomorphism on residue fields,

  3. $R_0$ is either isomorphic to $k[[X_1, \ldots , X_ d]]$ where $k$ is a field or $\Lambda [[X_1, \ldots , X_ d]]$ where $\Lambda $ is a Cohen ring.

Proof. Let $\Lambda $ be a coefficient ring of $R$. Since $R$ is a domain we see that either $\Lambda $ is a field or $\Lambda $ is a Cohen ring.

Case I: $\Lambda = k$ is a field. Let $d = \dim (R)$. Choose $x_1, \ldots , x_ d \in \mathfrak m$ which generate an ideal of definition $I \subset R$. (See Section 10.59.) By Lemma 10.95.9 we see that $R$ is $I$-adically complete as well. Consider the map $R_0 = k[[X_1, \ldots , X_ d]] \to R$ which maps $X_ i$ to $x_ i$. Note that $R_0$ is complete with respect to the ideal $I_0 = (X_1, \ldots , X_ d)$, and that $R/I_0R \cong R/IR$ is finite over $k = R_0/I_0$ (because $\dim (R/I) = 0$, see Section 10.59.) Hence we conclude that $R_0 \to R$ is finite by Lemma 10.95.12. Since $\dim (R) = \dim (R_0)$ this implies that $R_0 \to R$ is injective (see Lemma 10.111.3), and the lemma is proved.

Case II: $\Lambda $ is a Cohen ring. Let $d + 1 = \dim (R)$. Let $p > 0$ be the characteristic of the residue field $k$. As $R$ is a domain we see that $p$ is a nonzerodivisor in $R$. Hence $\dim (R/pR) = d$, see Lemma 10.59.12. Choose $x_1, \ldots , x_ d \in R$ which generate an ideal of definition in $R/pR$. Then $I = (p, x_1, \ldots , x_ d)$ is an ideal of definition of $R$. By Lemma 10.95.9 we see that $R$ is $I$-adically complete as well. Consider the map $R_0 = \Lambda [[X_1, \ldots , X_ d]] \to R$ which maps $X_ i$ to $x_ i$. Note that $R_0$ is complete with respect to the ideal $I_0 = (p, X_1, \ldots , X_ d)$, and that $R/I_0R \cong R/IR$ is finite over $k = R_0/I_0$ (because $\dim (R/I) = 0$, see Section 10.59.) Hence we conclude that $R_0 \to R$ is finite by Lemma 10.95.12. Since $\dim (R) = \dim (R_0)$ this implies that $R_0 \to R$ is injective (see Lemma 10.111.3), and the lemma is proved. $\square$

[1] This includes the condition that $\bigcap \mathfrak m^ n = (0)$; in some texts this may be indicated by saying that $R$ is complete and separated. Warning: It can happen that the completion $\mathop{\mathrm{lim}}\nolimits _ n R/\mathfrak m^ n$ of a local ring is non-complete, see Examples, Lemma 102.6.1. This does not happen when $\mathfrak m$ is finitely generated, see Lemma 10.95.3 in which case the completion is Noetherian, see Lemma 10.96.5.

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