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*}

102.17 Existence of bad local Noetherian rings

Let $(A, \mathfrak m, \kappa )$ be a Noetherian complete local ring. In [Lech] it was shown that $A$ is the completion of a Noetherian local domain if $\text{depth}(A) \geq 1$ and $A$ contains either $\mathbf{Q}$ or $\mathbf{F}_ p$ as a subring, or contains $\mathbf{Z}$ as a subring and $A$ is torsion free as a $\mathbf{Z}$-module. This produces many examples of Noetherian local domains with “bizarre” properties.

Applying this for example to $A = \mathbf{C}[[x, y]]/(y^2)$ we find a Noetherian local domain whose completion is nonreduced. Please compare with Section 102.15.

In [LLPY] conditions were found that characterize when $A$ is the completion of a reduced local Noetherian ring.

In [Heitmann-completion-UFD] it was shown that $A$ is the completion of a local Noetherian UFD $R$ if $\text{depth}(A) \geq 2$ and $A$ contains either $\mathbf{Q}$ or $\mathbf{F}_ p$ as a subring, or contains $\mathbf{Z}$ as a subring and $A$ is torsion free as a $\mathbf{Z}$-module. In particular $R$ is normal (Algebra, Lemma 10.119.11) hence the henselization of $R$ is a normal domain too (More on Algebra, Lemma 15.44.6). Thus $A$ as above is the completion of a henselian Noetherian local normal domain (because the completion of $R$ and its henselization agree, see More on Algebra, Lemma 15.44.3).

Apply this to find a Noetherian local UFD $R$ such that $R^\wedge \cong \mathbf{C}[[x, y, z, w]]/(wx, wy)$. Note that $\mathop{\mathrm{Spec}}(R^\wedge )$ is the union of a regular $2$-dimensional and a regular $3$-dimensional component. The ring $R$ cannot be universally catenary: Let

\[ X \longrightarrow \mathop{\mathrm{Spec}}(R) \]

be the blowing up of the maximal ideal. Then $X$ is an integral scheme. There is a closed point $x \in X$ such that $\dim (\mathcal{O}_{X, x}) = 2$, namely, on the level of the complete local ring we pick $x$ to lie on the strict transform of the $2$-dimensional component and not on the strict transform of the $3$-dimensional component. By Morphisms, Lemma 28.50.1 we see that $R$ is not universally catenary. Please compare with Section 102.16.

The ring above is catenary (being a $3$-dimensional local Noetherian UFD). However, in [Ogoma-example] the author constructs a normal local Noetherian domain $R$ with $R^\wedge \cong \mathbf{C}[[x, y, z, w]]/(wx, wy)$ such that $R$ is not catenary. See also [Heitmann-Ogoma] and [Lech-YAPO].

In [Heitmann-isolated] it was shown that $A$ is the completion of a local Noetherian ring $R$ with an isolated singularity provided $A$ contains either $\mathbf{Q}$ or $\mathbf{F}_ p$ as a subring or $A$ has residue characteristic $p > 0$ and $p$ cannot map to a nonzero zerodivisor in any proper localization of $A$. Here we say a Noetherian local ring $R$ has an isolated singularity if $R_\mathfrak p$ is a regular local ring for all nonmaximal primes $\mathfrak p \subset R$.

The papers [Nishimura-few] and [Nishimura-few-II] contain long lists of “bad” Noetherian local rings with given completions. In particular it constructs an example of a $2$-dimensional Nagata local normal domain whose completion is $\mathbf{C}[[x, y, z]]/(yz)$ and one whose completion is $\mathbf{C}[[x, y, z]]/(y^2 - z^3)$.

As an aside, in [Loepp] it was shown that $A$ is the completion of an excellent Noetherian local domain if $A$ is reduced, equidimensional, and no integer in $A$ is a zero divisor. However, this doesn't lead to “bad” Noetherian local rings as we obtain excellent ones!

Comments (2)

Comment #1505 by Kollar on

About the [Lech] paper. On math reviews it claims that one more condition is needed: if the prime ring is Z then A should be torsion free as a Z-module. (I did not check the paper). I think that the same may apply to the paper [Heitmann-completion-UFD].

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