## Tag `0AL7`

## 100.17. 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 100.15.

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 15.42.6). 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 15.42.3).

Apply this to find a Noetherian local UFD $R$ such that $R^\wedge \cong \mathbf{C}[[x, y, z, w]]/(wx, wy)$. Note that $\mathop{\rm Spec}(R^\wedge)$ is the union of a regular $2$-dimensional and a regular $3$-dimensional component. The ring $R$ cannot be universally catenary: Let $$ X \longrightarrow \mathop{\rm Spec}(R) $$ 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 28.49.1 we see that $R$ is not universally catenary. Please compare with Section 100.16.

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 paper [Nishimura-few] contains a long list 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!

The code snippet corresponding to this tag is a part of the file `examples.tex` and is located in lines 1221–1306 (see updates for more information).

```
\section{Existence of bad local Noetherian rings}
\label{section-bad}
\noindent
Let $(A, \mathfrak m, \kappa)$ be a Noetherian complete local ring.
In \cite{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.
\medskip\noindent
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 \ref{section-local-completion-nonreduced}.
\medskip\noindent
In \cite{LLPY} conditions were found that characterize when $A$ is
the completion of a reduced local Noetherian ring.
\medskip\noindent
In \cite{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 \ref{algebra-lemma-UFD-normal})
hence the henselization of $R$ is a normal domain too
(More on Algebra, Lemma \ref{more-algebra-lemma-henselization-normal}).
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 \ref{more-algebra-lemma-henselization-noetherian}).
\medskip\noindent
Apply this to find a Noetherian local UFD $R$ such that
$R^\wedge \cong \mathbf{C}[[x, y, z, w]]/(wx, wy)$.
Note that $\Spec(R^\wedge)$ is the
union of a regular $2$-dimensional and a regular $3$-dimensional component.
The ring $R$ cannot be universally catenary: Let
$$
X \longrightarrow \Spec(R)
$$
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 \ref{morphisms-lemma-dimension-formula}
we see that $R$ is not universally catenary. Please compare with
Section \ref{section-non-catenary-Noetherian-local}.
\medskip\noindent
The ring above is catenary (being a $3$-dimensional local Noetherian UFD).
However, in \cite{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 \cite{Heitmann-Ogoma} and
\cite{Lech-YAPO}.
\medskip\noindent
In \cite{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 $\mathfrak p \subset R$.
\medskip\noindent
The paper \cite{Nishimura-few} contains a long list 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 $\mathbf{C}[[x, y, z]]/(y^2 - z^3)$.
\medskip\noindent
As an aside, in \cite{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!
```

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