The Stacks Project


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!

    Comments (2)

    Comment #1505 by Kollar on June 15, 2015 a 8:52 pm UTC

    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].

    Comment #1509 by Johan (site) on June 16, 2015 a 5:19 pm UTC

    Yes indeed. Thanks for commenting. Fixed here.

    Add a comment on tag 0AL7

    Your email address will not be published. Required fields are marked.

    In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).

    All contributions are licensed under the GNU Free Documentation License.




    In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

    This captcha seems more appropriate than the usual illegible gibberish, right?