## 108.17 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 108.15.

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

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 29.50.1 we see that $R$ is not universally catenary. Please compare with Section 108.16.

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!

## Post a comment

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 toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #1505 by Kollar on

Comment #1509 by Johan on