## 110.19 A non catenary Noetherian local ring

Even though there is a successful dimension theory of Noetherian local rings there are non-catenary Noetherian local rings. An example may be found in [Appendix, Example 2, Nagata]. In fact, we will present this example in the simplest case. Namely, we will construct a local Noetherian domain $A$ of dimension $2$ which is not universally catenary. (Note that $A$ is automatically catenary, see Exercises, Exercise 111.18.3.) The existence of a Noetherian local ring which is not universally catenary implies the existence of a Noetherian local ring which is not catenary – and we spell this out at the end of this section in the particular example at hand.

Let $k$ be a field, and consider the formal power series ring $k[[x]]$ in one variable over $k$. Let

be a formal power series. We assume $z$ as an element of the Laurent series field $k((x)) = k[[x]][1/x]$ is transcendental over $k(x)$. Put

Note that $z = xz_1$. Let $R$ be the subring of $k[[x]]$ generated by $x$, $z$ and all of the $z_ j$, in other words

Consider the ideals $\mathfrak m = (x)$ and $\mathfrak n = (x - 1, z_1, z_2 + a_1, z_3 + a_1 + a_2, \ldots )$ of $R$.

We have $xz_{j + 1} + a_ j = z_ j$. Hence $R/\mathfrak m = k$ and $\mathfrak m$ is a maximal ideal. Moreover, any element of $R$ not in $\mathfrak m$ maps to a unit in $k[[x]]$ and hence $R_{\mathfrak m} \subset k[[x]]$. In fact it is easy to deduce that $R_{\mathfrak m}$ is a discrete valuation ring and residue field $k$.

We claim that

Namely, the relation above implies that $z_{j + 1} = z_ j - a_ j - (x - 1)z_{j + 1}$, and hence we may express the class of $z_{j + 1}$ in terms of $z_ j$ in the quotient $R/(x - 1)$. Since the fraction field of $R$ has transcendence degree $2$ over $k$ by construction we see that $z$ is transcendental over $k$ in $R/(x - 1)$, whence the desired isomorphism. Hence $\mathfrak n = (x - 1, z)$ and is a maximal ideal. In fact the map

is an isomorphism (since $x^{-1}$ is invertible in $R_{\mathfrak n}$ and since $z_{j + 1} = x^{-1}z_ j - a_ j = \ldots = f_ j(x, x^{-1}, z)$). This shows that $R_{\mathfrak n}$ is a regular local ring of dimension $2$ and residue field $k$.

Let $S$ be the multiplicative subset

and set $B = S^{-1}R$. We claim that

The ring $B$ is a $k$-algebra.

The maximal ideals of the ring $B$ are the two ideals $\mathfrak mB$ and $\mathfrak nB$.

The residue field at these maximal ideals is $k$.

We have $B_{\mathfrak mB} = R_{\mathfrak m}$ and $B_{\mathfrak nB} = R_{\mathfrak n}$ which are Noetherian regular local rings of dimensions $1$ and $2$.

The ring $B$ is Noetherian.

We omit the details of the verifications.

Whenever given a $k$-algebra $B$ with the properties listed above we get an example as follows. Take $A = k + \text{rad}(B) \subset B$ with $\text{rad}(B) = \mathfrak mB \cap \mathfrak nB$ the Jacobson radical. It is easy to see that $B$ is finite over $A$ and hence $A$ is Noetherian by Eakin's theorem (see [Eakin], or [Appendix A1, Nagata], or insert future reference here). Also $A$ is a local domain with the same fraction field as $B$ and residue field $k$. Since the dimension of $B$ is $2$ we see that $A$ has dimension $2$ as well, by Algebra, Lemma 10.112.4.

If $A$ were universally catenary then the dimension formula, Algebra, Lemma 10.113.1 would give $\dim (B_{\mathfrak mB}) = 2$ contradiction.

Note that $B$ is generated by one element over $A$. Hence $B = A[x]/\mathfrak p$ for some prime $\mathfrak p$ of $A[x]$. Let $\mathfrak m' \subset A[x]$ be the maximal ideal corresponding to $\mathfrak mB$. Then on the one hand $\dim (A[x]_{\mathfrak m'}) = 3$ and on the other hand

is a maximal chain of primes. Hence $A[x]_{\mathfrak m'}$ is an example of a non catenary Noetherian local ring.

## 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 (10)

Comment #4261 by Manuel Hoff on

Comment #4303 by David Speyer on

Comment #4431 by Johan on

Comment #6832 by David Speyer on

Comment #6833 by Laurent Moret-Bailly on

Comment #6972 by Johan on

Comment #7963 by Friedrich Knop on

Comment #8191 by Stacks Project on

Comment #8984 by Jake Levinson on

Comment #9202 by Stacks project on