In the [wikipedia webpage for "excellent ring"](https://en.wikipedia.org/wiki/Excellent_ring), one finds the following. 



> 
> If R is the subring of the polynomial ring k[x1,x2,...] in infinitely many generators generated by the squares and cubes of all generators, and S is obtained from R by adjoining inverses to all elements not in any of the ideals generated by some xn, then S is a 1-dimensional Noetherian domain that is not a J-1 ring as S has a cusp singularity at every closed point, so the set of singular points is not closed, though it is a G-ring. This ring is also universally catenary, as its localization at every prime ideal is a quotient of a regular ring.
> 
> 
> 


I am not sure what the "elements not in any of the ideals generated by some xn" are, because no xn lies in R. Also I am not able to prove noetherianity. In fact, I am not sure that the example has all the claimed properties.


In Exposé XIX of the volume "Travaux de Gabber" in Astérisque 363-364, there is an example of a one-dimensional noetherian domain whose regular locus is not open, with ordinary double points as singularities at closed points.


I understand this latter example, but it is much more complicated than the former and I'd really like to find an example simple enough to be presented in a colloquium-style talk.


Can anyone help me understand the wikipedia example, or find an example in the same vein?



Looking for a simple one-dimensional noetherian domain whose regular locus is not open
This example appears as Example 1 in the following reference:



> 
> Melvin Hochster, *Non-openness of loci in Noetherian rings,* Duke Math. J. **40** (1973), 215–219. [MR311653](https://mathscinet.ams.org/mathscinet-getitem?mr=311653). [ZBL0257.13015](https://zbmath.org/?q=an:0257.13015). DOI: [10.1215/S0012-7094-73-04020-9](https://doi.org/10.1215/S0012-7094-73-04020-9).
> 
> 
> 


In fact, the example in question is a special case of the general result quoted below.



> 
> […] $K$ denotes a field and all otherwise unspecified tensor products
>  are taken over $K$. If $R$ is a $K$-algebra, we say that $R$ is
>  *absolutely* Noetherian over $K$ if for every overfield $L \supset K$, $L \otimes R$ is Noetherian. If $R$ is a localization of a finitely
>  generated $K$-algebra, then $R$ is absolutely Noetherian over $K$. $R$
>  is *absolutely* a domain over $K$ if, likewise, each $L \otimes R$ is
>  a domain, and $P$ is absolutely prime if, equivalently, either $R/P$
>  is absolutely a domain or each $L \otimes P$ is prime (in $L \otimes R$).
> 
> 
> […]
> 
> 
> **Proposition 1.** *Let $\{R\_i\}\_{i \in I}$ be a family of absolutely Noetherian $K$-algebras indexed by an
>  infinite set $I$, and for each $i \in I$, let $P\_i$ be a nonzero
>  absolutely prime ideal of $R\_i$. Assume that each $R\_i$ is absolutely
>  a domain.*
> 
> 
> *Let $R' = \bigotimes\_{i \in I} R\_i$. Then $R'$ is a domain and for
>  each $i$, $P\_iR'$ is prime. Moreover, if $S = R' - (\bigcup\_i P\_iR')$
>  and $R = S^{-1}R'$, then $R$ is a Noetherian domain whose maximal
>  ideals are in one-to-one correspondence with $I$ via the map $i \mapsto P\_iR$. In addition, each nonzero element of $R$ belongs to
>  only finitely many maximal ideals, and for any maximal ideal $P\_iR$ of
>  $R$* $$R\_{P\_iR} \cong (L\_i \otimes R\_i)\_{P\_i^e},$$ *where $L\_i$ is a
>  certain extension field of $K$ and $P\_i^e$ is $P\_i(L\_i \otimes R\_i)$.*
> 
> 
> *In particular, if for each $i \in I$, $R\_i$ is a subring of a
>  polynomial ring over $K$ and is generated by a finite nonempty set of
>  forms of positive degree and $P\_i$ is the ideal generated by these
>  forms, then the hypotheses of the first paragraph are satisfied, and
>  the conclusions of the second paragraph hold. Moreover, the local
>  rings of $R$ are algebro-geometric in this case.*
> 
> 
> 


In Example 1, Hochster applies the Proposition to the situation where $I$ is the set of positive integers and $K$ is an arbitrary field, in which case he sets $R\_i = K[x\_i^2,x\_i^3]$ and $P\_i = (x\_i^2,x\_i^3)R\_i$ for each $i$. Then, $R\_{P\_iR} \cong L\_i[x\_i^2,x\_i^3]\_{P\_i^e}$ for every $i$, which is a non-normal local domain of dimension one. By the Proposition these are the localizations of $R$ at nonzero prime ideals, and the only regular point in $\operatorname{Spec}R$ is the generic point corresponding to the zero ideal.


It is useful to note the following consequence of Proposition 1, also in Hochster's paper:



> 
> **Proposition 2.** *Let $\mathscr{P}$ be a property of local rings and suppose there is an algebro-geometric local ring $(R\_1,P\_1)$ over a
>  field $K$ such that*
> 
> 
> 1. *$R\_1$ is absolutely a domain,*
> 2. *$P\_1$ is absolutely prime, and*
> 3. *for every overfield $L \supset K$* $$(L \otimes R\_1)\_{P\_1^e}$$ *fails to have property $\mathscr{P}$.*
> 
> 
> *Suppose that each field $L \supset K$ has $\mathscr{P}$. Then there is a locally algebro-geometric Noetherian domain $R$ over $K$ in which the $\mathscr{P}$ locus is not open.*
> 
> 
> 


This gives a nice way to construct rings that are locally excellent but not excellent, for instance.


