In the wikipedia webpage for "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. 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?

In the wikipedia webpage for "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?

In the wikipedia webpage for "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. 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 understanding the wikipedia example, or finding an example in the same vein?

In the wikipedia webpage for "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. 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?