# 1-dim noetherian domain with dense set of cusps diagram

Consists of a scheme $X$, satisfying the following properties:

• $X$ connected, affine, cohen-macaulay, not regular, not normal, locally noetherian, all stalks of the structure sheaf are domains, dimension 1, irreducible, separated, reduced, pure dimension 1, noetherian, integral, quasi-separated, not excellent, finite dimensional
By the theorems, we also have {"X": {"local rings are domains": true, "quasi-compact": true}}

## References

• https://mathoverflow.net/questions/339694/looking-for-a-simple-one-dimensional-noetherian-domain-whose-regular-locus-is-no
• Let $k$ be a field. For every integer $n \ge 0$, let $R_n = k[x^2, x^3]$ and $I_n$ the ideal $(x^2,x^3)$. Let $R'$ be the tensor product of all $R_n$ over $k$, and let $S$ be the multiplicative set of elements of $R'$ not belonging to any of the prime ideals $I_n R'$. Then the localization $R = S^{-1} R'$ is a noetherian domain of dimension 1. For every maximal ideal of $R$, the localization is the localization at $(x, y)$ of $k[x,y]/x^2-y^3$, i.e. a cusp. [Example 1, Melvin Hochster, Non-openness of loci in Noetherian rings, Duke Math. J. 40 (1973), 215-219]