Which posets are spectra of Noetherian rings? The case of height at most one

@misc{li2026which,
  title = {{Which posets are spectra of Noetherian rings? The case of height at most one}},
  author = {Xin Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.33675},
  eprint = {2610.33675},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.33675},
}

TY  - UNPB
TI  - Which posets are spectra of Noetherian rings? The case of height at most one
AU  - Xin Li
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.33675
UR  - https://arxiv.org/abs/2610.33675
AB  - A finite poset is the prime spectrum of a Noetherian ring exactly when it has no chain of three elements, that is, it is the poset of a finite bipartite graph. This answers Conjecture 5.3 of Alkass. The substance is a construction: minimal elements become curves y=g_i(x), maximal elements points, edges the incidences, arranged by interpolation. The coordinate ring of the union of the curves, localized at the chosen points, is essentially of finite type over Q and of dimension at most one. Dropping finiteness costs two conditions, both isolated by Colbert: a poset with no chain of three elements is a Noetherian spectrum exactly when it has finitely many minimal elements and any two distinct ones have finitely many common upper bounds. A variant keeping the shape of the ideals of Alkass but changing their exponents shows why that construction produces only nested neighborhoods. Constraining the ring destroys most of this freedom: normal rings leave finite disjoint unions of stars, domains a single star, local rings a point or a finite co-star. In height two the question is open in general, but the conditions conjectured by Wiegand and Wiegand are not sufficient
ER  -