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arXiv · 2610.33675

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

Xin Li
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Abstract

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

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Xin Li. 2026-10-24. Which posets are spectra of Noetherian rings? The case of height at most one. https://arxiv.org/abs/2610.33675

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