J. Sally's question and a conjecture of Y. Shimoda

TL;DR

This paper investigates a conjecture about the structure of prime ideals in Noetherian local rings and standard graded algebras, providing explicit descriptions and conditions under which certain prime ideals exist.

Contribution

It reduces Shimoda's conjecture to dimension three, describes prime ideals in regular local rings, and explores geometric and algebraic conditions for prime ideals in graded algebras.

Findings

Explicit description of prime ideals in regular local rings of dimension three

Additional hypotheses needed for non-regular rings to find similar prime ideals

Geometric approach yields definitive results for standard graded algebras

Abstract

In 2007, Y. Shimoda, in connection with a long-standing question of J. Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most two. In this paper, having reduced the conjecture to the case of dimension three, if the ring is regular and local of dimension three, we explicitly describe a family of prime ideals of height two minimally generated by three elements. Weakening the hypothesis of regularity, we find that, to achieve the same end, we need to add extra hypotheses, such as completeness, infiniteness of the residue field and the multiplicity of the ring being at most three. In the second part of the paper we turn our attention to the category of standard graded algebras. A geometrical approach via a double use of a Bertini Theorem, together with a result of A. Simis, B. Ulrich and W.V. Vasconcelos, allows us to obtain a definitive answer in this setting. Finally, by adapting work of M. Miller on prime Bourbaki ideals in local rings, we detail some more technical results concerning the existence in standard graded algebras of homogeneous prime ideals with an "excessive" number of generators.