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Sally’s question and a conjecture of shimoda

Published online by Cambridge University Press:  11 January 2016

Shiro Goto ,
Liam O’carroll  and
Francesc Planas-Vilanova
Shiro Goto
Affiliation:
Department of Mathematics, School of Science and Technology, Meiji University, Tama-Ku, Kawasaki-shi, 214-8571, Japan, goto@math.meiji.ac.jp
Liam O’carroll
Affiliation:
Maxwell Institute for Mathematical Sciences, School of Mathematics, University of Edinburgh, EH9 3JZ, Edinburgh, Scotland, L.O’Carroll@ed.ac.uk
Francesc Planas-Vilanova
Affiliation:
Departament de Matemàtica Aplicada 1, Universitat Politècnica de Catalunya, 08028 Barcelona, Catalunya, francesc.planas@upc.edu
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Abstract

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In 2007, Shimoda, in connection with a long-standing question of 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 2. In this paper, having reduced the conjecture to the case of dimension 3, if the ring is regular and local of dimension 3, we explicitly describe a family of prime ideals of height 2 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 3. 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 Simis, Ulrich, and Vasconcelos, allows us to obtain a definitive answer in this setting. Finally, by adapting work of 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 (as it were) excessive number of generators.

Keywords

13A1713F15
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 211 , September 2013 , pp. 137 - 161
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2013

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