Sally的问题和Y.下田
J. Sally's question and a conjecture of Y. Shimoda
摘要：
2007年，Y.下田，在与一个长期存在的问题J.萨利，问是否诺特局部环，使其所有的总理理想不同的最大理想是完全交叉，有克鲁尔维数最多两个。本文将猜想还原到三维情形，若环是正则的三维局部环，则我们明确地刻画了一族由三个元素极小生成的高度为2的素理想.通过弱化正则性的假设，我们发现，为了达到同样的目的，我们需要增加额外的假设，如完备性、剩余域的无限性和环的重数至多为3。在第二部分中，我们把注意力转向标准分次代数范畴。通过Bertini定理的双重应用给出了一种几何方法，并结合A.锡米斯湾Ulrich和W.V. Vasconcelos，使我们能够在这种情况下获得一个明确的答案。最后，通过对M.米勒总理布尔巴基理想的局部环，我们详细介绍了一些更技术性的结果，关于存在于标准的分次代数的齐次素理想与“过多”的发电机数量。
英文摘要：
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.
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