Algebro-Geometric Method in Commutative Algebra

交换代数中的代数几何方法

• 批准号: 13640005
• 负责人: HARA Nobuo
• 金额: $ 2.3万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640005/
• 关键词: Tight closure; F-singularity; test ideal; multiplier ideal; positive characteristic; commutative algebra; algebraic geometry; Rees algebra; 密着閉包(I-密着閉包); Subadditivity; 随伴束; 記号的ベキ乗; multiplier ideal; 特異点; F-正則; F-正則(F-regular); F-有理(F-rati

We reinterpreted the theory of tight closure in prime characteristic commutative algebra from the viewpoint of singularity theory and birational geometry. Namely, we generalized the concepts of tight closure and F-singularities, gave foundation to the theory thereof, and applied it to problems in commutative algebra and al-gebraic geometry. Our results are summarized as follows:1.Study of F-singularities of Rees algebras : There have been few researches on Rees algebras from a geometric viewpoint, although a Rees algebra is a geometric object in the sense that its "Proj" gives a blow-up. Taking this into account, we studied ring-theoretical and geometric aspects of Rees algebras R(1) associated to an in-primary ideal I of a normal local ring (R,m) in terms of miscellaneous methods such as F-singularities, blow-up and desingularization.2.A generalization of tight closure : We generalized the notion of the tight closure of an ideal in a ring R of characteristic p to those of "D-tight clo … More sure" associated to an effective Q-divisor D on Spec R and of "I-tight closure" associated to an ideal I of R. We established foundation of the theory of I-tight closure and the ideal r(I) defined via I-tight closure, and proved various properties of the ideal -r(I) such as Skoda's theorem, restriction theorem and subadditivity.3.Applications of I-tight closure : We considered the global generation of adjoint bundles K_X+nL of a polarized variety (X, L), as an application of a variant of Skoda's theorem in the canonical module of the section ring of (X,L). In particular, we obtained an alternative proof of K.E.Smith's result on a special case of Fujita's conjecture in characteristic p, assuming that Litself is spanned.We also constructed a characteristic p analog T(‖I.‖) of the asymptotic multiplier ideal associated to a filtration of ideals I.={I_n|n= 1,2,...}. As an application, we reinterpret the result on the uniform behavior of symbolic powers due to Ein-Lazarsfeld-Smith and Hochster-Huneke. Less

从奇点理论和二次几何的角度重新解释了素数特征交换代数中的紧闭包理论。即推广了紧闭包和F-奇点的概念，给出了紧闭包和F-奇点的理论基础，并将其应用于交换代数和代数几何中的问题。1.Rees代数的F-奇异性研究：虽然Rees代数是一个几何对象，因为它的“射影”是爆破的，但从几何的角度对Rees代数的研究很少。考虑到这一点，我们用F-奇点、爆破和去奇异等方法研究了正规局部环(R，m)的内素理想I所对应的Rees代数R(1)的环论和几何问题。2.紧闭包的推广：我们将特征为p的环R中理想的紧闭包的概念推广到D-紧…与Spec R上的一个有效q-因子D有关，以及与R的一个理想I有关的“I-紧闭包”。我们建立了I-紧闭包理论和由I-紧闭包定义的理想r(I)的理论基础，并证明了理想-r(I)的各种性质，如Skoda定理、限制定理和次可加性。3.I-紧闭包的应用：我们考虑了极化簇(X，L)的伴随丛K_X+nL的整体生成，作为Skoda定理的一个变形在(X，L)的截面环的典范模中的一个应用。特别地，在假设L本身是扩张的情况下，得到了K.E.Smith关于特征p中Fujita猜想的结果的另一种证明，并构造了与理想滤子有关的渐近乘子理想的特征p模拟T(‖I.‖)。作为应用，我们重新解释了Ein-Lazarsfeld-Smith和Hochster-Huneke关于符号幂的一致行为的结果。较少

项目成果

N.Hara: "A characteristic p analog of multiplier ideals and applications"Comm.in Algebra. 印刷中. (2004)

N.Hara：“乘数理想的特征 p 模拟和应用”Comm.in 代数 (2004)。

N.Hara, K.-i.Watanabe, K.Yoshida: "Rees algebras of F-regular type"J.Algebra. 247. 191-218 (2002)

N.Hara、K.-i.Watanabe、K.Yoshida：“F-正则类型的里斯代数”J.代数。

Nobuo Hara: "Kawachi's invariant for fat points"Journal of Pure and Applied Algebra. 165. 201-211 (2001)

原信夫：“河内脂肪点不变量”纯粹与应用代数杂志。

N.Hara, K.-i.Watanabe, K.Yoshida: "Rees algebras of F-regular type"J.of Algebra. 247. 191-218 (2002)

N.Hara、K.-i.Watanabe、K.Yoshida：“F-正则类型的里斯代数”J.of Algebra。

Nobuo Hara: "A characteristic p analog of multiplier ideals and applications"Communications in Algebra. (発表予定).

Nobuo Hara：“乘数理想和应用的特征 p 模拟”代数通讯（待提交）。