Algebro-Geometric Method in Commutative Algebra

交换代数中的代数几何方法

基本信息

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

项目摘要

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

我们从奇点理论和双有理几何的角度重新解释了素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，为其理论奠定了基础，并将其应用于交换问题中。我们的研究成果概括如下： 1.Rees代数的F奇异性研究：目前对Rees代数的研究还很少。从几何角度来看，尽管里斯代数是一个几何对象，但其“Proj”给出了爆炸，考虑到这一点，我们研究了与里斯代数 R(1) 相关的环理论和几何方面。正常局部环 (R,m) 的初级理想 I，采用 F 奇点、爆炸和去奇异化等各种方法。2.紧闭包的推广：我们推广了特征 p 的环 R 中的理想的紧密闭合的概念与与 Spec R 上的有效 Q-除数 D 相关的“D 紧密闭合……更确定”以及与我们建立了I-紧闭包的理论基础以及通过I-紧闭包定义的理想r(I)，并证明了理想-r(I)的各种性质，如斯柯达定理、限制定理和subadditivity.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 的全局生成，作为 Skoda 定理在截面环规范模中的变体应用(X,L) 特别是，我们在特征 p 的 Fujita 猜想的一个特例上获得了 K.E.Smith 结果的替代证明，假设 Litself 是我们还构造了与理想值 I.={I_n|n= 1,2,...} 过滤相关的渐近乘数理想的特征 p 模拟 T(‖I.‖) 作为应用，我们重新解释。 Ein-Lazarsfeld-Smith 和 Hochster-Huneke Less 关于符号权力统一行为的结果。