Algebro-Geometric Method in Commutative Algebra

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的紧闭包理论。即推广了紧闭包和F-奇点的概念，奠定了紧闭包和F-奇点的理论基础，并将其应用于交换代数和代数几何问题。主要结果如下：1. Rees代数的F-奇异性研究：虽然Rees代数是一个几何对象，它的“投影”给出了一个blow-up，但从几何角度对Rees代数的F-奇异性的研究却很少。考虑到这一点，我们用F-奇异性、爆破和去奇异化等方法研究了正规局部环（R，m）的准素内理想I所对应的Rees代数R（1）的环理论和几何问题。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-紧闭包”的概念。 ...更多信息 确定”与规范R上的有效Q因子D相关，“I紧闭合”与R的理想I相关。我们建立了I-紧闭包理论和由I-紧闭包定义的理想r（I）的基础，证明了理想r（I）的Skoda定理、限制定理和次可加性等性质。3. I-紧闭包的应用：考虑了极化簇（X，L）的伴随丛K_X+nL的整体生成，作为Skoda定理的一个变体在（X，L）的截环的典范模中的应用。特别地，我们得到了K. E. Smith关于Fujita猜想在特征p中的一个特殊情形的结果的另一种证明，假定L本身是张成的.我们还构造了与理想I.=的滤子相联系的渐近乘子理想的特征p模拟T（T_I. T_）. {I_n| n= 1，2，.}。作为应用，我们重新解释了Ein-Lazarsfeld-Smith和Hochster-Huneke关于符号权力一致行为的结果。少

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