Tight closure and primary decomposition in Commutative Algebra

交换代数中的紧闭闭与初等分解

• 批准号: 0700554
• 负责人: Yongwei Yao
• 金额: $ 8.39万
• 依托单位: Georgia State University Research Foundation, Inc.
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2012-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700554&HistoricalAwards=false
• 关键词: Tight; closure; primary; decomposition; Commutative

This project is in the field of Commutative Algebra, in which the PrincipalInvestigator (PI) will study commutative Noetherian rings. First of all, thePI plans to study commutative Noetherian rings of prime characteristic p with a focus on the tight closure and Frobenius closure of ideals (or, moregenerally, submodules). In particular, the PI plans to study the existenceof a uniform test exponent (for tight closure) for all modules of finitephantom projective dimension as well as the existence of a uniform testexponent (for Frobenius closure) for all ideals generated by systems ofparameter. In order to better understand the structure of modules of finitephantom projective dimension, the PI, in joint work with Mel Hochster, plans to show that any such module can be `weakly embedded' into another module of finite phantom projective dimension whose structure is much easier to understand. Using this technique, the PI hopes to prove the existence of auniform test exponent (for tight closure) for all modules of finite phantomprojective dimension. The PI also plans to study certain numericalinvariants, namely the F-rational signature, the F-signature and theHilbert-Kunz multiplicity of a local ring of prime characteristic. Each ofthe invariants carries important information about the ring, depending onwhether it is positive, zero, large or small. In particular, the notion ofF-rational signature is a new invariant defined in this project, whosepositivity characterizes the F-rationality of the ring. Other projects thatthe PI plans to carry out include the Hilbert-Kunz functions, the EmbeddingTheorem for modules of finite projective dimension (or, more generally,finite G-dimension), and the theory of primary decomposition concerning thelinear growth property for families of modules.Commutative Algebra is a branch of mathematics, which arose from a study of solutions of a set of polynomial equations. The study of the functions overthe solution sets led to the investigation of the structure of thecoordinate rings. In this sense, Commutative Algebra is closely related toAlgebraic Geometry. By reduction to prime characteristic p, many questionsin Commutative Algebra and in Algebraic Geometry have been answers by using the tight closure theory. For example, the tight closure theory has found its applications in the studies of singularities. Scientific research in Commutative Algebra will not only deepen our understanding in the areaitself, but will also benefit the related branches of Mathematics asCommutative Algebra is in constant interaction with Algebraic Geometry,Homological Algebra, Combinatorics, coding theory and Number Theory, etc. Given the importance of Mathematics in the development of modern sciences, the research shall be of importance to the overall advance of science and technology.

这个项目是在交换代数领域，其中首席研究员（PI）将研究交换诺特环。首先，PI计划研究素特征p的交换Noether环，重点是理想（或者更一般地说，子模）的紧闭包和Frobenius闭包。特别是，PI计划研究的存在性的一个统一的测试指数（紧封闭）的所有模块的有限幻影投影维以及存在性的一个统一的测试指数（Frobenius封闭）的所有理想所产生的系统的参数。为了更好地理解有限幻影投影维度的模块的结构，PI与Mel Hochster合作，计划证明任何这样的模块都可以“弱嵌入”到另一个有限幻影投影维度的模块中，其结构更容易理解。使用这种技术，PI希望证明存在一个统一的测试指数（紧封闭）的所有模的有限幻影投射维数。PI还计划研究某些数字不变量，即F-有理数签名，F-签名和局部素特征环的Hilbert-Kunz重数。每一个不变量都携带着关于环的重要信息，取决于它是正的、零的、大的还是小的。特别地，F-有理数签名是本项目中定义的一个新的不变量，它的正性刻画了环的F-有理数。PI计划开展的其他项目包括希尔伯特-昆兹函数、有限投影维数（或者更一般地说，有限G维）模的嵌入定理以及关于模族线性增长性质的初等分解理论。交换代数是数学的一个分支，它起源于对一组多项式方程解的研究。对解集上的函数的研究导致了对坐标环结构的研究。在这个意义上，交换代数与代数几何有着密切的联系。通过对素特征p的约化，利用紧闭包理论，解决了交换代数和代数几何中的许多问题。例如，紧闭包理论在奇点的研究中得到了应用。对交换代数的科学研究不仅可以加深我们对这一领域本身的认识，而且对数学的相关分支也会有所裨益，因为交换代数与代数几何、同调代数、组合数学、编码论和数论等学科之间存在着不断的相互影响。鉴于数学在现代科学发展中的重要性，对科学技术的全面进步具有重要意义。