Tight closure and primary decomposition in Commutative Algebra

这个项目是在交换代数领域，其中主要研究者（PI）将研究交换诺etherian环。首先，pi计划研究素特征p的交换noether环，重点研究理想的紧闭包和Frobenius闭包（或者更一般地说，子模）。特别是，PI计划研究所有有限幻影投影维的模块的一致测试指数（紧闭包）的存在性，以及由参数系统生成的所有理想的一致测试指数（Frobenius闭包）的存在性。为了更好地理解有限幻像投影维模块的结构，PI与Mel Hochster合作，计划证明任何这样的模块都可以“弱嵌入”到另一个结构更容易理解的有限幻像投影维模块中。使用这种技术，PI希望证明对于所有有限幻射影维数的模都存在一个一致的测试指数（对于紧闭）。PI还计划研究某些数值不变量，即素数特征局部环的f -有理签名、f -签名和hilbert - kunz多重性。每个不变量都携带着环的重要信息，这取决于它是正的、零的、大的还是小的。特别地，off -有理签名的概念是本项目中定义的一个新的不变量，它的正性表征了环的f -合理性。PI计划开展的其他项目包括Hilbert-Kunz函数，有限射影维（或者更一般地说，有限g维）模块的嵌入定理，以及关于模块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的解的研究。对解集上的函数的研究导致了对坐标环结构的研究。从这个意义上说，交换代数与代数几何密切相关。通过约简到素数特征p，交换代数和代数几何中的许多问题都可以用紧闭包理论得到解答。例如，紧闭理论已经在奇点的研究中得到了应用。交换代数与代数几何、同调代数、组合学、编码论、数论等学科相互影响，对交换代数的科学研究不仅可以加深我们对交换代数本身的理解，而且对数学的相关分支也有好处。鉴于数学在现代科学发展中的重要性，对数学的研究对科学技术的全面进步也应具有重要意义。

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.