Tight closure and primary decomposition in Commutative Algebra

这个项目是在交换代数领域，其中负责人(PI)将研究交换Notherian环。首先，PI计划研究素数特征p的交换Notherian环，重点研究理想(或者说，子模)的紧闭包和Frobenius闭包。特别是，PI计划研究有限元射影维模的一致检验指数(对于紧闭包)的存在性，以及由参数系统生成的所有理想的统一检验指数(对于Frobenius闭包)的存在性。为了更好地理解有限幻影射影维模的结构，PI与Mel Hochster合作，计划证明任何这样的模都可以弱嵌入到另一个结构更容易理解的有限幻影射影维模中。利用这一技巧，PI希望证明有限幻射维模的一致测试指数的存在性(对于紧闭包)。PI还计划研究某些数值不变量，即具有素数特征的局部环的F-有理签名、F-签名和Hilbert-kunz重数。每个不变量都携带着关于环的重要信息，这取决于环是正的、零的、大的还是小的。特别地，无理签名是本课题中定义的一个新的不变量，它的正性刻画了环的F-有理性质。PI计划实施的其他项目包括希尔伯特-昆兹函数、有限投射维模的嵌入定理(更广泛地说，有限G维模)以及关于模族的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，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同轴环的结构的研究。从这个意义上说，交换代数与代数几何密切相关。通过约化为素数特征p，用紧闭包理论回答了交换代数和代数几何中的许多问题。例如，紧闭包理论在奇点研究中得到了应用。交换代数的科学研究不仅加深了我们对这一领域的认识，而且由于交换代数与代数几何、同调代数、组合数学、编码论、数论等学科的不断互动，对数学相关学科的研究也是有益的。鉴于数学在现代科学发展中的重要性，这项研究对科学技术的全面进步具有重要意义。

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.