Topics in Commutative Algebra

本课题主要研究交换代数，特别是Notherian代数，特别是域上的多项式环，主要研究模为大素数的方程的渐近性及其约化。这种渐近性是通过局部上同调、希尔伯特-昆兹重数、符号幂和紧闭来捕捉的。所提出的方法部分是经典的方法，也是由提出者发展的方法。该项目还研究同调代数，特别是关于分次Betti数和有限Cohen-Macaulay型环的研究。提出了一种新的局部上同调的渐近长度函数，并将其应用于同调猜想，特别是单项猜想。在联络理论上提出了其他问题，特别是关于比较局部的和齐次的版本，以及理解Gorenstein联络类。交换代数产生于19世纪对多元多项式方程及其解的研究。多项式方程和几何之间的关系至少可以追溯到笛卡尔和坐标平面的思想。交换代数通过形成一个称为环的代数对象来研究这种多项式或幂级数方程的解，环由‘一般’解组成。这些一般解的代数性质使我们深入了解这些解的几何和代数性质。这一领域的一项重要技术是通过降低所有大素数的模素数系数来研究这类方程。一个特别的例子是，在过去的二十年里，紧密封闭理论得到了爆炸性的发展。交换代数结合了许多其他领域的技术，包括组合学、拓扑学和分析。

This project is in the field of commutative algebra, especially Noetherian algebras and more especially polynomial rings over fields.The research in this proposal is directed at understanding the asymptotics of equations and their reduction modulo large prime numbers. Such asymptotics are captured through local cohomology, Hilbert-Kunz multiplicities, symbolic powers, and tight closure.The methods proposed are in part classical methods as well as those being developed by the proposer. The project also studies homological algebra, especially in terms of the graded Betti numbers and in terms of rings of finite Cohen-Macaulay type. A new asymptotic length function for local cohomology is proposed with applications to the homological conjectures, especially the monomial conjecture. Additional problems are proposed on the theory of liaison, especially regarding comparing the local and homogeneous versions, as well as understanding Gorenstein liaison classes.Commutative algebra arose from the 19th century study of polynomial equations in many variables, and their solutions. The relationship between polynomial equations and geometry goes back at least to Descartes and the idea of coordinatizing the plane.Commutative algebra studies the solutions of such polynomial or power series equations by forming an algebraic object, called a ring, which consists of the 'generic' solutions.The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the solutions. An important technique in this field has been to study such equations by reducing the coefficients modulo prime numbers for all large primes. A particular example of this has been the explosive development of the theory of tight closure over the last twenty years. Commutative algebra combines techniques from a number of other areas including combinatorics, topology, and analysis.