Topics in Commutative Algebra

这个项目是在交换代数的领域，特别是noether代数，特别是多项式环在域上。本文的研究方向是理解方程的渐近性及其对大素数的约化模。这种渐近性是通过局部上同调、Hilbert-Kunz多重性、符号幂和紧闭性来捕获的。所提出的方法部分是经典的方法，以及那些正在发展的提议。本项目还研究了同调代数，特别是关于分级Betti数和有限Cohen-Macaulay型环的同调代数。提出了一个新的局部上同调的渐近长度函数，并将其应用于同调猜想，特别是单项式猜想。本文还提出了关于联络理论的其他问题，特别是关于比较局部版本和同构版本，以及理解Gorenstein联络类的问题。交换代数起源于19世纪对多变量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平面坐标的思想。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，该代数对象由“一般”解组成。这些泛型解的代数性质可以让我们深入了解解的几何和代数性质。该领域的一项重要技术是通过对所有大素数的模素数进行系数化简来研究这类方程。一个特别的例子是近二十年来紧闭理论的爆炸性发展。交换代数结合了许多其他领域的技术，包括组合学、拓扑学和分析学。

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.