Topics in Commutative Algebra

这个项目是在交换代数领域，特别是诺特代数和更特别的多项式环域。在这个建议的研究是针对理解方程的渐近性和他们的约简模大素数。这种渐近性是通过局部上同调，Hilbert-Kunz多重性，符号幂和紧闭包来捕获的。所提出的方法部分是经典方法，也是提出者正在开发的方法。该项目还研究同调代数，特别是在分次贝蒂数和有限科恩-麦考利型环方面。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，特别是单项式猜想。在联络理论上提出了额外的问题，特别是关于比较局部和齐次版本，以及理解Gorenstein联络类。交换代数起源于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化的思想。交换代数通过形成一个代数对象（称为环）来研究多项式或幂级数方程的解，该代数对象由“通用”解组成。这些通用解的代数性质然后让我们深入了解解的几何和代数性质。在这一领域的一个重要技术已研究这样的方程，减少系数模素数的所有大素数。这方面的一个特别的例子是在过去的二十年里紧封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。

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.