Mathematical Sciences: Rees Algebras, Associated Graded Rings, and Multiple Points of Finite Morphisms

9623259这项建议支持B.Ulrich教授在交换代数和代数几何问题上的工作。他打算研究Rees代数及其相关的分次环。特别是，这项拟议的研究调查了与理想环相关的分次环如何与Cohen-Macaulay性质交织在一起。他还打算研究围绕确定某些因子环的经典并簇和割簇的经典问题。最后，他打算研究与完全R模和某些Cohen-Macaulay环之间的有限态射的源和目标有关的方案。这项研究是关于交换代数和代数几何之间的边界的。这两个领域都起源于研究平面上由最简单的方程定义的数学对象的经典问题，即多项式。今天，这些领域不仅使用代数中的方法，而且还使用分析和拓扑学中的方法，相反，在这些领域中被广泛使用。此外，它在物理、理论计算机科学、密码学、编码理论和机器人学等领域都证明了自己的有用。

Ulrich 9623259 This proposal supports the work of Professor B. Ulrich to work on problems in commutative algebra and algebraic geometry. he intends to study the Rees algebra and its associated graded ring. In particular the proposed research investigates how graded rings associated with an ideal intertwine with the Cohen-Macaulay property. He also intends to study problems that circle around the classical problems of determining the classical join and secant varieties for certain factor rings. Finally, he intends to study the scheme associated to the source and target of finite morphisms between perfect R modules and certain Cohen-Macaulay rings. This research is on the boundary between commutative algebra and algebraic geometry. Both fields arose from the classical problem of studying mathematical objects defined in the plane by the simplest of equations, namely polynomials. Today, the fields use methods not only from algebra, but also from analysis and topology, and conversely are extensively used in those fields. Moreover it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.