Powers of Ideals

9970566在这个建议中的项目的共同线程是一个给定的理想的普通，Frobenius和符号幂的渐近性质。 背景动机是仍然悬而未决的问题，即严格封闭是否与本地化相一致。 提议者计划使用和进一步发展的主要工具是一大类赋值的里斯-泉型比较，以及理想幂的上核的初等分解和素过滤。 Rees-Izumi部分将继续与Reinhold Hubl合作完成。 主要的分解和主要的过滤部分是有关Huneke的阿廷里斯引理，并将包括合作与安娜Guerrieri对主要分解的特殊雅可比理想。 在仿射环的情形下，理想的Frobenius幂的渐近性质也可以通过Grobner基的次数的有界性来研究。 这一部分将与苏珊·赫米勒合作完成。 与寻找计算伴随的算法有关的部分将与提议者的研究生马克罗兹一起进行。 塞尔坎Hocsten，Bernd Sturmfels和提议者计划工作的稳定性的决议和相关的素数的权力monomial ideals.This建议是在交换代数领域的代数几何和计算代数中的应用。 主要的背景动机是证明一些关于紧闭包的基本性质，但所提出的技术也会在紧闭包理论之外产生影响。 紧封闭是一个年轻的理论，由于梅尔文Hochster和克雷格Huneke已在很短的时间内证明了许多新的结果，交换代数和代数几何，此外缩短和简化了几个旧的结果。 这个建议的主要研究对象是理想，特别是以某种自然方式相互关联的理想集，如普通的，Frobenius或特定理想的象征力量。 这个建议的主要目标是建立这些理想集的某些渐近性和稳定性。

9970566The common thread for the projects in this proposal are asymptotic properties of ordinary, Frobenius, and symbolic powers of a given ideal. The background motivation is the still open question of whether tight closure commutes with localization. The main tools that the proposer plans to use and develop further are the Rees-Izumi type comparisons of a large class of valuations, as well as the primary decompositions and the prime filtrations of the cokernels of the powers of an ideal. The Rees-Izumi part would be done in continued collaboration with Reinhold Hubl. The primary decompositions and prime filtrations part is related to Huneke's Artin-Rees lemma and would include collaboration with Anna Guerrieri on primary decompositions of special Jacobian ideals. In the case of affine rings, the asymptotic properties of Frobenius powers of ideals can also be studied via boundedness of the degrees of Grobner bases. This part would be done in collaboration with Susan Hermiller. The part concerned with finding algorithms for calculating adjoints would be undertaken with the proposer's graduate student Mark Rhodes. Serkan Hocsten, Bernd Sturmfels and the proposer plan to work on the stabilization properties of resolutions and of associated primes of powers of monomial ideals.This proposal is in the area of commutative algebra with applications in algebraic geometry and computational algebra. The main background motivation is proving some basic properties about tight closure, but the proposed techniques would have implications also outside the tight closure theory. Tight closure is a young theory due to Melvin Hochster and Craig Huneke which has in a short time proved many new results in commutative algebra and algebraic geometry and has in addition shortened and simplified several old results. The predominant object of study of this proposal are ideals, and especially sets of ideals which are related to each other in some natural way, such as ordinary, Frobenius, or symbolic powers of a particular ideal. The main goal of this proposal is then establishing certain asymptotic and stabilization properties of these sets of ideals.