Cores, regularity and principal ideal theorems

核心、正则性和主要理想定理

• 批准号: 0501011
• 负责人: Bernd Ulrich
• 金额: $ 18.8万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0501011&HistoricalAwards=false
• 关键词: Cores; regularity; principal; ideal; theorems

The core of an ideal in a commutative ring encodes information about allpossible reductions of the ideal. It also has a close connection toBriancon-Skoda type theorems and to a conjecture by Kawamata aboutsections of line bundles. The proposer intends to further explore thisinterplay by studying the relation between cores and adjoints ormultiplier ideals. Having worked on a formula for the core inequicharacteristic zero, he wishes to obtain a similar explicit expressionin positive characteristic, where the shape of the core is markedlydifferent. Likewise he would like to find a combinatorial description forthe core of monomial ideals. The investigator plans to continue his workon blowup algebras of ideals, most notably of zero-dimensional ideals inregular local rings. He asks whether the quasi-Gorenstein property of theextended Rees algebra implies the Gorensteinness of the associated gradedring. He also suggests that the normality or Cohen-Macaulayness of thespecial fiber ring of a Gorenstein ideal may force the ideal to be acomplete intersection. On a more computational note, he addresses theproblem of constructing the integral closure of algebras, in particular ofRees algebras. Passing to the integral closure of the Rees algebra of anideal is the first step towards resolution of singularities and the onlyknown general method for computing the integral closure of the ideal. Theproposer wishes to estimate the complexity of this process by finding boundson the number of generators of the integral closure, the degrees of thegenerators and the number of steps required in the computation. As anothermeasure of complexity he plans to study the Castelnuovo-Mumford regularityof powers and symmetric powers of homogeneous ideals having dimension atmost one. He expects that estimates on the regularity do not only persistwhen the ideal is raised to powers, but that they actually improve. Similarimproved bounds for the regularity of symmetric powers would help findingthe equations of Rees algebras and thereby lead to efficient algorithms inelimination theory. The proposer also intends to continue his work ongeneralized principal ideal theorems. The goal is to bound the codimensionand prove connectedness properties for degeneracy loci of maps of modulesthat are not necessarily free; here one has to assume that the maps are not`too generic'. The investigator proposes a weak version of this conditionby introducing a notion of ampleness for modules over local rings. Hehopes to prove principal ideal theorems that only require the weakerassumption, thus generalizing the known results in both local algebra andprojective geometry.The investigator works in Commutative Algebra, an area concerned with thequalitative study of systems of polynomial equations in several variables.Such systems arise in numerous applications outside of mathematics. Overthe past two decades commutative algebraists have become increasinglyinterested in computational aspects, thereby emphasizing connections toapplied areas such as computer algebra, robotics, cryptography and codingtheory. This investigator's research too has a strong computationalcomponent.Part of the project involving the collaboration with mathematicians in Brazil is funded by the NSFOffice of International Science and Engineering

交换环中理想的核心编码有关理想的所有可能约简的信息。它还与 Briancon-Skoda 型定理和 Kawamata 关于线束截面的猜想有密切的联系。提议者打算通过研究核与伴随或乘数理想之间的关系来进一步探索这种相互作用。在研究了核心不等特性零的公式后，他希望在正特性中获得类似的明确表达，其中核心的形状明显不同。同样，他希望找到单项式理想核心的组合描述。研究人员计划继续研究理想的爆炸代数，尤其是正则局部环中的零维理想。他询问扩展里斯代数的拟戈伦斯坦性质是否暗示了相关分级环的戈伦斯坦性。他还认为，戈伦斯坦理想的特殊纤维环的正态性或科恩-麦考利性可能会迫使理想成为完全相交。在更具计算性的方面，他解决了构造代数（特别是里斯代数）的积分闭包的问题。传递到理想里斯代数的积分封闭是解决奇点的第一步，也是计算理想积分封闭的唯一已知通用方法。提议者希望通过找到积分闭包的生成器数量、生成器的度数和计算所需步骤数的界限来估计该过程的复杂性。作为复杂性的另一种衡量标准，他计划研究维数至多为一的同质理想的幂和对称幂的 Castelnuovo-Mumford 正则性。他预计，对规律性的估计不仅会在理想化为幂时持续存在，而且实际上会有所改善。对称幂正则性的类似改进界限将有助于找到里斯代数方程，从而导致消除理论中的有效算法。提议者还打算继续他在广义理想定理方面的工作。目标是限制余维并证明不一定是自由的模块图的简并位点的连通性属性；这里我们必须假设这些地图不是“太通用”。研究人员通过引入本地环上模块的充足性概念，提出了该条件的弱版本。他希望证明只需要弱擦除的主要理想定理，从而推广局部代数和射影几何中的已知结果。研究人员从事交换代数工作，该领域涉及多变量多项式方程组的定性研究。此类系统出现在数学之外的许多应用。在过去的二十年中，交换代数学家对计算方面越来越感兴趣，从而强调与计算机代数、机器人学、密码学和编码理论等应用领域的联系。该研究人员的研究也具有强大的计算成分。该项目涉及与巴西数学家合作的部分内容由美国国家科学基金会国际科学与工程办公室资助