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

交换环中理想的核编码了关于理想的所有可能约化的信息。它也有一个密切的联系Briancon-Skoda型定理和猜想由Kawamata aboutsections线丛。作者试图通过研究核与伴随或乘子理想之间的关系来进一步探讨这种相互作用。在研究了核心不平等特征零的公式之后，他希望得到一个类似的明确的正特征表达式，其中核心的形状明显不同。同样，他想找到一个组合描述为核心的单项理想。调查计划继续他的工作爆破代数的理想，最显着的零维理想不规则的地方环。他问是否准Gorenstein财产的扩展里斯代数意味着Gorenstein的相关gradedring。他还指出，Gorenstein理想的特殊纤维环的正规性或Cohen-Macaulay性可能迫使理想成为完全交。在一个更计算注意，他解决了问题的建设积分封闭代数，特别是里斯代数。传递到一个理想的Rees代数的积分闭包是解决奇点的第一步，也是唯一已知的计算理想的积分闭包的一般方法。该提议者希望估计这个过程的复杂性，通过发现boundson的生成元的数量的积分闭包，thegenerators的程度和在计算中所需的步骤数。作为另一种复杂性的措施，他计划研究Castelnuovo-Mumford regularity的权力和对称权力的齐次理想的维数最多为1。他预计，对规律性的估计不仅在理想的幂次上保持不变，而且实际上会有所改善。对称幂的正则性的类似改进的界将有助于发现Rees代数的方程，从而导致消除理论的有效算法。提出者还打算继续他的工作对广义主理想定理。我们的目标是约束余维和证明连通性的性质退化轨迹的地图modulesthat不一定是免费的;在这里，人们必须假设的地图是不是“太一般”。研究者通过引入局部环上模的满度的概念，提出了这个条件的一个弱版本。他希望证明主要理想定理，只需要较弱的假设，从而推广了已知的结果在本地代数和射影几何。调查员工作在交换代数，一个领域关注的定性研究系统的多项式方程在几个变量。这样的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，从而强调与应用领域的联系，如计算机代数，机器人，密码学和编码理论。这个研究者的研究也有很强的计算能力。这个项目的一部分是由美国国家科学基金会国际科学与工程办公室资助的，