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

通勤环中理想的核心编码有关理想的可取性减少的信息。它还具有紧密的连接tobriancon-skoda型定理，并与川田束的猜想有关。提议者打算通过研究核心与伴随或杀变理想之间的关系来进一步探索此讲习班。在为核心核酸菌零的公式工作后，他希望获得类似的显式表达在阳性特征中，其中核心的形状明显不同。同样，他想为单一理想的核心找到组合描述。调查人员计划继续他的工作原理爆炸代数，最著名的是零维理想在本地环中的零维理想。他询问Theextended Rees代数的准戈伦斯坦特性是否意味着相关分级的戈伦斯坦。他还建议，戈伦斯坦理想的特征纤维环的正态性或cohen-racaulness可能迫使理想是完全完整的交集。从更加计算的说明中，他介绍了构建代数的整体封闭的问题，尤其是代数代数。传递到Anideal的REES代数的整体封闭是迈向奇异性的第一步，也是计算理想整体封闭的唯一一般方法。 Proposer希望通过找到BONDSON的积分闭合的发电机数量，代发电机的程度以及计算中所需的步骤数来估算此过程的复杂性。作为复杂性的另一种措施，他计划研究Castelnuovo-Mumford的规律性和对称能力的同质理想，具有最大的尺寸。他期望对规律性的估计不仅持续到理想提高到力量时，而且实际上会提高。相似的对称能力规则性的界限将有助于找到REES代数的方程，从而导致有效的算法不受欢迎的理论。提议者还打算继续他的综合主要理想定理。目的是绑定模块化地图的变性基因座的Codimension和证明连接性属性不一定是免费的。在这里，必须假设地图不是“通用”。研究人员提出了这种条件的薄弱版本，引入了模块比本地环的增强性概念。 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.在过去的二十年中，可交换的代数主义者已经越来越吸引计算方面，从而强调了与计算机代数，机器人，密码学和编码理论等领域的连接。这项研究者的研究也具有强大的计算能力。