Cores, regularity and principal ideal theorems

交换环中理想的核心编码有关理想的所有可能约化的信息。它还与Briancon-Skoda型定理和Kawamata关于线丛的截面的猜想有密切的联系。作者打算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。在研究了核的不等特征零点的公式之后，他希望在核的形状明显不同的正特征中得到类似的显式表达式。同样，他希望找到单项理想核心的组合描述。这位研究人员计划继续他在理想的爆破代数方面的工作，最著名的是正则局部环中的零维理想。他问扩展的Rees代数的拟Gorenstein性质是否蕴含着相关等级映射的Gorenstein性。他还指出，Gorenstein理想的特殊纤维环的正规性或Cohen-Macaulay性可能迫使该理想成为完全交集。在更具计算性的方面，他讨论了构造代数的积分闭包的问题，特别是Rees代数的积分闭包的构造问题。通过理想的Rees代数的积分闭包是奇点分解的第一步，也是唯一已知的计算理想的积分闭包的一般方法。提出者希望通过求积分闭包的生成元数、生成元阶和计算所需步数的界来估计这个过程的复杂性。作为复杂性的另一种度量，他计划研究具有至多一维的齐次理想的幂和对称幂的Castelnuovo-Mumford规则。他预计，对这种规律性的估计不仅在理想被提升为强国时会持续下去，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寻找Rees代数的方程，从而导致消去论中的有效算法。作者还打算继续推广主理想定理的工作。我们的目标是限制余维，并证明不一定是自由的模块的退化轨迹的连通性；在这里，人们必须假设这些映射不是“太一般的”。通过引入局部环上模的幅度的概念，研究者提出了这一条件的一个弱版本。他希望证明只需要较弱假设的主理想定理，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，这一领域涉及对多元多项式方程组的定性研究。这种系统出现在数学之外的许多应用中。在过去的二十年里，交换代数学家对计算方面越来越感兴趣，从而强调了与计算机代数、机器人学、密码学和编码理论等应用领域的联系。这位研究人员的研究也有很强的计算成分。该项目的一部分涉及与巴西数学家的合作，由NSF国际科学和工程办公室资助

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