Cores, regularity and principal ideal theorems

交换环中理想的核编码关于理想的所有可能约简的信息。它也与briancon - skoda型定理和Kawamata关于线束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法器理想之间的关系来进一步探讨这种相互作用。在研究了磁芯非等征零的公式后，他希望在磁芯形状明显不同的正征中得到类似的显式表达式。同样，他也想找到一种组合的描述，来描述单项理想的核心。这位研究者计划继续他的理想膨胀代数的研究，最著名的是零维理想不规则局部环。他问扩展Rees代数的拟gorenstein性质是否暗示了相关分级的gorenstein性。他还提出，戈伦斯坦理想的特殊纤维环的正态性或科恩-麦考利性可能迫使理想成为完全相交。在一个更计算的笔记中，他解决了构造代数的积分闭包的问题，特别是rees代数。过渡到理想的Rees代数的积分闭包是解决奇点的第一步，也是计算理想的积分闭包的唯一已知的一般方法。作者希望通过寻找积分闭包的生成器数量、生成器的程度和计算所需的步数的边界来估计该过程的复杂性。作为复杂性的另一个度量，他计划研究卡斯泰尔诺沃-芒福德幂和维数不超过1的齐次理想的对称幂的正则性。他预计，当理想被提升到权力时，对规律性的估计不仅会持续下去，而且会实际上得到改善。对对称幂正则性的类似改进将有助于找到里斯代数方程，从而导致消除理论中有效的算法。该提议者还打算继续他的工作，在广义的主要理想定理。目标是约束非自由模映射的简并轨迹的协维数并证明其连通性；在这里，人们必须假设这些地图不是“太通用”。研究者通过引入局部环上模的丰度概念，提出了这个条件的一个弱版本。他希望证明只需要较弱假设的主要理想定理，从而推广局部代数和射影几何中的已知结果。研究者研究交换代数，这是一个研究多变量多项式方程组定性研究的领域。这样的系统出现在数学以外的许多应用中。在过去的二十年里，交换代数学者对计算方面越来越感兴趣，因此强调与计算机代数、机器人、密码学和编码理论等应用领域的联系。这位研究者的研究也有很强的计算成分。与巴西数学家合作的部分项目由nsf国际科学与工程办公室资助

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