Local Cohomology and Singularities

局部上同调和奇点

基本信息

  • 批准号:
    1502282
  • 负责人:
  • 金额:
    $ 8.8万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2015
  • 资助国家:
    美国
  • 起止时间:
    2015-09-01 至 2019-08-31
  • 项目状态:
    已结题

项目摘要

The Principal Investigator plans to study geometric problems using different techniques in commutative algebra. This research focuses on spaces given by the set of points that satisfy certain polynomial equations in many variables. Since many phenomena can be described in terms of polynomial equations, these spaces appear in many fields of science and its applications. In such spaces most points are what is called "smooth", which, roughly speaking, means that after zooming in, their vicinity looks like a linear space. For instance, in a sphere every point is smooth and, just as the Earth, from a very close view its neighborhood looks like a plane. Then, those points that are not smooth present a particular behavior and, for that reason, are called "singular points". For instance, a cone has exactly one singular point at its vertex. The set of singular points can be described in terms of the derivatives of the polynomial equations that the points in the space satisfy. For many purposes, detecting singularities is not enough, as some are worse than others. For instance, the sharper vertices of cones are considered worse. To distinguish different singularities, one needs to use more sophisticated algebraic techniques. This research project seeks to study singularities using local cohomology modules, which can be seen as algebraic objects associated to a point. This has already proven to be a powerful tool to detect different kinds of singularities. The Principal Investigator plans to use local cohomology to study measurements of how bad a singular point is. The research includes long-standing problems in the study of singularities as well as new conjectures that could have theoretical and computational consequences. The project involves graduate students in the research. The Principal Investigator seeks to study the structure of local cohomology modules and singularities in positive and mixed characteristic. One of the main problems that one encounters while working with local cohomology modules is that they are usually very large and difficult to handle. However, these modules behave as if they were finitely generated over regular local rings that contain a field. An example of a regular ring in mixed characteristic for which injective dimension behaves differently from equal characteristic was recently found. Motivated by this result, the Principal Investigator intends to explore potential counter-examples for the properties regarding associated primes and Bass numbers of local cohomology modules over regular local rings of mixed characteristic. In addition, the Principal Investigator plans to work on the following related conjecture: the support of a local cohomology module is a Zariski closed set in the spectrum of the ring. Using local cohomology over rings containing a field, Lyubeznik introduced a family of invariants now called Lyubeznik numbers. These invariants have shown several connections with the algebraic and geometric properties of a ring. This inspired an analogous definition of these numbers in mixed characteristic. The project aims to compare the Lyubeznik numbers of rings that contain fields with those that do not. In particular, the research seeks a topological or arithmetic criterion that relates both notions of Lyubeznik numbers. In addition, the project seeks to find geometric properties encoded by the Lyubeznik numbers in mixed characteristic. Lastly, the Principal Investigator plans to work on singularities in positive characteristic via the Frobenius map. In particular, he is planning to work on the ACC conjecture for F-pure thresholds and its corollaries. In addition, the Principal Investigator and a collaborator will investigate a conjectured inequality that relates the F-pure thresholds with the Hilbert-Kunz multiplicities. If this project succeeds, the conjectured relation could have several computational and geometric consequences.
首席研究员计划使用交换代数中的不同技术来研究几何问题。本文主要研究满足多元多项式方程的点集所给出的空间。由于许多现象可以用多项式方程来描述,这些空间出现在科学及其应用的许多领域。在这样的空间中,大多数点都是所谓的“平滑”,粗略地说,这意味着放大后,它们的附近看起来像是一个线性空间。例如,球体中的每个点都是光滑的,就像地球一样,从非常近的地方看,它的邻域看起来像一个平面。然后,那些不光滑的点表示一种特定的行为,因此,称为“奇点”。例如,圆锥体的顶点处恰好有一个奇点。奇点集可以用空间中的点所满足的多项式方程的导数来描述。对于许多目的来说,仅仅检测奇点是不够的,因为一些奇点比另一些更糟糕。例如,圆锥体的顶点越锐利,就被认为越糟糕。要区分不同的奇点,需要使用更复杂的代数技术。这项研究项目试图使用局部上同调模来研究奇点,局部上同调模可以被视为与点相关的代数对象。这已经被证明是检测不同类型奇点的强大工具。首席调查员计划使用局部上同调来研究奇点有多糟糕的测量。这项研究包括奇点研究中长期存在的问题,以及可能产生理论和计算结果的新猜想。该项目邀请研究生参与这项研究。主要研究者试图研究局部上同调模的结构以及正特征和混合特征的奇点。在使用局部上同调模时遇到的主要问题之一是,它们通常非常大,很难处理。然而,这些模的行为就像它们是在包含域的正则局部环上有限生成的。最近发现了一个具有混合特征的正则环,它的内射维度表现出与等价特征不同的行为。受这一结果的启发,主要研究者打算探索混合特征正则局部环上的局部上同调模的相关素数和Bass数性质的潜在反例。此外,首席调查者计划研究以下相关猜想:局部上同调模的支撑是环的谱中的Zariski闭集。利用包含域的环上的局部上同调,Lyubeznik引入了一族不变量,现在称为Lyubeznik数。这些不变量已经显示出与环的代数和几何性质的几个联系。这启发了对这些数字的一个类似的混合特征的定义。该项目旨在比较包含和不包含场的环的Lyubeznik数。特别是,这项研究寻求一种拓扑或算术准则,将Lyubeznik数的两个概念联系起来。此外,该项目寻求寻找由混合特征的Lyubeznik数编码的几何性质。最后,首席调查员计划通过Frobenius映射研究正特征中的奇点。特别是,他正计划研究ACC关于F-纯阈值的猜想及其推论。此外,首席调查员和一名合作者将调查一个猜想的不等式,该不等式将F-纯阈值与希尔伯特-昆兹重数联系起来。如果这个项目成功,推测的关系可能会产生几个计算和几何后果。

项目成果

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Craig Huneke其他文献

Order ideals and a generalized Krull height theorem
  • DOI:
    10.1007/s00208-004-0513-6
  • 发表时间:
    2004-08-24
  • 期刊:
  • 影响因子:
    1.400
  • 作者:
    David Eisenbud;Craig Huneke;Bernd Ulrich
  • 通讯作者:
    Bernd Ulrich
Upper bound of multiplicity of F-rational rings and F-pure rings
F-有理环和 F-纯环的重数上限
Good ideals of 2-dimensional normal singularities
二维正态奇点的良好理想
  • DOI:
  • 发表时间:
    2013
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Craig Huneke;. Kei-ichi Watanabe;Kei-ichi Watanabe
  • 通讯作者:
    Kei-ichi Watanabe
The projective dimension of codimension two algebras presented by quadrics
  • DOI:
    10.1016/j.jalgebra.2013.06.038
  • 发表时间:
    2013-11-01
  • 期刊:
  • 影响因子:
  • 作者:
    Craig Huneke;Paolo Mantero;Jason McCullough;Alexandra Seceleanu
  • 通讯作者:
    Alexandra Seceleanu
Multiplicity bounds in graded rings
分级环中的重数界限
  • DOI:
  • 发表时间:
    2011
  • 期刊:
  • 影响因子:
    0.6
  • 作者:
    Craig Huneke;S. Takagi;Kei-ichi Watanabe
  • 通讯作者:
    Kei-ichi Watanabe

Craig Huneke的其他文献

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{{ truncateString('Craig Huneke', 18)}}的其他基金

Uniformity in Commutative Algebra
交换代数的一致性
  • 批准号:
    1460638
  • 财政年份:
    2015
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Studies in Commutative Algebra
交换代数研究
  • 批准号:
    1259142
  • 财政年份:
    2012
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Studies in Commutative Algebra
交换代数研究
  • 批准号:
    1063538
  • 财政年份:
    2011
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Travel support for an ICTP workshop
ICTP 研讨会的差旅支持
  • 批准号:
    1001133
  • 财政年份:
    2010
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Standard Grant
Topics in Commutative Algebra
交换代数主题
  • 批准号:
    0756853
  • 财政年份:
    2008
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Homological Methods and Ideal Closures in Commutative Algebra
交换代数中的同调方法和理想闭包
  • 批准号:
    0244405
  • 财政年份:
    2003
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Problems in Commutative Algebra
交换代数问题
  • 批准号:
    0098654
  • 财政年份:
    2001
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Characteristic p Methods in Commutative Algebra
交换代数中的特征 p 方法
  • 批准号:
    9996155
  • 财政年份:
    1999
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Characteristic p Methods in Commutative Algebra
交换代数中的特征 p 方法
  • 批准号:
    9731512
  • 财政年份:
    1998
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant
Mathematical Sciences: "Uniform Bounds in Noetherian Rings, The Theory of Tight Closure, and Big Cohen-Macaulay Algebras"
数学科学:“诺特环的一致界、紧闭理论和大科恩-麦考利代数”
  • 批准号:
    9301053
  • 财政年份:
    1993
  • 资助金额:
    $ 8.8万
  • 项目类别:
    Continuing Grant

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Dual complexes and weight filtrations: Applications to cohomology of moduli spaces and invariants of singularities
对偶复形和权重过滤:模空间上同调和奇点不变量的应用
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    0901123
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    2009
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Algebraic analysis of algebraic local cohomology and computational complex analysis of non-isolated singularities
代数局部上同调的代数分析和非孤立奇点的计算复分析
  • 批准号:
    21540167
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Torus actions, cohomology and singularities of Schubert varieties
舒伯特簇的环面作用、上同调和奇点
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