Homological Techniques in Commutative Algebra

交换代数中的同调技术

• 批准号: 1003384
• 负责人: Claudia Miller
• 金额: $ 12.5万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1003384&HistoricalAwards=false
• 关键词: Homological; Techniques; Commutative; Algebra

Commutative algebra is crucial in developing the foundations of algebraic geometry. Two central topics of research in these fields are multiplicities and the homological behavior of modules. This proposal concerns questions related to these topics and especially their interplay. The first project involves studying whether Hilbert-Kunz multiplicity, an intriguing and not well-understood measure of the complexity of a singularity in positive characteristic, has an interpretation in characteristic zero where it could be simpler or clearer to interpret geometrically (some known examples show that a limit as the characteristic goes to infinity might exist and seems to be a simpler number). The second concerns the homology of Koszul complexes, and although in this case the approaches are not exclusively from the realm of complete intersection rings, many of the same ideas arise since the Koszul complex is the first step in constructing a Tate resolution. Many directions are being examined, with several new approaches to old problems studied by Huneke, Simis, Vasconcelos and many others. The third concerns an older conjecture in rational homotopy theory, a field with which the ideas from resolutions of complete intersections and from differential graded resolutions have a long history of interaction, but have not yet been used for this particular problem; preliminary results of this approach look encouraging. The fourth concerns Serre's intersection multiplicity over non-regular rings. These projects are at the center of some of the main directions in commutative algebra, but with a view towards neighboring fields of mathematics or subfields of algebra. For example, the first and main project concerns multiplicities, which are numerical measures of the complexity of a non-smooth (that is, sharp) point on a curve, surface, or higher dimensional object. The investigator is especially interested in these in an algebraic or number theoretic setting that is not directly related to the physical geometry of an object. Known results show quite mysterious behavior of these numbers and the investigator hopes to shed some light on the situation by relating the setting to a more geometric one where classical geometric techniques might be applied. This is a delicate procedure and work in progress. Likewise, the third project involves a problem that would apply ideas from her abstract field of algebra to a part of topology, a field which deals with physical spaces. In summary, although the subjects of the projects are quite varied, they hold a common theme that may not be immediately evident: Namely, the investigator proposes to apply her experience in her own area to various problems further removed from this area and not traditionally studied in this way.

交换代数是发展代数几何基础的关键。在这些领域的研究的两个中心议题是多重性和同调行为的模块。这项建议涉及与这些专题有关的问题，特别是它们之间的相互作用。第一个项目涉及研究希尔伯特-昆兹多重性，一个有趣的和不好理解的措施的复杂性的奇点在积极的特点，是否有一个解释的特点零，它可以更简单或更清楚地解释几何（一些已知的例子表明，极限的特点走向无穷大可能存在，似乎是一个更简单的数字）。第二个是关于Koszul复形的同调性，虽然在这种情况下，这些方法并不完全来自完全相交环的领域，但由于Koszul复形是构建泰特解析的第一步，因此出现了许多相同的想法。许多方向正在审查，与一些新的方法来研究老问题的Huneke，Simis，Vasconcelos和许多其他人。第三个问题涉及理性同伦理论中的一个较老的猜想，在这个领域中，完全相交的解决方案和微分分级解决方案的思想有着悠久的相互作用历史，但尚未用于这个特定的问题;这种方法的初步结果看起来令人鼓舞。第四个问题是非正则环上的Serre交重数。这些项目是在交换代数的一些主要方向的中心，但对邻近领域的数学或代数的子领域的看法。例如，第一个和主要项目涉及多重性，这是曲线，曲面或高维对象上非光滑（即尖锐）点的复杂性的数值度量。调查人员特别感兴趣的是这些在代数或数论的设置是不直接相关的物理几何的一个对象。已知的结果表明这些数字的行为相当神秘，研究人员希望通过将设置与可能应用经典几何技术的更几何的设置相关联来阐明这种情况。这是一个微妙的过程和正在进行的工作。同样，第三个项目涉及的问题，将适用于思想从她的抽象领域的代数的一部分拓扑，一个领域，其中涉及物理空间。总之，虽然项目的主题是相当不同的，他们持有一个共同的主题，可能不会立即明显：即，研究人员建议将她在自己的领域的经验，进一步从这个领域，而不是传统上研究的各种问题。