The structure of free resolutions in commutative algebra and algebraic geometry

交换代数和代数几何中自由解析的结构

• 批准号: 1302057
• 负责人: Daniel Erman
• 金额: $ 13.26万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1302057&HistoricalAwards=false
• 关键词: structure; free; resolutions; commutative; algebra

The PI's research aims to provide a deeper understanding of the structure of free resolutions and their use algebra and geometry via several specified projects. One such project will continue the systematic development of Boij-Soederberg theory. This theory was recently developed by Eisenbud and Schreyer, and it provides results for understanding free resolutions in terms of specified atomic building blocks. This is a powerful new theory, and there remain many open questions about both the foundations and the reach of the theory. Another project aims to develop a new framework for homological algebra in the context of smooth toric varieties. The work of David Cox and others suggests the potential for a far-reaching algebraic/geometric dictionary that largely parallels the powerful dictionary between the algebra of the graded polynomial ring and the geometry of projective space. Yet many of the results from homological commutative algebra have no satisfying analogue in the context of toric varieties. A more robust homological theory would rely on connections between multigraded commutative algebra, algebraic geometry, and combinatorics. A third project involves the asymptotic structure of the free resolutions of high degree Veronese subrings. This aims to fill a gap in the literature: whereas the study of the free resolutions of high degree Veronese subrings for curves has been fruitful and widely applied, little is known about the free resolutions of higher dimensional varieties under a very positive embedding. Free resolutions are built from matrices of polynomials. They arise naturally in many algebraic contexts, with connections to topics in commutative algebra, algebraic geometry, computational algebra, and more. Matrices of polynomials have a richer structure than the matrices of scalars that arise in linear algebra, and so basic questions about free resolutions remain unresolved. The PI's research projects aim to provide overarching structural results about free resolutions, and to apply these new structural insights to open problems in commutative algebra and algebraic geometry. These research projects all offer potential for explicit computations, providing research opportunities for undergraduate and graduate students, as well as leading to new software packages to be developed for Macaulay2. The PI will also continue his outreach projects on K-12 education, which often involve collaborative efforts between research mathematicians, educators, and public or nonprofit groups.

PI的研究旨在通过多个指定项目更深入地了解自由决议的结构及其使用代数和几何形状。一个这样的项目将继续进行Boij-Soederberg理论的系统发展。 该理论最近是由Eisenbud和Schreyer开发的，它提供了根据指定的原子构建块理解自由决议的结果。 这是一个有力的新理论，关于理论的基础和覆盖范围仍然存在许多开放的问题。另一个项目旨在在光滑的感谢您的品种的背景下为同源代数开发一个新的框架。 大卫·考克斯（David Cox）和其他人的作品表明，有可能具有深远的代数/几何字典，这在很大程度上与分级多项式环的代数与投射空间的几何形状之间的强大词典相同。 然而，在复曲面的背景下，同源交换代数的许多结果都没有令人满意的类似物。 更强大的同源理论将依赖于多层交换代数，代数几何和组合学之间的联系。 第三个项目涉及高级佛罗伦斯子弹的自由分辨率的渐近结构。这旨在填补文献中的空白：虽然研究曲线的高度佛罗尼斯子弹的自由分辨率的研究是富有成果的，并且对较高尺寸品种在非常阳性的嵌入下的自由分辨率知之甚少。 免费分辨率是由多项式矩阵构建的。 它们在许多代数环境中自然出现，并与交换代数，代数几何，计算代数等主题有联系。 多项式的矩阵比线性代数中出现的标量矩阵具有更丰富的结构，因此有关自由分辨率的基本问题仍未解决。 PI的研究项目旨在提供有关自由决议的总体结构结果，并将这些新的结构见解应用于交换代数和代数几何形状的开放问题。 这些研究项目都为明确的计算提供了潜力，为本科生和研究生提供了研究机会，并为Macaulay2开发了新的软件包。 PI还将继续他关于K-12教育的推广项目，该项目通常涉及研究数学家，教育工作者以及公共或非营利组织之间的合作努力。