New Methods in Tensor Triangular Geometry

张量三角形几何的新方法

• 批准号: 1600032
• 负责人: Paul Balmer
• 金额: $ 15.8万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600032&HistoricalAwards=false
• 关键词: New; Methods; Tensor; Triangular; Geometry

Tensor triangular geometry, the theme of this research project, bridges a large class of specialized areas of mathematics, including stable homotopy theory, algebraic geometry, modular representation theory, motivic theory, and noncommutative topology. Tensor triangular geometry promises to serve as an overarching theory that facilitates the transposition of techniques and methods from one specialized area to neighboring ones, providing conceptual unification among those fields. Moreover, it displays steadily expanding influence in mathematics. For instance, via tensor triangular geometry, the subject of étale topology in modern algebraic geometry now bears new applications in modular representation theory, where it provides answers to longstanding problems about the relations between representations of finite groups of order a power of a prime number and representations of arbitrary general finite groups. This research project aims to broaden and deepen understanding in the field of tensor triangular geometry.This project studies the geometry of tensor-triangulated categories as they appear in several areas of mathematics. Tensor-triangulated categories are now in common use in algebraic geometry, in modular representation theory, stable homotopy theory, and its equivariant versions, in motivic theory, in equivariant noncommutative topology, and beyond. This project aims to develop new methods in tensor triangular geometry, in order to study those many different incarnations from a unified perspective. In recent years, much progress has been made in étale extensions (i.e., separable and commutative extensions) of tensor-triangulated categories and on the related theory of descent. The present project builds on these and related results and aims to apply descent to connect the algebraic geometry of projective support varieties with the modular representation theory of finite groups. Some applications of these techniques can be spelled out in concrete terms without tensor-triangular technicalities. For instance, in work on endotrivial modules the investigator introduced "weak homomorphisms." It appears that the very same weak homomorphisms are connected to complex line bundles on the Brown simplicial complex of p-subgroups. Further investigation of their role in algebraic geometry is part of the current project. Another general theme of the project is the computation of the "spectrum" of a tensor-triangulated category. A new general approach for such computations comes through filtering tensor-triangulated categories and understanding the successive strata via étale extensions. An early prototype of this method has been applied to compute the spectrum of the equivariant stable homotopy category of a finite group. The project aims to transport those ideas to new examples.

张量三角几何形状是该研究项目的主题，它桥接了许多数学专业领域，包括稳定的同型理论，代数几何，模块化代表理论，主题理论和非交互性拓扑。张量三角几何形状有望作为一种总体理论，促进了从一个专业区域到相邻区域的技术和方法的转换，从而提供了这些领域的概念统一。此外，它在数学中显示出刻板印象的扩展影响。例如，通过张量三角几何形状，现代代数几何形状中的od topology的主题现在在模块化表示理论中具有新的应用，在其中为有限的阶数级表示势力的表示的关系提供了长期问题的答案。该研究项目旨在扩大和加深张量三角几何学领域的理解。该项目研究了张量三角形类别的几何形状，因为它们出现在数学的几个领域。现在，在代数几何形状，模块化表示理论，稳定同义理论及其等效版本中，在基序理论中，在等效的非共同拓扑及其他方面的等效版本中，张量三角的类别在代数几何形状中常用。该项目旨在开发张量三角几何形状的新方法，以便从统一的角度研究许多不同的化身。近年来，在张量 - 三角形类别和相关的下降理论的odale扩展（即单独和交换扩展）中取得了很多进展。本项目以这些和相关的结果为基础，并旨在应用下降以将投影支持变化的代数几何形状与有限群体的模块化表示理论联系起来。这些技术的某些应用可以用具体的术语来阐明，而无需张张三角技术。例如，在内属培训模块的工作中，研究人员引入了“弱同构”。看来，同一弱的同态同态与棕色简单的p子组复合物上的复杂线束相关。他们在代数几何形状中的作用进一步投资是当前项目的一部分。该项目的另一个一般主题是计算张量三角仪类别的“频谱”。通过过滤张张三角量的类别并通过étale扩展了解成功的Stratta，一种新的一般方法是通过过滤张量的三角形类别来实现的。该方法的早期原型已应用于计算有限基团的等效稳定同性恋类别的光谱。该项目旨在将这些想法运送到新的例子上。

项目成果

Relative stable categories and birationality

• DOI: 10.1112/jlms.12474
• 发表时间: 2020-03
• 期刊: Journal of the London Mathematical Society
• 作者: Paul Balmer;Greg Stevenson

Nilpotence theorems via homological residue fields

• DOI: 10.2140/tunis.2020.2.359
• 发表时间: 2017-10
• 期刊: Tunisian Journal of Mathematics
• 影响因子: 0.9
• 作者: Paul Balmer