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.

张量三角几何是该研究项目的主题，它跨越了一大类数学专业领域，包括稳定同伦理论、代数几何、模表示论、动机理论和非交换拓扑，有望成为一种总体理论。它促进了技术和方法从一个专业领域到邻近领域的转换，提供了这些领域之间的概念统一，例如，通过张量三角形，它在数学中的影响力不断扩大。几何学是现代代数几何中的基本拓扑学科，现在在模表示论中有了新的应用，它为有关素数次幂的有限群表示与任意一般有限群表示之间关系的长期存在的问题提供了答案该研究项目旨在扩大和加深对张量三角形几何领域的理解。该项目研究张量三角形类别的几何形状，因为它们出现在数学的多个领域中。代数几何、模表示论、稳定同伦理论及其等变版本、动机理论、等变非交换拓扑等。该项目旨在开发张量三角几何的新方法，以便研究这些不同的形式。近年来，张量三角范畴的étale扩展（即可分离扩展和可交换扩展）以及相关的研究取得了很大进展。本项目建立在这些和相关结果的基础上，旨在应用下降将射影支持簇的代数几何与有限群的模表示理论联系起来，这些技术的一些应用可以在没有张量的情况下以具体术语阐明。 - 三角技术细节，例如，在研究微内模时，研究者引入了“弱同态”。对它们在代数几何中的作用的进一步研究是该项目的另一个主题是张量三角范畴的“谱”的计算，这种计算的新通用方法是通过过滤。张量三角化类别并通过 étale 扩展理解连续层。该方法的早期原型已应用于计算有限群的等变稳定同伦类别的谱。这些想法到新的例子。

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