ETALE TOPOLOGY IN TENSOR TRIANGULAR GEOMETRY

张量三角形几何中的ETALE拓扑

• 批准号: 1303073
• 负责人: Paul Balmer
• 金额: $ 34.74万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303073&HistoricalAwards=false
• 关键词: ETALE; TOPOLOGY; TENSOR; TRIANGULAR; GEOMETRY

This project is part of the PI's long-term program of "Tensor Triangular Geometry," which studies the geometry of tensor triangulated categories as they occur throughout mathematics. The specific goal of the present project is to expand etale topology from algebraic geometry into tensor triangular geometry. The starting point is the observation that restriction of modular representations from a finite group to a subgroup is nothing but an extension-of-scalars, with respect to a suitable ring object. Moreover, that ring object is commutative and separable and compact (finite dimensional), so it deserves to be called a "tensor-triangular etale" ring object. In short, such enhanced etale extensions not only generalize the classical etale extensions of algebraic geometry but also cover restriction to subgroups in representation theory. In algebraic geometry, the power of extension-of-scalars resides in the ability to apply descent theory. Extending descent to tensor triangular geometry and specializing to representation theory yields a machinery which allows one to decide when modular representations of a p-group extend to an arbitrary finite group containing our p-group as a Sylow subgroup. In modular representation theory of finite groups, abstract tensor triangular etale topology materializes into the so-called "sipp topology" on the category of finite G-sets and leads to stacks of derived and stable categories. These ideas then lead to (sipp) cohomology theory, which is one of the main avenues of the present project. Sipp cohomology theory can be related to classical group cohomology, to classical algebro-geometric etale cohomology and therefore to Galois cohomology. The first concrete range of application is the description of endotrivial representations over arbitrary finite groups, extending from the Carlson-Thevenaz classification over p-groups. Beyond this first major achievement, the sipp topology is actually perfectly suited for treating gluing problems from p-local to global. This is particularly interesting for derived and stable categories, in view of a series of long-standing conjectures, like Broue's Abelian Defect Group Conjecture for instance.One of the driving forces of research in Fundamental Sciences, including Mathematics, is the constant attempt to unify specialized theories into more fundamental principles. The first goal is the discovery of deeper scientific beauties but the second, more collective, goal is the transposition of techniques and methods from one specialized area to other neighboring ones, thus creating new applications and further interaction and innovation. Tensor Triangular Geometry covers a large class of specialized areas of Mathematics, ranging from Algebra to Analysis: It appears in Algebraic Geometry, in Modular Representation Theory, in Stable Homotopy Theory, in Motivic Theory, in Noncommutative Topology, and more. Tensor Triangular Geometry provides a growing number of new theorems and great conceptual unification between those fields. Moreover, it displays a steadily expanding collection of applications. For instance, via Tensor Triangular Geometry, the deep beauty of Grothendieck's famous etale topology, with its origins in Galois theory and its full manifestations in modern Algebraic Geometry, now surfaces again in Modular Representation Theory. There, it provides answers to long-standing problems about the relations between representations of finite groups of order a power of a prime number and representations of arbitrary general finite groups.

该项目是 PI 长期计划“张量三角几何”的一部分，该计划研究数学中出现的张量三角类别的几何形状。本项目的具体目标是将 etale 拓扑从代数几何扩展到张量三角几何。出发点是观察到，从有限群到子群的模表示的限制只不过是相对于合适的环对象的标量扩展。而且，该环对象是可交换的、可分离的、紧致的（有限维），因此它值得被称为“张量三角 etale”环对象。简而言之，这种增强的 etale 扩展不仅概括了代数几何的经典 etale 扩展，而且还涵盖了表示论中对子群的限制。在代数几何中，标量扩张的力量在于应用下降理论的能力。将下降扩展到张量三角几何并专门研究表示论，产生了一种机制，允许人们决定 p 群的模表示何时扩展到包含我们的 p 群作为 Sylow 子群的任意有限群。在有限群的模表示论中，抽象张量三角etale拓扑具体化为有限G集范畴上的所谓“sipp拓扑”，并导致派生的稳定范畴的堆栈。这些想法随后引出了（sipp）上同调理论，这是当前项目的主要途径之一。 Sipp 上同调理论可以与经典群上同调、经典代数几何 etale 上同调相关，从而与伽罗瓦上同调相关。第一个具体的应用范围是任意有限群上的内琐表示的描述，从 p 群上的 Carlson-Thevenaz 分类扩展而来。除了这第一个重大成就之外，sipp 拓扑实际上非常适合处理从 p 局部到全局的粘合问题。考虑到一系列长期存在的猜想，例如布鲁尔的阿贝尔缺陷群猜想，这对于派生范畴和稳定范畴尤其有趣。包括数学在内的基础科学研究的驱动力之一是不断尝试统一将专业理论转化为更基本的原则。第一个目标是发现更深层次的科学之美，但第二个更集体的目标是将技术和方法从一个专业领域转移到其他邻近领域，从而创造新的应用以及进一步的互动和创新。张量三角几何涵盖了从代数到分析的一大类数学专业领域：它出现在代数几何、模表示理论、稳定同伦理论、动机理论、非交换拓扑等等中。张量三角几何提供了越来越多的新定理以及这些领域之间的伟大概念统一。此外，它还显示了不断扩大的应用程序集合。例如，通过张量三角几何，格洛腾迪克著名的 etale 拓扑的深刻之美，起源于伽罗瓦理论，并在现代代数几何中得到充分体现，现在在模表示理论中再次浮现出来。在那里，它为有关素数次方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。