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，该计划研究张量三角范畴的几何，因为它们在整个数学中出现。本课题的具体目标是将代数几何中的代数拓扑推广到张量三角几何中。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标量的扩展，关于一个合适的环对象。此外，这个环对象是可交换的、可分的和紧致的（有限维），所以它应该被称为“张量三角形环”环对象。简而言之，这种增强的代数扩张不仅推广了代数几何中的经典代数扩张，而且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。扩展下降到张量三角几何和专门的表示论产生了一个机器，它允许一个决定当模块表示的p-群扩展到任意有限群包含我们的p-群作为一个西洛子群。在有限群的模表示理论中，抽象张量三角形标拓扑在有限G-集范畴上具体化为所谓的“sipp拓扑”，并导致导出范畴和稳定范畴的堆叠。这些想法，然后导致（sipp）上同调理论，这是本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上同调以及伽罗瓦上同调联系起来。第一个具体的应用范围是描述任意有限群上的endovrive表示，扩展自p群上的Carlson-Thevenaz分类。除了第一个主要成就，sipp拓扑实际上非常适合处理从p-局部到全局的胶合问题。这对于导出的和稳定的范畴来说特别有趣，因为有一系列长期存在的猜想，例如布鲁的阿贝尔亏群猜想。基础科学研究的驱动力之一，包括数学，是不断尝试将专业理论统一为更基本的原则。第一个目标是发现更深层次的科学之美，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相邻领域，从而创造新的应用和进一步的互动和创新。张量三角几何涵盖了数学的一大类专业领域，从代数到分析：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。张量三角几何提供了越来越多的新定理和这些领域之间的伟大概念统一。此外，它显示了一个稳步扩大的应用程序集合。例如，通过张量三角几何，Grothendieck著名的etale拓扑的深层美，其起源于伽罗瓦理论，并在现代代数几何中得到充分体现，现在再次出现在模表示论中。在那里，它提供了答案，长期存在的问题之间的关系表示有限群的顺序幂的素数和表示任意一般有限群。