ETALE TOPOLOGY IN TENSOR TRIANGULAR GEOMETRY

这个项目是PI长期项目“张量三角几何”的一部分，该项目研究在数学中出现的张量三角分类的几何。本计画的具体目标是将代数几何扩展为张量三角几何。起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，相对于一个合适的环对象。此外，该环对象是可交换的、可分离的和紧致的（有限维），因此它应该被称为“张量-三角形”环对象。简而言之，这种增强的正则扩展不仅推广了代数几何的经典正则扩展，而且涵盖了表征理论中对子群的限制。在代数几何中，标量扩展的力量在于应用下降理论的能力。将下降扩展到张量三角形几何，专门研究表示理论，产生一种机制，允许人们决定何时p群的模表示扩展到包含我们的p群作为Sylow子群的任意有限群。在有限群的模表示理论中，抽象张量三角矩阵拓扑在有限g集的范畴上具体化为所谓的“sipp拓扑”，并导致一堆派生的稳定范畴。这些想法然后导致（sipp）上同源理论，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任意有限群上的内平凡表示的描述，由p群上的Carlson-Thevenaz分类扩展而来。除了这第一个主要成就之外，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Broue的Abelian缺陷群猜想，这对于衍生的和稳定的范畴来说是特别有趣的。基础科学（包括数学）研究的推动力之一是不断尝试将专门的理论统一为更基本的原理。第一个目标是发现更深层次的科学之美，第二个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，从而创造新的应用和进一步的互动和创新。张量三角形几何涵盖了从代数到分析的大量数学专业领域：它出现在代数几何、模表示理论、稳定同伦理论、动机理论、非交换拓扑等中。张量三角形几何提供了越来越多的新定理和这些领域之间伟大的概念统一。此外，它还显示稳步扩展的应用程序集合。例如，通过张量三角形几何，Grothendieck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显现出来。在那里，它回答了长期存在的关于质数的一阶幂有限群的表示与任意一般有限群的表示之间的关系的问题。

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.