New Methods in Tensor Triangular Geometry

张量三角几何是这个研究项目的主题，它连接了一大类专门的数学领域，包括稳定同伦理论、代数几何、模表示理论、动机理论和非交换拓扑学。张量三角几何有望成为一个支配性的理论，促进技术和方法从一个专门领域向邻近领域的转移，在这些领域之间提供概念上的统一。此外，它在数学方面的影响也在稳步扩大。例如，通过张量三角几何，现代代数几何中的埃尔塔尔拓扑学的主题现在在模表示理论中有了新的应用，它回答了关于素数的幂的阶有限群的表示与任意一般有限群的表示之间的关系的长期存在的问题。这项研究项目旨在拓宽和加深对张量三角几何领域的理解。本项目研究张量三角范畴的几何，因为它们出现在数学的几个领域。张量三角范畴目前在代数几何、模表示理论、稳定同伦理论及其等变形式、动机理论、等变非交换拓扑学等领域都有广泛的应用。这个项目旨在发展张量三角几何的新方法，以便从一个统一的角度研究这许多不同的化身。近年来，张量三角范畴的内部扩张(即可分扩张和交换扩张)以及相关的下降理论都取得了很大的进展。本项目建立在这些和相关结果的基础上，目的是应用下降将射影支撑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用具体的术语来阐明，而不需要张量三角技术。例如，在关于内模的工作中，研究者引入了“弱同态”。证明了P-子群的Brown单纯复群上的弱同态与复线丛是完全相同的。进一步研究它们在代数几何中的作用是当前项目的一部分。该项目的另一个总体主题是计算张量三角范畴的“谱”。这种计算的一种新的通用方法是通过过滤张量三角化类别并通过étale扩展来理解连续的地层。该方法的一个早期原型已被用于计算有限群的等变稳定同伦范畴的谱。该项目旨在将这些想法转化为新的例子。

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.