ETALE TOPOLOGY IN TENSOR TRIANGULAR GEOMETRY

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

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.