The spectrum of global representations for families of bounded rank and VI-modules

TL;DR

This paper systematically studies the derived category of global representations for families of finite groups, computing its spectrum and classifying finitely generated modules, revealing complex geometric structures.

Contribution

It introduces a tt-geometry approach to global representations, computes the Balmer spectrum for key families, and classifies finitely generated VI-modules.

Findings

Balmer spectrum has infinite Krull dimension for abelian p-groups

Complete tt-theoretic classification of finitely generated VI-modules

Complex geometric phenomena in the spectrum of global representations

Abstract

A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family $\mathscr{U}$. These arise in classical representation theory, in the study of representation stability, as well as in global homotopy theory. In this paper we begin a systematic study of the derived category $\mathsf{D}(\mathscr{U};k)$ of global representations over fields $k$ of characteristic zero, from the point-of-view of tensor-triangular geometry. We calculate its Balmer spectrum for various infinite families of finite groups including elementary abelian $p$-groups, cyclic groups, and finite abelian $p$-groups of bounded rank. We then deduce that the Balmer spectrum associated to the family of finite abelian $p$-groups has infinite Krull dimension and infinite Cantor--Bendixson rank, illustrating the complex phenomena we encounter. As a concrete application, we provide a complete tt-theoretic classification of finitely generated derived VI-modules. Our proofs rely on subtle information about the growth behaviour of global representations studied in a companion paper, as well as novel methods from non-rigid tt-geometry.