Global representation theory: Homological foundations

TL;DR

This paper develops the homological foundations of the derived category of global representations, connecting classical representation theory with new categorical and tensor-triangular structures.

Contribution

It establishes the DG-projectivity of complexes of projective global representations and explores their tensor-triangular properties, including the construction of torsion-free classes.

Findings

Derived category admits an explicit model as the homotopy category of projective global representations.

Few dualizable objects exist in the tensor-triangular structure, with many more compact objects.

Torsion-free classes encode growth properties in the family of groups.

Abstract

A global representation is a compatible collection of representations of the outer automorphism groups of the groups belonging to some collection of finite groups $\mathscr{U}$. Global representations assemble into an abelian category $\mathsf{A}(\mathscr{U})$, simultaneously generalising classical representation theory and the category of VI-modules appearing in the representation theory of the general linear groups. In this paper we establish homological foundations of its derived category $\mathsf{D}(\mathscr{U})$. We prove that any complex of projective global representations is DG-projective, and hence conclude that the derived category admits an explicit model as the homotopy category of projective global representations. We show that from a tensor-triangular perspective it exhibits some unusual features: for example, there are very few dualizable objects and in general many more compact objects. Under more restrictive conditions on the family $\mathscr{U}$, we then construct torsion-free classes for global representations which encode certain growth properties in $\mathscr{U}$. This lays the foundations for a detailed study of the tensor-triangular geometry of derived global representations which we pursue in forthcoming work.