Abstract

A global representation is a compatible collection of representations of the outer automorphism groups of the members of some collection of finite groups $\mathcal{𝒰}$. Global representations assemble into an abelian category $A(\mathcal{𝒰})$, 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 $D(\mathcal{𝒰})$. 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 dualisable objects and in general many more compact objects. Under more restrictive conditions on the family $\mathcal{𝒰}$, we then construct torsion-free classes for global representations which encode certain growth properties in $\mathcal{𝒰}$. This lays the foundations for a detailed study of the tensor-triangular geometry of derived global representations which we pursue in forthcoming work.

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