Tensor Abelian Geometry of VI-modules

In this short note, we study the spectrum of prime Serre ideals of global representations for noetherian families. In particular, we prove that the spectrum of prime Serre ideals of finitely generated VI-modules is homeomorphic to N^{*}, the one-point compactification of N, which differs from the Balmer spectrum of derived VI-modules. Our method could also be applied to the category of finitely generated FI-modules and the category of global representations for the family of cyclic p-groups.