Tensor Abelian geometry of VI-modules
Peng Xu
TL;DR
The paper computes the spectrum of prime Serre ideals in the tensor abelian category of finitely generated global VI-modules arising from noetherian families, using the framework of tensor abelian geometry. It shows that for any noetherian and $\mathbb{N}$-stable family $\mathcal{U}$, the spectrum $\mathrm{Spc}(\mathrm{A}(\mathcal{U})^{c})$ is homeomorphic to the one-point compactification $\mathbb{N}^{*}$, with a distinguished non-group prime $P_{\infty}$ and group primes $P_{G}$ corresponding to objects supported away from $G$. The result yields an explicit, topology-preserving description of the prime spectrum, including maximal primes $P_{n}$ and the minimal $P_{\infty}$, and highlights that the abelian spectrum can differ from the Balmer spectrum of the derived category. The methods extend to FI-modules and to global representations for cyclic $p$-groups, underscoring the broad applicability of tensor abelian geometry to representation-theoretic families.
Abstract
In this short note, we study the spectrum of prime Serre ideals of global represen tations 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.