$\mathrm{VI}$ modules in non-describing characteristic, Part I

TL;DR

This paper investigates the structure of functor categories from the VI category over finite fields to modules over a noetherian ring, establishing key properties like finiteness and series descriptions crucial for classifying certain group representations.

Contribution

It provides a structure theorem, proves finiteness of regularity, and describes the Hilbert series for functors from VI over fields where q is invertible in the ring.

Findings

Structure theorem for functor categories

Finiteness of regularity established

Explicit description of Hilbert series

Abstract

Let $\mathrm{VI}$ be the category of finite dimensional $\mathbb{F}_q$-vector spaces whose morphisms are injective linear maps, and let $\mathbf{k}$ be a noetherian ring. We study the category of functors from $\mathrm{VI}$ to $\mathbf{k}$-modules in the case when $q$ is invertible in $\mathbf{k}$. Our results include a structure theorem, finiteness of regularity, and a description of the Hilbert series. These results are crucial in the classification of smooth irreducible $\mathbf{GL}_{\infty}(\mathbb{F}_q)$-representations in non-describing characterisitic which is contained in Part II of this paper.