VI-modules in nondescribing characteristic, part I

Abstract

Let VI be the category of finite dimensional ${\mathbb{F}}_q$-vector spaces whose morphisms are injective linear maps and let $k$ be a noetherian ring. We study the category of functors from VI to $k$-modules in the case when $q$ is invertible in $k$. 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 ${GL}_{\infty }\left({\mathbb{F}}_q\right)$-representations in nondescribing characteristic which is contained in Part II of this paper (VI-modules in nondescribing characteristic, part II, arxiv:1810.04592).