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

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.

Errata added to Corollary 4.23 of the published version

References in corpus (4)

• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• GL-equivariant modules over polynomial rings in infinitely many variables. II
• Periodicity in the cohomology of symmetric groups via divided powers
• Regularity of FI-modules and local cohomology

Cited by in corpus (5)

• Stability in the high-dimensional cohomology of congruence subgroups
• Categorifications of rational Hilbert series and characters of $F{S}^{op}$ modules
• GL-algebras in positive characteristic I: the exterior algebra
• Periodicity in the cohomology of finite general linear groups via q-divided powers
• The generalized Auslander-Reiten duality on a module category