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

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arxiv 1709.07591 v4 pith:ZC2AY4KN submitted 2017-09-22 math.RT math.AC

classification math.RTmath.AC
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. GL-algebras in positive characteristic III: the divided power algebra

    math.AC 2026-08 conditional novelty 7.0 of 10

    The divided power algebra Div(k^∞) over a field of characteristic p is GL-coherent, and its bounded derived category of finitely presented modules has a semi-orthogonal decomposition into pieces generated by D^(r) ⊗ L_λ.

  2. A torsion theoretic interpretation for sheaves of modules and Grothendieck topologies on directed categories

    math.RT 2025-06 conditional novelty 7.0 of 10

    Sheaves of modules are exactly the J-saturated presheaves, every Grothendieck topology on a noetherian EI directed category is rigid, and all topologies on type N/Z categories are classified.

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