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Filtered fiber functors over a general base

Paul Ziegler

TL;DR

The paper studies filtered fiber functors on $\mathrm{Rep}^{\circ} G$ over general bases and proves that, provided $G$ has enough dualizable representations, every such functor is splittable on affine bases, i.e., it arises from a graded fiber functor via $\mathrm{fil}\circ\mathrm{gr}$. The approach combines a robust base-change framework for exact/tensor categories with an étale-local splitting argument: any filtered functor locally admits a splitting by a cocharacter, and the resulting torsor under a unipotent group is trivial on affine bases. Central to the method is the construction of automorphism group schemes $P(\varphi)$, $L(\varphi)$, $U(\varphi)$ and the link between graded and filtered structures through $D^{\Gamma}$-cocharacters. The results extend prior work on splittings (FFF) and have potential applications in Shimura varieties and Tannakian contexts over general bases.

Abstract

We prove that every filtered fiber functor on the category of dualizable representations of a smooth affine group scheme with enough dualizable representations comes from a graded fiber functor.

Filtered fiber functors over a general base

TL;DR

The paper studies filtered fiber functors on over general bases and proves that, provided has enough dualizable representations, every such functor is splittable on affine bases, i.e., it arises from a graded fiber functor via . The approach combines a robust base-change framework for exact/tensor categories with an étale-local splitting argument: any filtered functor locally admits a splitting by a cocharacter, and the resulting torsor under a unipotent group is trivial on affine bases. Central to the method is the construction of automorphism group schemes , , and the link between graded and filtered structures through -cocharacters. The results extend prior work on splittings (FFF) and have potential applications in Shimura varieties and Tannakian contexts over general bases.

Abstract

We prove that every filtered fiber functor on the category of dualizable representations of a smooth affine group scheme with enough dualizable representations comes from a graded fiber functor.
Paper Structure (12 sections, 30 theorems, 20 equations)

This paper contains 12 sections, 30 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Acknowledgment
  3. Preliminaries
  4. Exact Categories
  5. Tensor Categories
  6. Base change
  7. Categories of representations of group schemes
  8. Graded Fiber Functors
  9. Filtered Fiber Functors
  10. Filtered Modules
  11. Filtered Fiber Functors
  12. Splittings

Key Result

Theorem 1.1

If $G$ has enough dualizable representations, then every filtered fiber functor on $\mathop{\text{\tt Rep}}\nolimits^\circ G$ over an affine $R$-scheme $S$ is splittable.

Theorems & Definitions (68)

  • Theorem 1.1: c.f. Theorem \ref{['MainThm']}
  • Lemma 2.1: Buhler
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: c.f. Buhler
  • Definition 2.5
  • Proposition 2.6: MR3303245
  • Definition 2.7
  • Lemma 2.8
  • Definition 2.9
  • ...and 58 more