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Homogeneous spaces in tensor categories

Kevin Coulembier, Alexander Sherman

TL;DR

The paper addresses the problem of defining and understanding homogeneous spaces within tensor categories of moderate growth by proving the existence of quotients $\mathcal{G}/\mathcal{H}$ for algebraic groups $\mathcal{G}$ with a subgroup $\mathcal{H}$. It develops Frobenius twists and Frobenius kernels to reduce questions about quotients in general tensor categories to the classical theory of algebraic groups, and proves that $\mathcal{G}/\mathcal{H}$ exists, is of finite type, and is algebraic/separated; moreover, its geometric type (quasi-affine, affine, or proper) agrees with that of the body quotient $\mathcal{G}_0/\mathcal{H}_0$. The results yield a robust framework for equivariant sheaves and induction, with a detailed treatment in the Ver_p setting where the category is semisimple, allowing explicit presentations of the quotients. The work advances the program of understanding representation theory and geometric structures in incompressible tensor categories, and provides tools for extending classical geometric methods to the tensor-categorical landscape, including applications to Ver_p and related categories. The methodology blends Takian-style Hopf-algebra techniques, fppf-descent formalism, and Frobenius-twisted reductions to connect categorical geometry with traditional algebraic geometry.

Abstract

Let $\mathscr{C}$ be a symmetric tensor category of moderate growth, and let $\mathcal{H}\subseteq\mathcal{G}$ be algebraic groups in $\mathscr{C}$. We prove that the homogeneous space $\mathcal{G}/\mathcal{H}$ exists and is of finite type when $\mathscr{C}$ satisfies (GR) and (MN1-2), which are conjecturally equivalent to incompressibility. A key tool is the introduction of a Frobenius kernel of an group scheme. We further show that while $\mathcal{G}_0/\mathcal{H}_0$ and $(\mathcal{G}/\mathcal{H})_0$ need not be the same, they are close enough, so that $\mathcal{G}/\mathcal{H}$ is quasi-affine/affine/proper if and only if $\mathcal{G}_0/\mathcal{H}_0$ is.

Homogeneous spaces in tensor categories

TL;DR

The paper addresses the problem of defining and understanding homogeneous spaces within tensor categories of moderate growth by proving the existence of quotients for algebraic groups with a subgroup . It develops Frobenius twists and Frobenius kernels to reduce questions about quotients in general tensor categories to the classical theory of algebraic groups, and proves that exists, is of finite type, and is algebraic/separated; moreover, its geometric type (quasi-affine, affine, or proper) agrees with that of the body quotient . The results yield a robust framework for equivariant sheaves and induction, with a detailed treatment in the Ver_p setting where the category is semisimple, allowing explicit presentations of the quotients. The work advances the program of understanding representation theory and geometric structures in incompressible tensor categories, and provides tools for extending classical geometric methods to the tensor-categorical landscape, including applications to Ver_p and related categories. The methodology blends Takian-style Hopf-algebra techniques, fppf-descent formalism, and Frobenius-twisted reductions to connect categorical geometry with traditional algebraic geometry.

Abstract

Let be a symmetric tensor category of moderate growth, and let be algebraic groups in . We prove that the homogeneous space exists and is of finite type when satisfies (GR) and (MN1-2), which are conjecturally equivalent to incompressibility. A key tool is the introduction of a Frobenius kernel of an group scheme. We further show that while and need not be the same, they are close enough, so that is quasi-affine/affine/proper if and only if is.
Paper Structure (34 sections, 72 theorems, 84 equations)

This paper contains 34 sections, 72 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Tensor categories
  5. Commutative algebra and scheme theory
  6. Finite morphisms
  7. Properties of schemes and morphisms
  8. Infinitesimal schemes
  9. Affine group schemes
  10. Subgroups and images
  11. A filtration
  12. Connected components
  13. Distribution Hopf algebra
  14. Semi-direct products
  15. The group $\operatorname{gr}\mathcal{G}$
  16. ...and 19 more sections

Key Result

Lemma 2.8

Let $A$ be a finitely generated commutative algebra with nilpotent ideal $I$, and write $C:=A/I$. If $\psi:B\to A$ is an algebra morphism such that the composition $B\to C$ is finite, then $\psi$ is finite.

Theorems & Definitions (162)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Corollary 2.9
  • ...and 152 more