Homogeneous spaces in tensor categories

Technical Summary: Homogeneous Spaces in Tensor Categories

Problem Statement
The paper addresses the existence and structural properties of homogeneous spaces $G/H$ (quotient schemes) within the context of symmetric tensor categories $\mathcal{C}$ of moderate growth. While the representation theory of affine group schemes in characteristic zero is well-understood via Deligne's theorem (which identifies such categories with supergroup representations), the structure theory in positive characteristic $p > 0$ remains incomplete. Specifically, it is conjectured that incompressible tensor categories in characteristic $p$ are subcategories of $\text{Ver}_{p^\infty}$. A fundamental gap in the geometric theory of these categories is the rigorous existence of quotient schemes $G/H$ for algebraic groups $G$ and subgroups $H \leq G$. Without established quotient schemes, key representation-theoretic tools, such as induction functors and the study of equivariant sheaves, lack a solid geometric foundation.

Methodology
The authors develop a scheme theory for tensor categories satisfying two specific technical conditions: Geometric Reductivity (GR) and Maximal Nilpotency (MN). These conditions are posited as intrinsic characterizations of incompressible categories (e.g., $\text{Ver}_{p^\infty}$).

The methodology relies heavily on the interplay between the "body" of the group (the underlying classical algebraic group) and infinitesimal structures, utilizing the Frobenius morphism available only in positive characteristic. Key methodological steps include:

1. Frobenius Twists and Kernels: The authors define the Frobenius twist $A^{[1]}$ for algebras in $\mathcal{C}$ and the corresponding Frobenius kernels $G_r$ for algebraic groups. A central structural result (Proposition 4.19) establishes that for sufficiently large $r$, the quotient $G/G_r$ is a "purely even" algebraic group (an ordinary group scheme over $k$), and $G_r$ is an infinitesimal group scheme.
2. Reduction to Classical and Infinitesimal Cases: The existence of $G/H$ is reduced to two cases:
   • The classical case: Quotients of the "body" groups $G_0/H_0$ in the category of vector spaces ($\text{Vec}$).
   • The infinitesimal case: Quotients by infinitesimal subgroups, which are shown to exist and be affine.
3. Sheaf-Theoretic Construction: The authors utilize the theory of faisceaux (sheaves in the fppf topology) to construct the quotient. They demonstrate that the quotient in the category of schemes exists if it can be constructed locally and glued, leveraging the fact that the map from the group to the quotient is faithfully flat and affine under specific conditions.
4. Distribution Algebras: The paper employs distribution algebras ($\Gamma_G$) and their filtrations to analyze the structure of algebraic groups, showing how $\Gamma_G$ is generated by the distributions of the body $G_0$ and the Frobenius kernels.

Key Contributions and Results

• Existence of Homogeneous Spaces: The primary result (Theorem 7.5) proves that for an algebraic group $G$ and a subgroup $H$ in a tensor category $\mathcal{C}$ satisfying (GR) and (MN), the homogeneous space $G/H$ exists as a scheme of finite type. Furthermore, $G/H$ is separated.
• Relationship with the "Body": The paper establishes a precise relationship between the quotient $G/H$ and the quotient of the underlying classical groups $G_0/H_0$.
  • There exists a natural closed embedding $G_0/H_0 \hookrightarrow (G/H)_0$ which is a universal homeomorphism (Theorem 7.5).
  • While the equality $(G/H)_0 = G_0/H_0$ holds in the super setting (where $\mathcal{C} \subseteq \text{Ver}_p$), it may fail in more general incompressible categories (Example 6.1). However, the geometric properties (affine, quasi-affine, proper) of $G/H$ are equivalent to those of $G_0/H_0$ (Theorem 7.5).
• Structure of Quotients in $\text{Ver}_p$: For the specific case $\mathcal{C} = \text{Ver}_p$, the authors provide an explicit local construction of $G/H$. They show that $(G/H)_0 = G_0/H_0$ exactly, and the scheme is locally isomorphic to the spectrum of a symmetric algebra over a specific vector bundle derived from the Lie algebra filtration (Theorem 9.9).
• Affine Quotients and Injectivity: In Section 8, the paper characterizes when affine quotients exist. If $\mathcal{C}$ is semisimple, $G/H$ is affine if and only if the coordinate ring $k[G]$ is an injective $k[H]$-comodule (Theorem 8.1). This generalizes classical results from Schneider to the tensor category setting.
• Equivariant Sheaves and Induction: The existence of $G/H$ allows for the definition of the induction functor $\text{Ind}_H^G$. The paper proves an equivalence between the category of $H$-equivariant quasi-coherent sheaves on $G$ and quasi-coherent sheaves on $G/H$ (Proposition 7.11). Furthermore, it establishes an isomorphism between the derived induction functors and the sheaf cohomology of $G/H$ (Lemma 7.13).

Significance and Claims
The paper claims to resolve the foundational question of whether homogeneous spaces exist as schemes in the relevant tensor categories, thereby enabling the application of classical geometric techniques to the representation theory of groups in these categories.

• Foundational: The existence of $G/H$ is presented as a prerequisite for developing a robust geometric theory of algebraic groups in tensor categories, analogous to the role of flag varieties in classical representation theory.
• Structural Insight: The decomposition of algebraic groups into "infinitesimal" and "ordinary" parts via Frobenius kernels (Proposition 4.19) is highlighted as a fundamental structural result with potential applications beyond the immediate scope of homogeneous spaces.
• Representation Theory: The paper explicitly links the geometric properties of $G/H$ (specifically properness) to the preservation of compact objects under the induction functor (Corollary 7.14). This is crucial for computing characters of irreducible representations, a task where homogeneous supervarieties have been previously used in supergroup theory.
• Generality: The results apply to any tensor category satisfying (GR) and (MN), which includes $\text{Ver}_{p^\infty}$ and its subcategories, thus supporting the conjecture that these conditions characterize incompressible categories.

The authors maintain a modest tone regarding the scope, noting that while the equality $(G/H)_0 = G_0/H_0$ fails in general, the universal homeomorphism property is sufficient to transfer many geometric properties. They also note that the development of a full theory of Harish-Chandra pairs in these categories remains a subject for future work, as the current framework requires replacing the Lie algebra with a larger infinitesimal group.