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On finite group scheme-theoretical categories, I

Shlomo Gelaki, Guillermo Sanmarco

Abstract

Let $\mathscr {C}(G,H,ψ)$ be a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$ as introduced by the first author. For any indecomposable exact module category over $\mathscr {C}(G,H,ψ)$, we classify its simple objects and provide an expression for their projective covers, in terms of double cosets and projective representations of certain closed subgroup schemes, which upgrades a result by Ostrik for group-theoretical fusion categories. As a byproduct, we describe the simples and indecomposable projectives of $\mathscr {C}(G,H,ψ)$, and parametrize the Brauer-Piccard group of ${\rm Coh}(G)$ for any finite connected group scheme $G$. Finally, we apply our results to describe the blocks of the center of ${\rm Coh}(G)$.

On finite group scheme-theoretical categories, I

Abstract

Let be a finite group scheme-theoretical category over an algebraically closed field of characteristic as introduced by the first author. For any indecomposable exact module category over , we classify its simple objects and provide an expression for their projective covers, in terms of double cosets and projective representations of certain closed subgroup schemes, which upgrades a result by Ostrik for group-theoretical fusion categories. As a byproduct, we describe the simples and indecomposable projectives of , and parametrize the Brauer-Piccard group of for any finite connected group scheme . Finally, we apply our results to describe the blocks of the center of .
Paper Structure (32 sections, 40 theorems, 231 equations)

This paper contains 32 sections, 40 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sheaves on finite schemes
  4. Finite group schemes
  5. Quotients and free actions
  6. Equivariant sheaves
  7. Equivariant sheaves and comodules
  8. Equivariant morphisms
  9. Principal homogeneous spaces
  10. Twisting by the inverse $2$-cocycle
  11. Module categories over ${\rm Coh}(G)$
  12. Equivariant sheaves on group schemes
  13. The equivalence $\mathrm{Rep}_k(A,\xi^{-1}) \cong \mathrm{Coh}^{(A\times B,\xi^{-1}\times \Psi)}(B)$
  14. The equivalence $\mathrm{Coh}^{(A\times B,\xi^{-1}\times \Psi)}(B) \cong {\rm Coh}^{(B,\Psi)}(A\backslash B)$
  15. The first equivalence ${\rm Rep}_k(A,\xi^{-1}) \cong {\rm Coh}^{(B,\Psi)}(A\backslash B)$
  16. ...and 17 more sections

Key Result

Theorem 1.1

G The assignment $(H,\psi)\mapsto {\mathscr M}(H,\psi)$ determines a bijection between conjugacy classes of pairs $(H,\psi)$, where $H\subset G$ is a closed subgroup scheme and $\psi\in C^2(H,\mathbb G_m)$ is normalized such that $d \psi=\iota_H^{\sharp \otimes 3}(\omega)$, and equivalence classes o

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Example 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • ...and 89 more