On finite group scheme-theoretical categories, II

Abstract

Let $\C:=\C(G,ω,H,ψ)$ be a finite group scheme-theoretical category over an algebraically closed field of characteristic $p\ge 0$. For every indecomposable exact module category over $\C$, we classify the simple objects by double cosets and projective representations of their stabilizers. We construct the projective covers using finite-dimensional algebras of twisted equivariant sheaves on the full infinitesimal neighborhoods of the orbits, and determine their underlying sheaves and dimensions. We prove that $\C$ is unimodular if and only if ${\delta }_G{|}_H={\delta }_H^2$, where ${\delta }_G$ and ${\delta }_H$ are the modular characters. In particular, unimodularity is independent of the admissible twisting. We also describe the orbit-indexed Serre summands and relative-center equivariantization of the center of $\Coh(G,ω)$. In characteristic two, a calculation for $G={\alpha }_2$ shows that a nontrivial associator can change both the Loewy length of a projective cover and the dimension of the self-extension space of the simple object.

52 pages; mistakes corrected, new results added, exposition improved and expanded