Ore's theorem for thick subcategories

TL;DR

This paper characterizes finite groups with distributive lattices of thick subcategories in their derived categories, identifying them as exactly the p-nilpotent groups, and provides criteria for similar properties in algebraic contexts.

Contribution

It introduces a characterization of p-nilpotent groups via the distributive lattice property of their derived categories' thick subcategories.

Findings

Finite groups with distributive lattices are exactly p-nilpotent groups.

Provides criteria for distributive lattices in derived categories of finite dimensional algebras.

Establishes conditions for perfect complexes to have distributive lattices.

Abstract

We characterize those finite groups for which the bounded derived category of finite dimensional representations over an algebraically closed field of characteristic $p$ has distributive lattice of thick subcategories: they are precisely the $p$-nilpotent groups. Along the way we give necessary and sufficient criteria for the bounded derived category and perfect complexes of a finite dimensional $k$-algebra to have distributive lattices of thick subcategories.