Algebraic groups and compact generation of their derived categories of representations

TL;DR

This paper characterizes when the derived categories of representations of group schemes over a field are compactly generated and relates the cohomological dimension of algebraic stacks to their stabilizer groups.

Contribution

It provides a characterization of group schemes over a field for which the derived category of their classifying stack is compactly generated and describes algebraic stacks with finite cohomological dimension.

Findings

Characterization of group schemes with compactly generated derived categories

Description of algebraic stacks with finite cohomological dimension

Relation between stabilizer groups and cohomological properties

Abstract

Let $k$ be a field. We characterize the group schemes $G$ over $k$, not necessarily affine, such that $\mathsf{D}_{\mathrm{qc}}(B_kG)$ is compactly generated. We also describe the algebraic stacks that have finite cohomological dimension in terms of their stabilizer groups.