Θ-Completeness for Classifying Stacks

Abstract

We provide a criterion for the classifying stack BGto be Θ-completeover an algebraically closed field of characteristic 0. It is known that the classifying stack BG is Θ-complete when G is a unipotent algebraic group or reductive group (hence the product of a unipotent group and a reductive group). We prove that this is in fact a necessary condition. That is the Θ-completeness of the classifying stack BG implies that G must be the trivial product of a unipotent group and a reductive group.