Affine quotients of supergroups

TL;DR

This paper characterizes when sheaf quotients of affine superschemes by affine supergroups are affine, establishing conditions and properties of quotients and normal supersubgroups in the supergroup context.

Contribution

It provides necessary and sufficient conditions for affine quotients of affine supergroups to be affine and proves that normal supersubgroups lead to affine supergroup quotients.

Findings

Sheaf quotients of affine superschemes by affine supergroups are affine under specific conditions.

Normal supersubgroups of affine supergroups produce affine supergroup quotients.

Algebraic affine supergroups have algebraic quotient sheaves that coincide with their sheaf quotients.

Abstract

In this article we consider sheaf quotients of affine superschemes by affine supergroups that act on them freely. The necessary and sufficient conditions for such quotients to be affine are given. If $G$ is an affine supergroup and $H$ is its normal supersubgroup, then we prove that a dur $K$-sheaf $\tilde{\tilde{G/H}}$ is again affine supergroup. Additionally, if $G$ is algebraic, then a $K$-sheaf $\tilde{G/H}$ is also algebraic supergroup and it coincides with $\tilde{\tilde{G/H}}$. In particular, any normal supersubgroup of an affine supergroup is faithfully exact.