On quotients of affine superschemes over finite supergroups

TL;DR

This paper proves that when a finite supergroup acts freely on an affine superscheme, the quotient is also an affine superscheme, with the coordinate ring being the invariants, and the original ring being finitely presented and projective over the invariants.

Contribution

It establishes that quotients of affine superschemes by finite supergroups are affine superschemes with well-behaved coordinate rings, extending classical quotient results to the supergeometry setting.

Findings

The quotient sheaf is an affine superscheme.

The coordinate ring of the quotient is the invariants of the original ring.

The original ring is finitely presented and projective over the invariants.

Abstract

In this article we consider sheaf quotients of affine superschemes by finite supergroups that act on them freely. More precisely, if a finite supergroup $G$ acts on an affine superscheme $X$ freely, then the quotient $K$-sheaf $\tilde{X/G}$ is again an affine superscheme $Y$, where $K[Y]\simeq K[X]^G$. Besides, $K[X]$ is a finitely presented projective $K[X]^G$-module.