Geometric construction of quotients $G/H$ in supersymmetry

Abstract

It was proved by the first-named author and Zubkov [13] that given an affine algebraic supergroup $\mathbb{G}$ and a closed sub-supergroup $\mathbb{H}$ over an arbitrary field of characteristic $\ne 2$, the faisceau $\mathbb{G} \tilde{/} \mathbb{H}$ (in the fppf topology) is a superscheme, and is, therefore, the quotient superscheme $\mathbb{G}/\mathbb{H}$, which has desirable properties, in fact. We reprove this, by constructing directly the latter superscheme $\mathbb{G}/\mathbb{H}$. Our proof describes explicitly the structure sheaf of $\mathbb{G}/\mathbb{H}$, and reveals some new geometric features of the quotient, that include one which was desired by Brundan [2], and is shown in general, here for the first time.