Superschemes and supermanifolds with free $\mathbb{A}^{0|n}$-action

Source abstract

We prove that given an analytic supermanifold $X$ over a complete field on which the additive supergroup $\mathbb{A}^{0|n}$ of super-dimension $0|n$ acts freely (in an appropriate sense), the quotient $X/\mathbb{A}^{0|n}$ exists as an analytic supermanifold, and the canonical morphism $X \to X/\mathbb{A}^{0|n}$ is an $\mathbb{A}^{0|n}$-principal bundle. A stronger version of this result is proven for $C^{\infty}$-supermanifolds, whereby $X \to X/\mathbb{A}^{0|n}$ is a trivial $\mathbb{A}^{0|n}$-principal bundle. Previously, Polishchuk [18] established an algebro-geometric analogue, which proves that a superscheme $X$ with free $\mathbb{A}^{0|n}$-action gives rise to a locally trivial $\mathbb{A}^{0|n}$-torsor $X \to X/\mathbb{A}^{0|n}$. This result is proven here in an alternative, elementary way. To all of these, a Hopf-algebraic approach is effectively employed.