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

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