Affine algebraic super-groups with integral

Abstract

We generalize to the super context, the known fact that if an affine algebraic group $G$ over a commutative ring $k$ acts freely (in an appropriate sense) on an affine scheme $X$ over $k$, then the dur sheaf $X\widetilde{\stackrel{~}{/}}G$ of $G$-orbits is an affine scheme in the following two cases: (I) $G$ is finite; (II) $k$ is a field, and $G$ is linearly reductive. An emphasize is put on the more difficult generalization in the second case; the replaced assumption then is that an affine algebraic super-group $G$ over an arbitrary field has an integral. Those super-groups which satisfy the assumption are characterized, and are seen to form a large class if $\mathrm{char}k=0$. Hopf-algebraic techniques including bosonization are applied to prove the results.

Made very minor changes and corrections; final version to appear in Comm. Algebra

References in corpus (3)

• Semisimple algebraic tensor categories
• Hopf algebraic techniques applied to super algebraic groups
• Borel-Weil Theorem for Algebraic Supergroups

Cited by in corpus (1)

• Frobenius kernels of algebraic supergroups and Steinberg's tensor product theorem