Homogeneous spaces in tensor categories

Abstract

Let $\mathscr{C}$ be a symmetric tensor category of moderate growth, and let $\mathcal{H}\leq\mathcal{G}$ be algebraic groups in $\mathscr{C}$. We prove that the homogeneous space $\mathcal{G}/\mathcal{H}$ exists as a scheme and is of finite type when $\mathscr{C}$ is geometrically reductive and maximally nilpotent, conditions that are conjecturally equivalent to incompressibility. A key tool is the introduction of a Frobenius kernel of an group scheme. We further show that while $\mathcal{G}_0/\mathcal{H}_0$ and $(\mathcal{G}/\mathcal{H})_0$ need not be the same, they are close enough, so that $\mathcal{G}/\mathcal{H}$ is quasi-affine/affine/proper if and only if $\mathcal{G}_0/\mathcal{H}_0$ is.