Homogeneous spaces in tensor categories

TL;DR

This paper proves the existence and finite type property of homogeneous spaces in symmetric tensor categories under certain conditions, introducing a Frobenius kernel and analyzing their geometric properties.

Contribution

It establishes conditions for the existence and finite type of homogeneous spaces in tensor categories and introduces a Frobenius kernel as a key tool.

Findings

Homogeneous spaces exist and are of finite type under (GR) and (MN1-2) conditions.

The paper introduces a Frobenius kernel of a group scheme.

Quasi-affineness, affineness, and properness are characterized via the reduced parts.

Abstract

Let $\mathscr{C}$ be a symmetric tensor category of moderate growth, and let $\mathcal{H}\subseteq\mathcal{G}$ be algebraic groups in $\mathscr{C}$. We prove that the homogeneous space $\mathcal{G}/\mathcal{H}$ exists and is of finite type when $\mathscr{C}$ satisfies (GR) and (MN1-2), which 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.