Homogeneous spaces in tensor categories
Kevin Coulembier, Alexander Sherman

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 be a symmetric tensor category of moderate growth, and let be algebraic groups in . We prove that the homogeneous space exists and is of finite type when 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 and need not be the same, they are close enough, so that is quasi-affine/affine/proper if and only if is.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Advanced Algebra and Geometry
