Normal closed subgroups of reductive groups

A smooth geometrically connected normal closed subgroup of a reductive affine group is reductive, over any field and in every characteristic. After extension to an algebraic closure, the subgroup's unipotent radical is normal in the ambient group, and hence lies in its trivial unipotent radical. The quotient base-change isomorphism then identifies this with the geometric radical of the original subgroup.

Smoothness and geometric connectedness of the subgroup remain explicit: normal subgroups of reductive groups can be non-smooth or disconnected. The theorem applies in particular to the derived subgroup once its smoothness and geometric connectedness have been established, supplying the reductivity step toward its semisimplicity.

References

• A. Borel, Linear Algebraic Groups, §14.
• J. S. Milne, Algebraic Groups (2017), §19.

theorem TauCeti.reductiveCommHopfAlgProperty.quotient_of_isNormal {k : Type u} [Field k] {H : FiniteTypeCommHopfAlgCat k} (hH : reductiveCommHopfAlgProperty k H) (I : HopfIdeal k ↑H.obj) (hI : I.IsNormal) (hsmooth : Algebra.Smooth k ↑(H.quotient I).obj) (hconnected : geometricallyConnectedCommHopfAlgProperty k (H.quotient I).obj) : reductiveCommHopfAlgProperty k (H.quotient I)

A smooth geometrically connected normal closed subgroup of a reductive affine group is reductive. The quotient of coordinate Hopf algebras represents the closed subgroup.