Unipotent radicals of normal closed subgroups

Over an algebraically closed field, the unipotent radical of a normal closed subgroup of a reduced finite-type affine group is normal in the ambient group. Consequently it lies in the ambient unipotent radical. In particular, triviality of the ambient radical implies triviality of the subgroup's radical, without any smoothness or connectedness assumption on that subgroup.

Conjugation restricts to an automorphism of the normal subgroup, and isomorphism invariance of the unipotent radical makes its radical stable under that automorphism. Reducedness of the ambient group and smoothness of the radical allow rational-point stability to detect scheme-theoretic normality. The argument uses the quotient-isomorphism interface in TauCeti.Algebra.AlgebraicGroup.HopfIdeal.Quotient.Basic and the radical invariance in TauCeti.Algebra.AlgebraicGroup.Unipotent.Radical.Isomorphism.

References

• A. Borel, Linear Algebraic Groups, §11.21.
• J. S. Milne, Algebraic Groups (2017), §§6.45–6.46 and 19.

theorem TauCeti.FiniteTypeCommHopfAlgCat.isNormal_comapOfSurjective_unipotentRadicalDefiningIdeal {k : Type u} [Field k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) [IsReduced ↑H.obj] (I : HopfIdeal k ↑H.obj) (hI : I.IsNormal) : ((H.quotient I).unipotentRadicalDefiningIdeal.comapOfSurjective (CommHopfAlgCat.Hom.hom (H.obj.mkQuotient I)) ⋯).IsNormal

The unipotent radical of a normal closed subgroup of a reduced finite-type affine group over an algebraically closed field is normal in the ambient group.

theorem TauCeti.FiniteTypeCommHopfAlgCat.unipotentRadicalDefiningIdeal_le_comapOfSurjective_quotient {k : Type u} [Field k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) [IsReduced ↑H.obj] (I : HopfIdeal k ↑H.obj) (hI : I.IsNormal) : H.unipotentRadicalDefiningIdeal ≤ (H.quotient I).unipotentRadicalDefiningIdeal.comapOfSurjective (CommHopfAlgCat.Hom.hom (H.obj.mkQuotient I)) ⋯

The unipotent radical of a normal closed subgroup lies in the ambient unipotent radical. The inclusion is reversed on defining Hopf ideals.

theorem TauCeti.FiniteTypeCommHopfAlgCat.unipotentRadicalDefiningIdeal_quotient_eq_augmentation_of_isNormal {k : Type u} [Field k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) [IsReduced ↑H.obj] (I : HopfIdeal k ↑H.obj) (hI : I.IsNormal) (hH : H.unipotentRadicalDefiningIdeal = HopfIdeal.augmentation k ↑H.obj) : (H.quotient I).unipotentRadicalDefiningIdeal = HopfIdeal.augmentation k ↑(H.quotient I).obj

A normal closed subgroup of a reduced finite-type affine group with trivial unipotent radical also has trivial unipotent radical, over an algebraically closed field.