Unipotent radicals from geometrically semisimple quotients #
Let H represent a finite-type affine group and let a morphism from a coordinate algebra with
geometrically semisimple points to H represent a quotient homomorphism from that group. If its
kernel is connected, normal, smooth, and unipotent, then that kernel is the unipotent radical.
Indeed, the kernel is contained in the radical by maximality. In the other direction, the image of the smooth unipotent radical in the geometrically semisimple target is reduced and unipotent, hence trivial. This forces the radical to lie in the kernel.
Main declaration #
TauCeti.FiniteTypeCommHopfAlgCat. unipotentRadicalDefiningIdeal_eq_kernelHopfIdeal_of_geometricallySemisimple: a unipotent kernel with geometrically semisimple target is the unipotent radical.
References #
- J. S. Milne, Algebraic Groups (2017), Proposition 12.40 and §§6.45--6.46.
- A. Borel, Linear Algebraic Groups, §11.21.
theorem
TauCeti.FiniteTypeCommHopfAlgCat.unipotentRadicalDefiningIdeal_eq_kernelHopfIdeal_of_geometricallySemisimple
{k : Type u}
[Field k]
(H D : FiniteTypeCommHopfAlgCat k)
(hD : geometricallySemisimplePointsCommHopfAlgProperty k D.obj)
(f : D.obj ⟶ H.obj)
(hf : HopfIdeal.IsUnipotentRadicalCandidate H (CommHopfAlgCat.kernelHopfIdeal f))
:
A connected normal smooth unipotent kernel of a homomorphism to a group with geometrically semisimple points is the unipotent radical.