Central closed subgroups of semisimple affine groups

Let H be the coordinate Hopf algebra of a semisimple affine group over a field k. This file proves that every smooth geometrically connected central closed subgroup of the geometric fibre is trivial.

A central Hopf ideal is normal, and its quotient coordinate Hopf algebra is cocommutative. Consequently the represented subgroup has a commutative, hence solvable, group of geometric points. The defining universal property of semisimplicity then identifies its defining ideal with the augmentation ideal.

The theorem deliberately retains smoothness. In positive characteristic a semisimple group can have a non-smooth connected central subgroup scheme, such as an infinitesimal subgroup of its centre, so removing that hypothesis would be false. The result is the central-subgroup input for the finite-centre and adjoint-form steps: once a smooth connected central subgroup has been constructed, no separate solvability argument is needed to show that it is trivial.

Main declarations

• TauCeti.semisimpleCommHopfAlgProperty.eq_augmentation_of_isCentral: every smooth geometrically connected central closed subgroup of a semisimple group's geometric fibre is trivial.

References

• J. S. Milne, Algebraic Groups (2017), §§21.10 and 21.15.
• T. A. Springer, Linear Algebraic Groups, §8.1.

This advances Layer 6, "Reductive and semisimple groups", of the ReductiveGroups roadmap. It is the central-subgroup triviality step used in proving that the centre of a semisimple group is finite and in constructing its adjoint form.

theorem TauCeti.semisimpleCommHopfAlgProperty.eq_augmentation_of_isCentral {k : Type u} [Field k] {H : FiniteTypeCommHopfAlgCat k} (hH : semisimpleCommHopfAlgProperty k H) (I : HopfIdeal (AlgebraicClosure k) ↑H.baseChange.obj) (hI : I.IsCentral) (hconnected : geometricallyConnectedCommHopfAlgProperty (AlgebraicClosure k) (H.baseChange.quotient I).obj) (hsmooth : Algebra.Smooth (AlgebraicClosure k) ↑(H.baseChange.quotient I).obj) : I = HopfIdeal.augmentation (AlgebraicClosure k) ↑H.baseChange.obj

A smooth geometrically connected central closed subgroup of a semisimple affine group's geometric fibre is trivial.

The Hopf ideal I cuts out the subgroup contravariantly. Thus triviality is the equality of I with the augmentation ideal. Centrality supplies both normality and cocommutativity of the quotient; the latter makes its geometric point group commutative and therefore solvable.