Connectedness of a subgroup generated by connected affine groups

Over an algebraically closed field, the closed subgroup scheme generated by a family of connected affine group schemes is connected. We use the common-kernel construction of the generated subgroup: a connected generator factors through the identity component of the generated subgroup. Maximality of the common-kernel Hopf ideal then forces that identity component to be the whole generated subgroup.

This applies to root-subgroup and split-torus generators when their carrier is presented by the common-kernel construction over the algebraically closed field. It does not require the generated subgroup to be smooth or reduced.

Reference

• J. S. Milne, Algebraic Groups (2017), Propositions 2.37 and 2.48, and §7.

theorem TauCeti.CommHopfAlgCat.connectedSpace_commonKernelQuotient {k : Type u} [Field k] [IsAlgClosed k] {H : CommHopfAlgCat k} [Algebra.FiniteType k ↑H] {ι : Type w} {K : ι → CommHopfAlgCat k} (f : (i : ι) → H ⟶ K i) (hK : ∀ (i : ι), ConnectedSpace (PrimeSpectrum ↑(K i))) : ConnectedSpace (PrimeSpectrum ↑(quotient H (commonKernelHopfIdeal f)))

The common-kernel quotient of a finite-type affine group by a family of maps to connected affine groups is connected. It is the coordinate ring of the closed subgroup scheme generated by the images of the corresponding connected affine group schemes.

theorem TauCeti.CommHopfAlgCat.geometricallyConnectedCommHopfAlgProperty_commonKernelQuotient {k : Type u} [Field k] [IsAlgClosed k] {H : CommHopfAlgCat k} [Algebra.FiniteType k ↑H] {ι : Type w} {K : ι → CommHopfAlgCat k} (f : (i : ι) → H ⟶ K i) (hK : ∀ (i : ι), ConnectedSpace (PrimeSpectrum ↑(K i))) : geometricallyConnectedCommHopfAlgProperty k (quotient H (commonKernelHopfIdeal f))

The subgroup generated by connected affine groups over an algebraically closed field is geometrically connected.