The characters of a geometrically reduced, geometrically connected affine group are torsion free

A geometric character of an affine group Spec H is a group-like element of the coordinate Hopf algebra of its base change to an algebraic closure. This file proves that when H is geometrically reduced and geometrically connected, no geometric character has finite order, and records the corollary for a smooth geometrically connected affine group.

Over the algebraic closure the argument is a reduction to the diagonalizable case. Group-like elements of a Hopf algebra over a field are linearly independent, so evaluation embeds the group algebra on them into H. A subring of a reduced ring is reduced, and connectedness of a prime spectrum descends along an injective ring homomorphism, so both hypotheses pass to that group algebra, where TauCeti.isMulTorsionFree_of_isReduced_monoidAlgebra_of_connectedSpace already rules out torsion.

Neither hypothesis can be dropped. Connectedness alone fails in characteristic p, where the coordinate Hopf algebra of μ_p is connected, non-reduced, and carries a character of order p. Reducedness alone fails for the constant group ℤ/n over a field containing a primitive n-th root of unity: its coordinate algebra is reduced but disconnected, and it has characters of order n.

Contravariantly, a homomorphism from Spec H to the diagonalizable group D(M) is a morphism of coordinate bialgebras k[M] ⟶ H. Torsion-freeness therefore says that a smooth geometrically connected affine group admits no nontrivial homomorphism to a diagonalizable group on a torsion group, so in particular no nontrivial μ_n-quotient.

Main declarations

• TauCeti.CommHopfAlgCat.isMulTorsionFree_geometricCharacterGroup: the geometric character group of a geometrically reduced, geometrically connected commutative Hopf algebra is torsion free.
• TauCeti.CommHopfAlgCat.isMulTorsionFree_geometricCharacterGroup_of_smooth: its corollary for a smooth geometrically connected affine group, whose coordinate algebra is geometrically reduced.
• TauCeti.CommHopfAlgCat.isAddTorsionFree_additiveCharacterGroup and TauCeti.CommHopfAlgCat.isAddTorsionFree_additiveCharacterGroup_of_smooth: their additive forms.

References

• J. S. Milne, Algebraic Groups (2017), Definitions 12.14 and 12.17.
• W. C. Waterhouse, Introduction to Affine Group Schemes, Chapter 2.
• T. A. Springer, Linear Algebraic Groups, Theorem 6.3.1.

theorem TauCeti.CommHopfAlgCat.isMulTorsionFree_geometricCharacterGroup {k : Type u} [Field k] (H : CommHopfAlgCat k) (hred : geometricallyReducedCommHopfAlgProperty k H) (hconn : geometricallyConnectedCommHopfAlgProperty k H) : IsMulTorsionFree (geometricCharacterGroup H)

The geometric character group of a geometrically reduced, geometrically connected commutative Hopf algebra is torsion free.

theorem TauCeti.CommHopfAlgCat.isAddTorsionFree_additiveCharacterGroup {k : Type u} [Field k] (H : CommHopfAlgCat k) (hred : geometricallyReducedCommHopfAlgProperty k H) (hconn : geometricallyConnectedCommHopfAlgProperty k H) : IsAddTorsionFree (additiveCharacterGroup H)

The additive character group of a geometrically reduced, geometrically connected commutative Hopf algebra has no additive torsion.

theorem TauCeti.CommHopfAlgCat.isMulTorsionFree_geometricCharacterGroup_of_smooth {k : Type u} [Field k] (H : CommHopfAlgCat k) (hsmooth : smoothCommHopfAlgProperty k H) (hconn : geometricallyConnectedCommHopfAlgProperty k H) : IsMulTorsionFree (geometricCharacterGroup H)

The character lattice of a smooth geometrically connected affine group is torsion free. Over a field, smoothness implies geometric reducedness of its coordinate algebra.

theorem TauCeti.CommHopfAlgCat.isAddTorsionFree_additiveCharacterGroup_of_smooth {k : Type u} [Field k] (H : CommHopfAlgCat k) (hsmooth : smoothCommHopfAlgProperty k H) (hconn : geometricallyConnectedCommHopfAlgProperty k H) : IsAddTorsionFree (additiveCharacterGroup H)

The additive character lattice of a smooth geometrically connected affine group has no additive torsion.