Normality of reduced closed subgroups from rational points

Over an algebraically closed field k, let H be a reduced finite-type commutative Hopf algebra and I a Hopf ideal with reduced quotient. Then the closed subgroup cut out by I is normal as soon as conjugation by every rational point preserves it.

Normality asks that the coordinate conjugation c♯ : H → H ⊗ H send I into H ⊗ I, the kernel of H ⊗ H → H ⊗ (H ⧸ I). The algebra H ⊗ (H ⧸ I) is reduced and of finite type, so an element of it vanishes once it vanishes at every rational point. A rational point of H ⊗ (H ⧸ I) is a pair (g, h) with h a point of the subgroup, and evaluating c♯ x at it gives x (g h g⁻¹). That value is zero for x ∈ I when conjugation by g preserves I.

Without reducedness the criterion fails: rational points do not see infinitesimal structure.

Main declarations

• TauCeti.HopfIdeal.isNormal_of_forall_le_conjugate: stability under conjugation by rational points gives normality.
• TauCeti.HopfIdeal.isNormal_iff_forall_le_conjugate: the resulting characterization of normal reduced closed subgroups.
• TauCeti.HopfIdeal.IsNormal.comapOfSurjective_of_forall_iso: a reduced characteristic closed subgroup of a normal closed subgroup is normal in the ambient group.

References

• J. S. Milne, Algebraic Groups (2017), §1.d, for the rational points of reduced algebraic schemes, and §3.5, for normal closed subgroups.

theorem TauCeti.HopfIdeal.isNormal_of_forall_le_conjugate {k : Type u} [Field k] [IsAlgClosed k] {H : Type v} [CommRing H] [HopfAlgebra k H] [Algebra.FiniteType k H] [IsReduced H] (I : HopfIdeal k H) [IsReduced (H ⧸ I.toIdeal)] (hI : ∀ (g : WithConv (H →ₐ[k] k)), I ≤ I.conjugate g) : I.IsNormal

Normality from rational points. Over an algebraically closed field, a reduced closed subgroup of a reduced finite-type affine group is normal when conjugation by every rational point maps it into itself.

theorem TauCeti.HopfIdeal.isNormal_iff_forall_le_conjugate {k : Type u} [Field k] [IsAlgClosed k] {H : Type v} [CommRing H] [HopfAlgebra k H] [Algebra.FiniteType k H] [IsReduced H] (I : HopfIdeal k H) [IsReduced (H ⧸ I.toIdeal)] : I.IsNormal ↔ ∀ (g : WithConv (H →ₐ[k] k)), I ≤ I.conjugate g

Over an algebraically closed field, a reduced closed subgroup of a reduced finite-type affine group is normal exactly when conjugation by every rational point maps it into itself.

theorem TauCeti.HopfIdeal.IsNormal.comapOfSurjective_of_forall_iso {k : Type u} [Field k] [IsAlgClosed k] {H : Type v} [CommRing H] [HopfAlgebra k H] [Algebra.FiniteType k H] [IsReduced H] {I : HopfIdeal k H} (hI : I.IsNormal) (J : HopfIdeal k ↑((↧H).quotient I)) [IsReduced (H ⧸ (J.comapOfSurjective (CommHopfAlgCat.Hom.hom ((↧H).mkQuotient I)) ⋯).toIdeal)] (hJ : ∀ (e : (↧H).quotient I ≅ (↧H).quotient I), J.comapOfSurjective (CommHopfAlgCat.Hom.hom e.hom) ⋯ = J) : (J.comapOfSurjective (CommHopfAlgCat.Hom.hom ((↧H).mkQuotient I)) ⋯).IsNormal

A reduced characteristic closed subgroup of a normal closed subgroup is normal in the ambient reduced finite-type affine group over an algebraically closed field. Here characteristic means that every automorphism of the quotient coordinate Hopf algebra preserves its ideal.