Linearly reductive subgroups are cut out by their invariant functions

Let G be an affine group over a field k with coordinate Hopf algebra H, and let N be a closed subgroup cut out by the Hopf ideal I. Write B = H^{co H/I} for the functions on G invariant under right translation by N, the candidate coordinate ring of G / N. If N is linearly reductive, then I is generated by the elements of B vanishing at the identity (TauCeti.HopfIdeal.toIdeal_eq_span_coinvariants_counit_eq_zero_of_isLinearlyReductive). Equivalently, by TauCeti.HopfIdeal.canonicalMap_injective_iff_toIdeal_eq_span_coinvariants_counit_eq_zero, the canonical map

H ⊗[B] H → H ⊗[k] (H ⧸ I),    a ⊗ b ↦ (a ⊗ 1) (id ⊗ π)(Δ b)

is bijective (TauCeti.HopfIdeal.canonicalMap_bijective_of_isLinearlyReductive): the morphism (g, n) ↦ (g, g n) identifies G × N with G ×_{Spec B} G. This is the torsor half of the identification of the homogeneous space G / N with the affine scheme Spec B when N is linearly reductive; faithful flatness of H over B is a separate statement.

No normality, smoothness, connectedness or finite-type assumption is made: linear reductivity enters only through exactness of invariants.

The proof uses exactness of invariants. For x ∈ I, the element x ⊗ 1 of the cofree comodule H ⊗ (H ⧸ I) is invariant, and it is a finite sum of values of the comodule morphisms I → H ⊗ (H ⧸ I), m ↦ (b ⊗ 1) (id ⊗ π)(Δ m), at elements of I: this is where the Hopf ideal axioms enter, through TauCeti.HopfIdeal.leftTensorIdeal_le_map_comul_sup_rightTensorIdeal. Over a linearly reductive coalgebra the arguments can be chosen invariant (TauCeti.Comodule.Hom.exists_fixed_sum_eq_of_isLinearlyReductive), and an invariant element of I is a coinvariant with vanishing counit. Applying the counit to the second factor then writes x as an H-linear combination of such coinvariants.

References

• M. Takeuchi, A correspondence between Hopf ideals and sub-Hopf algebras, Manuscripta Math. 7 (1972), 251–270.
• H.-J. Schneider, Principal homogeneous spaces for arbitrary Hopf algebras, Israel J. Math. 72 (1990), 167–195.
• J. S. Milne, Algebraic Groups (2017), §7.d and Chapter 12.

theorem TauCeti.HopfIdeal.toIdeal_eq_span_coinvariants_counit_eq_zero_of_isLinearlyReductive {k : Type u} {H : Type v} [Field k] [CommRing H] [HopfAlgebra k H] (I : HopfIdeal k H) (hlr : Coalgebra.IsLinearlyReductive k (H ⧸ I.toIdeal)) : I.toIdeal = Ideal.span {h : H | h ∈ I.coinvariants ∧ CoalgebraStruct.counit h = 0}

A linearly reductive closed subgroup is cut out by its invariant functions. If the closed subgroup N cut out by the Hopf ideal I is linearly reductive, then I is generated by the functions on G which are invariant under right translation by N and vanish at the identity. Geometrically, N is the scheme-theoretic fibre over the identity coset of G → Spec O(G)^N.

theorem TauCeti.HopfIdeal.canonicalMap_bijective_of_isLinearlyReductive {k : Type u} {H : Type v} [Field k] [CommRing H] [HopfAlgebra k H] (I : HopfIdeal k H) (hlr : Coalgebra.IsLinearlyReductive k (H ⧸ I.toIdeal)) : Function.Bijective ⇑I.canonicalMap

The canonical map of a linearly reductive closed subgroup is bijective. For a linearly reductive closed subgroup N of an affine group G over a field, with B = O(G)^N, the canonical map H ⊗[B] H → H ⊗[k] (H ⧸ I) is an isomorphism: G × N ≅ G ×_{Spec B} G by (g, n) ↦ (g, g n).