Étale affine groups and their Lie algebras

An affine group of finite type over a field is étale exactly when its Lie algebra at the identity is zero. No smoothness, reducedness, or connectedness assumption is needed. This criterion detects infinitesimal structure in finite group schemes, and applies to the scheme-theoretic kernel of a homomorphism via the kernel of its differential.

Over an algebraically closed field, a zero Lie algebra forces the identity component to be trivial. The component-group classification then identifies the group with a finite constant group. The general criterion descends from the algebraic closure: vanishing of the augmentation cotangent space is idempotence of the augmentation ideal, which is preserved by base change.

References

• J. S. Milne, Algebraic Groups (2017), §§2 and 10.
• W. C. Waterhouse, Introduction to Affine Group Schemes, §§6 and 11.

theorem TauCeti.FiniteTypeCommHopfAlgCat.identityComponentHopfIdeal_eq_augmentation_of_finrank_lie_eq_zero {k : Type u} [Field k] [IsAlgClosed k] (H : FiniteTypeCommHopfAlgCat k) (h : Module.finrank k (Derivation k (↑H.obj) (Bialgebra.CounitAlgebra k (↑H.obj) k)) = 0) : HopfAlgebra.identityComponentHopfIdeal = HopfIdeal.augmentation k ↑H.obj

A finite-type affine group with zero Lie algebra has trivial identity component.

theorem TauCeti.HopfAlgebra.algebraEtale_iff_finrank_lie_eq_zero {k H : Type u} [Field k] [CommRing H] [HopfAlgebra k H] [Algebra.FiniteType k H] : Algebra.Etale k H ↔ Module.finrank k (Derivation k H (Bialgebra.CounitAlgebra k H k)) = 0

A finite-type affine group over a field is étale if and only if its Lie algebra is zero.