Étale kernels and injective differentials #
The scheme-theoretic kernel of a morphism of affine groups of finite type over a field is étale exactly when the differential at the identity is injective. In particular this detects whether the kernel of an isogeny has infinitesimal structure. Neither the source nor the target is assumed smooth, and the ground field need not be perfect.
The Lie algebra of the kernel is identified with the kernel of the differential by
CommHopfAlgCat.kernelLieEquiv. The zero-Lie-algebra criterion for étaleness then applies.
References #
- J. S. Milne, Algebraic Groups (2017), §10.
theorem
TauCeti.CommHopfAlgCat.algebraEtale_quotient_kernelHopfIdeal_iff
{k : Type u}
[Field k]
{H K : CommHopfAlgCat k}
[Algebra.FiniteType k ↑K]
(f : H ⟶ K)
:
Algebra.Etale k (↑K ⧸ (kernelHopfIdeal f).toIdeal) ↔ Function.Injective ⇑(derivationCompLieHom (CommHopfAlgCat.Hom.hom f))
The kernel of an affine group morphism is étale exactly when its differential is injective.
Only the source group scheme, represented by K, is required to be of finite type.