Smooth subgroups generated by geometrically reduced affine groups #
A finite-type closed subgroup generated by geometrically reduced affine groups over a field is smooth, including over imperfect fields. The generator coordinate algebras need not be of finite type. This applies in particular to groups generated by root subgroups and split tori.
References #
- W. C. Waterhouse, Introduction to Affine Group Schemes, §11.4.
- J. S. Milne, Algebraic Groups (2017), Proposition 1.26 and Corollary 1.27.
theorem
TauCeti.CommHopfAlgCat.smoothCommHopfAlgProperty_quotient_commonKernelHopfIdeal_of_geometricallyReduced
{k : Type u}
[Field k]
{H : CommHopfAlgCat k}
{ι : Type w}
{K : ι → CommHopfAlgCat k}
(f : (i : ι) → H ⟶ K i)
[∀ (i : ι), Algebra.IsGeometricallyReduced k ↑(K i)]
[Algebra.FiniteType k ↑(quotient H (commonKernelHopfIdeal f))]
:
A finite-type subgroup generated by geometrically reduced affine groups is smooth over any field. No perfectness assumption or finite-type assumption on the generators is needed.