Borel subgroups and reduction #
Over a perfect field the nilradical of a commutative Hopf algebra is a Hopf ideal. Every Borel candidate is smooth, hence reduced, so it factors through the reduction. Consequently a finite-type affine group and its reduction have the same Borel subgroups over an algebraically closed field. Combined with the identity-component comparison, this passes Borel subgroups to a reduced connected ambient group.
The proof uses HopfIdeal.reduction_le_of_isReduced_quotient and the surjective-coordinate
comparison in Borel.Subgroup.
References #
- W. C. Waterhouse, Introduction to Affine Group Schemes, §11.4.
- J. S. Milne, Algebraic Groups (2017), §17.a.
theorem
TauCeti.HopfIdeal.isBorelOverAlgClosed_comap_reduction_iff
{k : Type u}
[Field k]
[PerfectField k]
{H : FiniteTypeCommHopfAlgCat k}
(J : HopfIdeal k ↑(H.quotient (reduction k ↑H.obj)).obj)
:
IsBorelOverAlgClosed k H (J.comapOfSurjective (Bialgebra.Quotient.mkBialgHom (reduction k ↑H.obj).toIdeal) ⋯) ↔ IsBorelOverAlgClosed k (H.quotient (reduction k ↑H.obj)) J
A closed subgroup of the reduction is Borel there exactly when it is Borel in the original affine group. The statement also applies to nonreduced ambient groups.