Reducedness of closed subgroups generated by reduced group schemes #
The closed subgroup scheme generated by affine group schemes is the common-kernel Hopf quotient: the largest Hopf ideal killed by all generator maps. If its coordinate ring had a nilpotent, its reduction would give a further Hopf ideal killed by the same maps, contradicting that maximality.
The reduction is a Hopf ideal whenever the tensor square of the reduced quotient is reduced. Over an algebraically closed field, finite type supplies this tensor condition automatically. The result applies to root-subgroup and torus generators in constructions of split reductive groups. In that setting reducedness also makes the generated subgroup scheme smooth.
References #
- W. C. Waterhouse, Introduction to Affine Group Schemes, §11.4.
- J. S. Milne, Algebraic Groups (2017), Theorem 1.45 and §2.h.
The common-kernel Hopf quotient is reduced when each generator map has radical kernel and the tensor square of the quotient's reduction is reduced. The tensor condition makes the nilradical a Hopf ideal, to which the common-kernel universal property applies.
The common-kernel Hopf quotient of reduced generator algebras is reduced when the tensor square of its reduction is reduced.
Over an algebraically closed field, a finite-type common-kernel quotient is reduced if every generator map has radical kernel.
A subgroup scheme generated by reduced affine group schemes is reduced over an algebraically closed field. The quotient coordinate algebra is of finite type; the generators need not be of finite type.
A finite-type common-kernel quotient with radical generator kernels is smooth over an algebraically closed field.
A subgroup scheme generated by reduced affine group schemes is smooth when its coordinate algebra is of finite type over an algebraically closed field.