Normal unipotent subgroups of linearly reductive affine groups #
Let H be a reduced, finite-type coordinate Hopf algebra of an affine group over an
algebraically closed field k, and let I be a normal Hopf ideal cutting out a closed subgroup
N whose coordinate ring is reduced, of finite type, and has only unipotent points. If H is
linearly reductive then N is the identity subgroup, that is, I is the augmentation ideal.
The argument is the standard one. Fix a finite-dimensional representation V of the ambient
group. Its N-invariants are stable under the ambient group because I is normal, and geometric
point separation promotes that stable subspace to a subcomodule of V. Complete reducibility
supplies an ambient complement, which then contains no nonzero N-fixed vector; Kolchin's
theorem, already available for unipotent groups, forces such a complement to vanish. So N acts
trivially on every finite-dimensional representation of the ambient group. Applying this to the
finite-dimensional subcomodules of the regular representation, which exhaust H, shows that the
quotient morphism H ⟶ H ⧸ I sends h to ε h • 1, hence that I is the kernel of the counit.
The whole-group case — a linearly reductive unipotent group is trivial — is
TauCeti.HopfAlgebra.counitBialgEquivOfIsLinearlyReductiveOfForallIsUnipotentPoint, and it does
not suffice here: a closed subgroup of a linearly reductive group is not visibly linearly
reductive. Complete reducibility is therefore used on representations of the ambient group, and
normality is what makes the invariants of the subgroup an ambient subrepresentation.
For the object properties this gives one half of the Layer 6 comparison: a smooth, geometrically connected finite-type affine group whose geometric fibre is linearly reductive is reductive. The converse implication in characteristic zero is a separate development.
Main declarations #
mkQuotient_coact_eq_tmul_one_of_isNormal_of_forall_isUnipotentPoint_of_isCompletelyReducible: over an algebraically closed field, a normal unipotent closed subgroup of a reduced finite-type affine group acts trivially on every completely reducible finite-dimensional representation.mkQuotient_eq_counit_smul_one_of_isNormal_of_forall_isUnipotentPoint_of_isLinearlyReductive: under the same hypotheses, the quotient morphism of a linearly reductive group ish ↦ ε h • 1.eq_augmentation_of_isNormal_of_forall_isUnipotentPoint_of_isLinearlyReductive: such a normal unipotent closed subgroup is trivial.quotientCounitBialgEquivOfIsNormalOfForallIsUnipotentPointOfIsLinearlyReductive: the same conclusion as a bialgebra equivalence between the subgroup's coordinate ring and the ground field.of_smooth_of_geometricallyConnected_of_baseChange_linearlyReductive: a smooth, geometrically connected finite-type affine group with linearly reductive geometric fibre is reductive.
References #
- J. S. Milne, Algebraic Groups (2017), Corollary 12.45 and §22.42.
- W. C. Waterhouse, Introduction to Affine Group Schemes, §3.2 and §8.3.
- J. C. Jantzen, Representations of Algebraic Groups, I.2.
This advances the Layer 6 milestone "Reductive and semisimple groups" of the ReductiveGroups roadmap, whose instruction is to "provide both definitions and the char-0 equivalence so downstream work can pick either": this is the implication from linear reductivity to reductivity, and it holds in every characteristic.
Over an algebraically closed field, a normal closed subgroup whose coordinate ring is reduced, of finite type and has only unipotent points acts trivially on every completely reducible finite-dimensional representation of an ambient affine group with reduced, finite-type coordinate ring.
Normality makes the subgroup's fixed subspace an ambient subcomodule, complete reducibility supplies an ambient complement, and Kolchin's theorem finds a nonzero fixed vector inside a nonzero complement.
Over an algebraically closed field, the regular representation of a linearly reductive,
reduced finite-type affine group is fixed by a normal unipotent closed subgroup: comultiplication
followed by the quotient morphism in the second factor sends h to h ⊗ 1.
Every element lies in a finite-dimensional subcoalgebra, hence in a finite-dimensional subcomodule of the regular comodule, where the previous theorem applies.
Over an algebraically closed field, the quotient morphism of a linearly reductive, reduced
finite-type affine group by a normal unipotent closed subgroup is h ↦ ε h • 1.
Contracting the previous identity against the counit in the first factor removes comultiplication.
Over an algebraically closed field, a linearly reductive affine group with reduced, finite-type coordinate ring has no nontrivial normal unipotent closed subgroup.
A normal Hopf ideal whose quotient coordinate ring is reduced, of finite type and has only unipotent points is the augmentation ideal; contravariantly, the closed subgroup it cuts out is the identity subgroup. Connectedness of the subgroup is not needed.
Over an algebraically closed field, the coordinate Hopf algebra of a normal unipotent closed subgroup of a linearly reductive affine group with reduced, finite-type coordinate ring has zero augmentation ideal.
Over an algebraically closed field, a normal unipotent closed subgroup of a linearly reductive affine group with reduced, finite-type coordinate ring is trivial: its coordinate Hopf algebra is bialgebra-equivalent to the ground field via the counit.
Equations
Instances For
The triviality equivalence of a normal unipotent closed subgroup is its counit.
The inverse of the triviality equivalence is the structure map.
A smooth, geometrically connected finite-type affine group whose geometric fibre is linearly reductive is reductive.
This is the implication from linear reductivity to reductivity in Layer 6 of the ReductiveGroups roadmap. Linear reductivity is asked of the geometric fibre because reductivity is defined after extension to an algebraic closure.