Dominant affine group morphisms on algebraically closed points

A dominant homomorphism between finite-type affine group schemes over an algebraically closed field is surjective on rational points. No smoothness, reducedness, or flatness assumption is needed. In particular, an injective coordinate homomorphism gives a surjection on points valued in any algebraically closed extension field. The injective-coordinate-map case supplies the point-surjectivity step toward faithful flatness. Topological dominance alone does not imply flatness for nonreduced groups.

Chevalley's theorem gives a dense open subset contained in the spectral image. Rational points of the source are dense in the target; translating this open subset then expresses each target point as a quotient of two image points.

References

• J. S. Milne, Algebraic Groups (2017), §5, homomorphism theorems.

theorem TauCeti.CommHopfAlgCat.mapPointsFunctor_app_surjective_of_dominant {k : Type u} [Field k] [IsAlgClosed k] {H K : CommHopfAlgCat k} [Algebra.FiniteType k ↑H] [Algebra.FiniteType k ↑K] (f : H ⟶ K) (hf : DenseRange (PrimeSpectrum.comap (↑(CommHopfAlgCat.Hom.hom f)).toRingHom)) : Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom ((mapPointsFunctor f).app ↧k))

A dominant morphism of finite-type affine group schemes over an algebraically closed field is surjective on rational points, without a flatness or reducedness hypothesis.

theorem TauCeti.CommHopfAlgCat.mapPointsFunctor_app_surjective_of_injective {k : Type u} [Field k] {H K : CommHopfAlgCat k} [Algebra.FiniteType k ↑H] [Algebra.FiniteType k ↑K] (L : Type w) [Field L] [Algebra k L] [IsAlgClosed L] (f : H ⟶ K) (hf : Function.Injective ⇑(CommHopfAlgCat.Hom.hom f)) : Function.Surjective ⇑(CategoryTheory.ConcreteCategory.hom ((mapPointsFunctor f).app ↧L))

An injective homomorphism of finite-type commutative Hopf algebras over a field is surjective contravariantly on points valued in any algebraically closed extension field.