Finite dominant homomorphisms to geometrically reduced groups #
Over any field, a finite dominant homomorphism to a geometrically reduced affine group of finite type is an isogeny: faithful flatness follows from finiteness and dominance. The source need not be reduced, and the field need not be perfect. This criterion lets quotient and isogeny constructions use geometric hypotheses instead of assuming flatness.
References #
- J. S. Milne, Algebraic Groups (2017), Propositions 1.65(a) and 1.70.
theorem
TauCeti.CommHopfAlgCat.isIsogeny_iff_finite_and_dominant
{k : Type u}
[Field k]
{H K : CommHopfAlgCat k}
[Algebra.FiniteType k ↑H]
[Algebra.IsGeometricallyReduced k ↑H]
(f : H ⟶ K)
:
IsIsogeny f ↔ (↑(CommHopfAlgCat.Hom.hom f)).Finite ∧ DenseRange (PrimeSpectrum.comap (↑(CommHopfAlgCat.Hom.hom f)).toRingHom)
A homomorphism to a geometrically reduced finite-type affine group over a field is an isogeny exactly when it is finite and dominant. No reducedness assumption on the source or perfection assumption on the field is needed.