Faithful flatness of the morphism onto an affine group image #
For a homomorphism of finite-type affine groups with geometrically reduced source over a field,
the morphism from the source to its scheme-theoretic image is faithfully flat. In coordinates,
the injective factor imageι f : H / ker f ⟶ K is faithfully flat. The ambient target may be
nonreduced, the field may be imperfect, and the homomorphism need not be finite.
This supplies the flatness needed to identify the image with the fppf quotient by the kernel.
References #
- J. S. Milne, Algebraic Groups (2017), §5.a and Proposition 1.70.
- W. C. Waterhouse, Introduction to Affine Group Schemes, §§14--16.
theorem
TauCeti.CommHopfAlgCat.faithfullyFlat_imageι
{k : Type u}
[Field k]
{H K : CommHopfAlgCat k}
[Algebra.FiniteType k ↑H]
[Algebra.FiniteType k ↑K]
[Algebra.IsGeometricallyReduced k ↑K]
(f : H ⟶ K)
:
(↑(CommHopfAlgCat.Hom.hom (imageι f))).FaithfullyFlat
The homomorphism from a geometrically reduced finite-type affine group onto its scheme-theoretic image is faithfully flat, over any field.