Fppf quotients by kernels of finite dominant homomorphisms #
A finite dominant homomorphism to a geometrically reduced finite-type affine group over any field presents the target as the fppf quotient by its scheme-theoretic kernel. Neither flatness nor reducedness of the source is assumed. In particular, inseparable maps and nonreduced kernels are allowed.
The comparison is the existing kernelFppfQuotientHom; its invertibility follows from the
finite dominant isogeny criterion and the fppf first isomorphism theorem. Finiteness over
the noetherian target coordinate algebra supplies finite presentation.
References #
- J. S. Milne, Algebraic Groups (2017), §5.c.
- W. C. Waterhouse, Introduction to Affine Group Schemes, §15.
theorem
TauCeti.CommHopfAlgCat.isIso_kernelFppfQuotientHom_of_finite_of_dominant
{k : Type u}
[Field k]
{H K : CommHopfAlgCat k}
[Algebra.FiniteType k ↑H]
[Algebra.IsGeometricallyReduced k ↑H]
(f : H ⟶ K)
(hfin : (↑(CommHopfAlgCat.Hom.hom f)).Finite)
(hdom : DenseRange (PrimeSpectrum.comap (↑(CommHopfAlgCat.Hom.hom f)).toRingHom))
:
A finite dominant homomorphism to a geometrically reduced finite-type affine group presents its target as the fppf quotient by its kernel, over an arbitrary field.