Fppf lifting of affine-valued points

A surjective, flat morphism locally of finite presentation admits lifts of every affine-valued point after one faithfully flat, finitely presented extension of its value ring. Such a morphism is open, so finitely many affine opens of the pullback of the point have images covering the affine source of the point. The extension is the coordinate ring of their disjoint union. This works even when the target scheme is nonaffine and the value ring is nonreduced.

This lifting statement makes geometric fppf covers usable in comparisons of functors of points, for example when realizing a homogeneous quotient as a projective orbit.

References

• J. S. Milne, Algebraic Groups (2017), §5, quotient sheaves.

theorem AlgebraicGeometry.Scheme.Hom.exists_faithfullyFlat_lift {X Y : Scheme} (f : X ⟶ Y) [Flat f] [Surjective f] [LocallyOfFinitePresentation f] {A : CommRingCat} (y : Spec A ⟶ Y) : ∃ (B : CommRingCat) (φ : A ⟶ B) (z : Spec B ⟶ X), (CommRingCat.Hom.hom φ).FaithfullyFlat ∧ (CommRingCat.Hom.hom φ).FinitePresentation ∧ CategoryTheory.CategoryStruct.comp z f = CategoryTheory.CategoryStruct.comp (Spec.map φ) y

Every affine-valued point of the target of an fppf cover lifts after a faithfully flat, finitely presented extension of the value ring.