Local unit rescaling and equality of projective morphisms

Two systems of global homogeneous coordinates on an affine scheme define the same projective morphism precisely when, on an affine open cover, their degree-n coordinates differ by the nth power of a unit. The degree-one coordinates of the first system must generate the unit ideal. The units may vary between opens.

This criterion compares scheme morphisms, including their structure-sheaf maps. It therefore detects infinitesimal fibers and applies to nonreduced schemes. It combines the degree-one chart criterion Proj.SpecMap_awayLift_awayι_eq_iff with descent of equality along an open cover.

References

• J. S. Milne, Algebraic Groups (2017), §§7.d–7.f.

theorem AlgebraicGeometry.Proj.fromOfGlobalSections_eq_of_openCover_unit_rescaling {A σ : Type u} [CommRing A] [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] {X : Scheme} (f g : A →+* ↑(X.presheaf.obj (Opposite.op ⊤))) (hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤) (hg : Ideal.map g (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤) (𝒰 : X.OpenCover) (h : ∀ (i : 𝒰.I₀), ∃ (c : (↑((𝒰.X i).presheaf.obj (Opposite.op ⊤)))ˣ), ∀ (n : ℕ), ∀ a ∈ 𝒜 n, (CategoryTheory.ConcreteCategory.hom (Scheme.Hom.appTop (𝒰.f i))) (g a) = ↑c ^ n * (CategoryTheory.ConcreteCategory.hom (Scheme.Hom.appTop (𝒰.f i))) (f a)) : fromOfGlobalSections 𝒜 f hf = fromOfGlobalSections 𝒜 g hg

Locally unit-proportional homogeneous coordinates define the same projective morphism. The source scheme and the members of the cover need not be affine.

theorem AlgebraicGeometry.Proj.fromOfGlobalSections_eq_iff_exists_affineOpenCover_unit_rescaling {A σ : Type u} [CommRing A] [SetLike σ A] [AddSubgroupClass σ A] (𝒜 : ℕ → σ) [GradedRing 𝒜] {X : Scheme} [IsAffine X] (f g : A →+* ↑(X.presheaf.obj (Opposite.op ⊤))) (hf : Ideal.map f (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤) (hg : Ideal.map g (HomogeneousIdeal.irrelevant 𝒜).toIdeal = ⊤) (hspan : Ideal.span (⇑f '' ↑(𝒜 1)) = ⊤) : fromOfGlobalSections 𝒜 f hf = fromOfGlobalSections 𝒜 g hg ↔ ∃ (𝒰 : X.OpenCover), (∀ (i : 𝒰.I₀), IsAffine (𝒰.X i)) ∧ ∀ (i : 𝒰.I₀), ∃ (c : (↑((𝒰.X i).presheaf.obj (Opposite.op ⊤)))ˣ), ∀ (n : ℕ), ∀ a ∈ 𝒜 n, (CategoryTheory.ConcreteCategory.hom (Scheme.Hom.appTop (𝒰.f i))) (g a) = ↑c ^ n * (CategoryTheory.ConcreteCategory.hom (Scheme.Hom.appTop (𝒰.f i))) (f a)

On an affine source covered by its degree-one coordinates, equality of projective morphisms is equivalent to unit rescaling of homogeneous coordinates on an affine open cover. No reducedness assumption is made.