Surjective graded ring homomorphisms induce closed immersions of Proj

A surjective homomorphism f : A →+* B of ℕ-graded rings preserving degrees induces a closed immersion Proj B ⟶ Proj A: the projective spectrum of a graded quotient is a closed subscheme. On the standard chart D₊(s) of Proj A, for s homogeneous of positive degree, the morphism Proj.map f restricts to Spec B_{(f s)} ⟶ Spec A_{(s)}, which is Spec of the induced map of homogeneous localizations, and that map is surjective (HomogeneousLocalization.Away.map_surjective_of_grade_subset_range). Being a closed immersion is local on the target. Since Proj only sees positive degrees, it suffices that every homogeneous element of B of positive degree lies in the image of f; the degree-zero part of B may be larger than the image of that of A.

In particular, if the degree-one part of a graded R-algebra A generates every homogeneous element of positive degree, then the symmetric algebra Sym_R M of any module M mapping onto a generating set of degree-one elements maps onto A in positive degrees (SymmetricAlgebra.gradedLift), so Proj A is a closed subscheme of the projective space Proj (Sym_R M). When M is a finite R-module, Proj (Sym_R M) is proper over Spec R (TauCeti.SymmetricAlgebra.projToSpec). This is how the projective spectrum of a graded algebra generated in degree one, for instance a section ring of a divisor on a curve after passing to a Veronese subring, becomes a closed subscheme of a projective space. Generation is only needed in positive degrees: the degree-zero part of the section ring ⨁ₙ L(nD) over R = k is the field of constants, which is larger than k when k is not algebraically closed in the function field.

Main results

• AlgebraicGeometry.Proj.isClosedImmersion_map_of_grade_subset_range: Proj.map f is a closed immersion for every graded ring homomorphism f whose image contains every homogeneous element of positive degree.
• AlgebraicGeometry.Proj.isClosedImmersion_map: Proj.map f is a closed immersion for every surjective graded ring homomorphism f.
• AlgebraicGeometry.Proj.isClosedImmersion_map_gradedLift: if degree-one elements f m, m ∈ M, generate every homogeneous element of positive degree of a graded R-algebra A, then Proj A is a closed subscheme of Proj (Sym_R M).

References

• R. Hartshorne, Algebraic Geometry, GTM 52, Springer, 1977, Exercise II.3.12(a).

theorem AlgebraicGeometry.Proj.isClosedImmersion_map_of_grade_subset_range {A B σ τ : Type u} [CommRing A] [CommRing B] [SetLike σ A] [SetLike τ B] [AddSubgroupClass σ A] [AddSubgroupClass τ B] {𝒜 : ℕ → σ} {ℬ : ℕ → τ} [GradedRing 𝒜] [GradedRing ℬ] (f : 𝒜 →+*ᵍ ℬ) (hf : HomogeneousIdeal.irrelevant ℬ ≤ HomogeneousIdeal.map f (HomogeneousIdeal.irrelevant 𝒜)) (hsurj : ∀ (i : ℕ), 0 < i → ↑(ℬ i) ⊆ Set.range ⇑f) : IsClosedImmersion (map f hf)

A graded ring homomorphism surjective in positive degrees induces a closed immersion of projective spectra: if the image of f : 𝒜 →+*ᵍ ℬ contains every homogeneous element of ℬ of positive degree, then Proj.map f : Proj ℬ ⟶ Proj 𝒜 is a closed immersion. The containment ℬ₊ ≤ 𝒜₊.map f needed to form Proj.map f always holds for such an f (HomogeneousIdeal.irrelevant_le_map_of_grade_subset_range).

theorem AlgebraicGeometry.Proj.isClosedImmersion_map {A B σ τ : Type u} [CommRing A] [CommRing B] [SetLike σ A] [SetLike τ B] [AddSubgroupClass σ A] [AddSubgroupClass τ B] {𝒜 : ℕ → σ} {ℬ : ℕ → τ} [GradedRing 𝒜] [GradedRing ℬ] (f : 𝒜 →+*ᵍ ℬ) (hf : HomogeneousIdeal.irrelevant ℬ ≤ HomogeneousIdeal.map f (HomogeneousIdeal.irrelevant 𝒜)) (hsurj : Function.Surjective ⇑f) : IsClosedImmersion (map f hf)

A surjective graded ring homomorphism induces a closed immersion of projective spectra: if f : 𝒜 →+*ᵍ ℬ is surjective, then Proj.map f : Proj ℬ ⟶ Proj 𝒜 is a closed immersion. The containment ℬ₊ ≤ 𝒜₊.map f needed to form Proj.map f always holds for a surjective f (HomogeneousIdeal.irrelevant_le_map_of_surjective).

theorem AlgebraicGeometry.Proj.isClosedImmersion_map_gradedLift {A : Type u} [CommRing A] {R M : Type u} [CommRing R] [AddCommMonoid M] [Module R M] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] {f : M →ₗ[R] ↥(𝒜 1)} (hf : ∀ (i : ℕ), 0 < i → ↑(𝒜 i) ⊆ ↑(Algebra.adjoin R (Set.range fun (m : M) => ↑(f m)))) : IsClosedImmersion (map (SymmetricAlgebra.gradedLift R 𝒜 f) ⋯)

A graded algebra generated in degree one is a closed subscheme of a projective space: if f : M →ₗ[R] 𝒜 1 is a linear map whose values generate every homogeneous element of positive degree of the graded R-algebra A as an R-algebra, then the projective spectrum of A is a closed subscheme of Proj (Sym_R M), via the morphism induced by the graded lift Sym_R M → A of f.