Veronese subrings of finitely generated graded algebras

Let A = ⨁ₙ 𝒜 n be an ℕ-graded commutative R-algebra which is generated over its degree-zero part 𝒜 0 by finitely many homogeneous elements of positive degree. Then for every sufficiently divisible e > 0 the Veronese subring ⨁ₘ 𝒜 (m e) is generated by its degree-one part 𝒜 e:

𝒜 (m e) = (𝒜 e) ^ m for every m ≥ 1,

as R-submodules of A. Explicitly, if s is a finite set of homogeneous generators whose degrees divide h, this holds for every multiple e of h with #s · h ≤ e.

The reason is a pigeonhole count. A monomial in the generators of degree greater than #s · h contains some generator x at least h / deg x times, so it is divisible by a monomial x ^ (h / deg x) of degree h. Hence 𝒜 (h + N) = 𝒜 h · 𝒜 N once N ≥ #s · h, and iterating splits a component of degree m e into m components of degree e.

If a graded algebra B over a ring B₀ is generated by its degree-one part, then Proj B is a closed subscheme of the projective space Proj (Sym_{B₀} M) for any B₀-module M mapping onto a set of degree-one generators (AlgebraicGeometry.Proj.isClosedImmersion_map_gradedLift). When B is finitely generated over B₀, M can be taken finite, so this projective space is of finite type over Spec B₀. Passing to a Veronese subring reduces a finitely generated graded algebra, such as the section ring of a divisor on a curve, to this case.

Main results

• TauCeti.GradedAlgebra.grade_add_eq_mul_of_card_mul_le: 𝒜 (e + N) = 𝒜 e · 𝒜 N when h ∣ e and #s · h ≤ N.
• TauCeti.GradedAlgebra.grade_mul_eq_pow_of_card_mul_le: 𝒜 (m e) = (𝒜 e) ^ m for m ≠ 0, when h ∣ e and #s · h ≤ e.
• TauCeti.GradedAlgebra.exists_grade_mul_eq_pow: for a graded algebra of finite type over 𝒜 0, some e > 0 satisfies 𝒜 (m e') = (𝒜 e') ^ m for every multiple e' of e and every m ≠ 0.

References

• A. Grothendieck, Éléments de géométrie algébrique II, Publ. Math. IHÉS 8 (1961), §2.1.
• N. Bourbaki, Commutative Algebra, Chapter III, §1.

theorem TauCeti.GradedAlgebra.grade_add_eq_mul_of_card_mul_le {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommSemiring A] [Algebra R A] {𝒜 : ℕ → Submodule R A} {s : Finset A} {deg : A → ℕ} {h : ℕ} [GradedAlgebra 𝒜] (hs : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤) (hdeg : ∀ x ∈ s, x ∈ 𝒜 (deg x)) (hdvd : ∀ x ∈ s, deg x ∣ h) {e N : ℕ} (he : h ∣ e) (hN : s.card * h ≤ N) : 𝒜 (e + N) = 𝒜 e * 𝒜 N

Splitting off a component of degree e. Let A be generated over 𝒜 0 by a finite set s of homogeneous elements whose degrees divide h. If h ∣ e and N ≥ #s · h, then 𝒜 (e + N) = 𝒜 e · 𝒜 N.

theorem TauCeti.GradedAlgebra.grade_mul_eq_pow_of_card_mul_le {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommSemiring A] [Algebra R A] {𝒜 : ℕ → Submodule R A} {s : Finset A} {deg : A → ℕ} {h : ℕ} [GradedAlgebra 𝒜] (hs : Algebra.adjoin ↥(𝒜 0) ↑s = ⊤) (hdeg : ∀ x ∈ s, x ∈ 𝒜 (deg x)) (hdvd : ∀ x ∈ s, deg x ∣ h) {e : ℕ} (he : h ∣ e) (hse : s.card * h ≤ e) {m : ℕ} (hm : m ≠ 0) : 𝒜 (m * e) = 𝒜 e ^ m

The Veronese subring in degree e is generated in degree one. Let A be generated over 𝒜 0 by a finite set s of homogeneous elements whose degrees divide h. If h ∣ e and #s · h ≤ e, then 𝒜 (m e) = (𝒜 e) ^ m for every m ≠ 0.

theorem TauCeti.GradedAlgebra.exists_grade_mul_eq_pow {R : Type u_1} {A : Type u_2} [CommSemiring R] [CommSemiring A] [Algebra R A] (𝒜 : ℕ → Submodule R A) [GradedAlgebra 𝒜] [Algebra.FiniteType (↥(𝒜 0)) A] : ∃ (e : ℕ), 0 < e ∧ ∀ ⦃e' : ℕ⦄, e ∣ e' → ∀ ⦃m : ℕ⦄, m ≠ 0 → 𝒜 (m * e') = 𝒜 e' ^ m

Veronese subrings of a graded algebra of finite type are eventually generated in degree one (EGA II, §2.1). If A is a finitely generated algebra over 𝒜 0, then there is e > 0 such that for every multiple e' of e, 𝒜 (m e') = (𝒜 e') ^ m for all m ≠ 0.