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The section ring of an ample invertible sheaf is finitely generated

Statement

Assume AC as inherited from the projective and sheaf-cohomology suppliers. Let $X$ be a projective scheme over $C$ and $L$ an ample invertible sheaf on $X$ (Absolute ampleness by affine section opens). Then the section ring $R (X, L) = ⨁_{n \geq 0} Γ (X, L^{\otimes n})$ is a finitely generated graded $C$ -algebra, hence a Noetherian ring.

Facts & Assumptions

Given: A projective $C$ -scheme $X$ , an ample invertible sheaf $L$ on $X$ , and the section ring $R (X, L)$ .

[F1]

Very ample positive power. Applied to the proper finite-type morphism $X \to Spec C$ and the ample sheaf $L$ , the very-ampleness theorem gives an integer $m \geq 1$ , an integer $N \geq 0$ and global sections of $L^{\otimes m}$ generating $L^{\otimes m}$ whose associated morphism is a closed immersion $i : X ↪ P_{C}^{N}$ with $i^{*} O (1) ≅ L^{\otimes m}$ . (High powers of an ample line bundle embed a proper scheme, Generating line-bundle sections define a morphism to projective space, Global generation by the evaluation map)

[F2]

Graded sections of a coherent sheaf. For a coherent sheaf $F$ on $P_{C}^{N}$ the graded $S$ -module $Γ_{*} (F) = ⨁_{k \geq 0} Γ (P^{N}, F (k))$ has a finitely generated tail: there is $k_{0}$ with $⨁_{k \geq k_{0}} Γ (P^{N}, F (k))$ finitely generated over $S = C [x_{0}, \dots, x_{N}]$ ; the extended sheaf $i_{*} G$ along the closed immersion $i$ is coherent. Each individual space $Γ (P^{N}, F (k))$ is finite-dimensional over $C$ . (High-degree section module is finite graded, Finite-dimensional coherent cohomology over a field)

[F3]

The coordinate ring image. Let $S = C [x_{0}, \dots, x_{N}]$ and let $A$ be the homogeneous coordinate ring of the closed immersion $i : X ↪ P^{N}$ , namely the image of the graded restriction map $S \to M_{0} = ⨁_{k \geq 0} Γ (X, O_{X} (k))$ . It is a finitely generated graded $C$ -algebra, being a quotient of $S$ ; the restriction maps need not be surjective onto every space of global sections. (Closed subschemes of projective space and saturated ideals)

[F4]

Hilbert basis. A finitely generated algebra over a field is Noetherian. (Hilbert basis theorem: if $R$ is Noetherian then $R [x]$ is Noetherian)

Proof

technique · direct

1.1F1given

If $X = \emptyset$ then $R (X, L) = 0$ is generated by the empty set and is Noetherian; assume henceforth $X \neq \emptyset$ . Fix the integer $m \geq 1$ and the closed immersion $i : X ↪ P_{C}^{N}$ of [F1], so that $O_{X} (1)$ in the following denotes $i^{*} O_{P^{N}} (1) ≅ L^{\otimes m}$ and $L^{\otimes (m k + j)} ≅ i^{*} O (k) \otimes L^{\otimes j}$ for all $k \geq 0$ , $0 \leq j < m$ .

2.1F1F2step 1.1

The graded modules $M_{j}$ . For $0 \leq j < m$ put $M_{j} = ⨁_{k \geq 0} Γ (X, L^{\otimes (m k + j)})$ . Under the identification of step 1.1 and the projection formula for the finite morphism $i$ , $M_{j} = Γ_{*} (F_{j})$ for the coherent sheaf $F_{j} = i_{*} (L^{\otimes j})$ on $P_{C}^{N}$ ; hence by [F2] each $M_{j}$ has a finitely generated tail over $S = C [x_{0}, \dots, x_{N}]$ . A graded $S$ -module whose tail is finitely generated is finitely generated: the missing finite initial part is a finite-dimensional $C$ -vector space by [F2], and an extension of a finitely generated module by a finite-dimensional one is finitely generated. So each $M_{j}$ is a finitely generated graded $S$ -module.

3.1F2F3step 2.1algebra

The section modules over the coordinate ring image. Let $A \subseteq M_{0}$ be the image in [F3]. It is a finitely generated $C$ -algebra. For $p \in S$ in the kernel of $S \to A$ , its restriction is the zero section on $X$ , so multiplication by $p$ is zero on every $Γ (X, L^{\otimes (m k + j)})$ ; hence the $S$ -action on each $M_{j}$ factors through $A$ . The finite $S$ -module generators of step 2.1 therefore also generate $M_{j}$ as an $A$ -module. In particular $M_{0}$ and each of the finitely many $M_{j}$ are finite $A$ -modules.

4.1F3F4step 3.1algebra∎

Conclusion. The decomposition $R (X, L) = ⨁_{j = 0}^{m - 1} M_{j}$ and step 3.1 show that $R (X, L)$ is a finite module over the finitely generated $C$ -algebra $A$ . A finite set of algebra generators of $A$ together with a finite set of $A$ -module generators of $R (X, L)$ generates $R (X, L)$ as a $C$ -algebra. Hence $R (X, L)$ is a finitely generated $C$ -algebra, and it is Noetherian by [F4].

Remarks

The proof uses only the ample power. Neither the very-ampleness of $L$ itself nor a Hilbert--Mumford criterion is used; the finite generation comes from the graded-module theorem on projective space applied to the coherent sheaves $i_{*} (L^{\otimes j})$ .
No separatedness issue. $X$ is a projective $C$ -scheme, in particular proper and of finite type over $Spec C$ ; all cited suppliers are stated in that register.

Depends on

Used by

The projective GIT quotient from the invariant section ring Theorem

Dependency tree · two levels

96 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Victoria Hoskins, Moduli Problems and Geometric Invariant Theory, FU Berlin lecture notes (2015/16) (standard reference, not scraped)
P. E. Newstead, Geometric Invariant Theory, lecture notes, CIMAT Guanajuato 2006 (CEL/HAL; archived copy) (standard reference, not scraped)