LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-30

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High-degree section module is finite graded

Statement

Assume the Axiom of Choice as inherited from the cited suppliers (The Axiom of Choice). Let $A$ be a Noetherian commutative ring with $1$ (Noetherian commutative rings and modules), let $n \geq 0$ , write $S = A [x_{0}, \dots, x_{n}]$ for the polynomial ring in the total-degree grading (The polynomial ring $R [x_{i} : i \in I]$ as finitely supported coefficient families on monomials), let $X = P_{A}^{n}$ with twisting sheaves $O (m)$ (Relative projective space from standard charts, Projective space is Proj of a polynomial ring), and let $F$ be a coherent $O_{X}$ -module (Coherent module sheaves). For every $m$ put $F (m) = F \otimes_{O_{X}} O (m)$ (Twists of a quasi-coherent sheaf), so that $Γ_{*} (F) = ⨁_{m \geq 0} Γ (X, F (m))$ is a graded $S$ -module with the multiplication induced by the maps $O (m) \otimes_{O_{X}} O (k) \to O (m + k)$ of the twisting sheaves (Twisting sheaf on Proj, Invertible twists for degree-one generated rings).

Then there is an integer $m_{0} \geq 0$ such that the truncated graded module $Γ_{\geq m_{0}} (F) = ⨁_{m \geq m_{0}} Γ (X, F (m))$ is a finitely generated $S$ -module (Generated submodule, cyclic and finitely generated modules, module basis and free module).

Moreover, if $i : Y ↪ X$ is a closed subscheme and $G$ a coherent $O_{Y}$ -module, then the extension by zero $i_{*} G$ is a coherent $O_{X}$ -module and the same conclusion holds for it; that is, for a suitable $m_{0}$ the graded module $⨁_{m \geq m_{0}} H^{0} (Y, G \otimes_{O_{Y}} i^{*} O (m))$ is finitely generated over $S$ . The zero module, the zero ring $A = 0$ , the empty projective space and the case $n = 0$ are included.

Facts & Assumptions

Given: The Axiom of Choice as inherited, a Noetherian commutative ring $A$ , an integer $n \geq 0$ , the graded polynomial ring $S = A [x_{0}, \dots, x_{n}]$ , the projective space $X = P_{A}^{n}$ , and a coherent module $F$ on $X$ .

[F1]

The twisting sheaves satisfy $O (m) = \tilde{S (m)}$ (Twisting sheaf on Proj), each $O (m)$ is invertible with $O (m) \otimes_{O_{X}} O (k) ≅ O (m + k)$ and $F (m) = F \otimes_{O_{X}} O (m)$ (Invertible twists for degree-one generated rings, Twists of a quasi-coherent sheaf); on a standard chart $D_{+} (x_{i})$ the twist is trivial, so a twist of a finite type or quasi-coherent module is again of the same kind. Tensoring a short exact sequence of $O_{X}$ -modules by an invertible sheaf preserves exactness: on stalks the invertible sheaf is free of rank one over the local ring, tensoring with a free module is exact, and exactness is stalkwise. (Tensor product of sheaves of modules, The stalk of a tensor product sheaf is the tensor product of the stalks, Under the stated choice boundary, free modules are projective and hence flat, A sequence of abelian sheaves is exact exactly when it is exact on every stalk, Invertible sheaves)

[F2]

$O (1)$ is ample on $P_{A}^{n}$ in the absolute sense: the identity $P_{A}^{n} \to P_{A}^{n}$ is a quasi-compact closed immersion over $Spec A$ pulling $O (1)$ back to $O (1)$ , so $O (1)$ is closed H-very ample relative to the affine base and hence ample. (Relative very ampleness in the finite projective-space convention, Relative very ampleness implies relative ampleness, Absolute ampleness by affine section opens)

[F3]

Since $P_{A}^{n}$ is projective over the Noetherian ring $A$ in the H-projective convention (the identity is a closed immersion over $Spec A$ ) and $O (1)$ is ample, there is $m_{1}$ such that $F (m)$ is globally generated for every $m \geq m_{1}$ . (Eventual generation of coherent projective twists, Global generation by the evaluation map)

[F4]

A globally generated quasi-coherent module $H$ of finite type on a quasi-compact scheme is a quotient of $O^{\oplus N}$ for some finite $N$ : for every point $x$ there are finitely many global sections generating the stalk at $x$ , the locus where a fixed finite family of global sections generates is the complement of the support of the cokernel of $O^{\oplus N} \to H$ , which is closed because that cokernel is quasi-coherent of finite type, and a finite subcover of the quasi-compact space is extracted from these loci. (Global generation by the evaluation map, Support of a finite-type quasi-coherent sheaf is closed, Quasi-compact and quasi-separated schemes)

[F5]

On the locally Noetherian scheme $P_{A}^{n}$ the kernel of a morphism of coherent modules is coherent, and the same holds for twists of coherent modules. (Coherent sheaves on a locally Noetherian scheme, Locally Noetherian and Noetherian schemes, Coherent module sheaves)

[F6]

For a coherent module $H$ on $P_{A}^{n}$ with $A$ Noetherian there is $m_{2}$ such that $H^{q} (X, H (m)) = 0$ for every $q > 0$ and every $m \geq m_{2}$ . Indeed Projective coherent finiteness and large twist vanishing gives a bound for each $q = 1, \dots, n$ ; take the maximum of these finitely many bounds and $0$ . For $q > n$ all twists vanish by Projective n-space has quasi-coherent cohomological dimension at most n, so this maximum works for every $q > 0$ , also for $n = 0$ .

[F7]

For $k \geq 0$ the canonical map $S_{k} \to H^{0} (X, O (k))$ is an isomorphism, and these identifications are compatible with the multiplication maps, so that $Γ_{*} (O)$ is the graded ring $S$ ; consequently for each fixed $a$ and all large $m$ , $Γ (X, O (m + a))$ is identified with $S_{m + a}$ . (Cohomology of O(d) on projective space, Twisting sheaf on Proj)

[F8]

For a fixed integer $c$ and $m_{0} + c \geq 0$ , the shifted graded $S$ -module $⨁_{m \geq m_{0}} S_{m + c}$ is generated in degree $m_{0}$ by the finitely many monomials of polynomial degree $m_{0} + c$ , so it is finitely generated; the ring $S$ is Noetherian, and a quotient of a finitely generated graded module over $S$ is finitely generated. (If $R$ is Noetherian then $R [x_{1}, \dots, x_{n}]$ is Noetherian for every $n \in N$ , Hilbert basis theorem: if $R$ is Noetherian then $R [x]$ is Noetherian, Generated submodule, cyclic and finitely generated modules, module basis and free module)

[F9]

For a closed immersion $i : Y \to X$ into the locally Noetherian scheme $X$ and a coherent $O_{Y}$ -module $G$ , the pushforward $i_{*} G$ is coherent (Closed immersion preserves cohomology and coherent pushforward), and on an affine chart $U = Spec R \subseteq X$ with $i^{- 1} (U) = Spec (R / K)$ the identity $i_{*} (G \otimes_{O_{Y}} i^{*} O (m)) ≅ (i_{*} G) \otimes_{O_{X}} O (m)$ holds, because both sides are, on the basic opens of the chart, the localisations of the same $R / K$ -module; hence $Γ (X, (i_{*} G) (m)) ≅ H^{0} (Y, G \otimes_{O_{Y}} i^{*} O (m))$ . (Closed immersions are affine quotients and survive base change, Direct image of a sheaf along a continuous map, Tensor product of sheaves of modules)

Proof

technique · direct: global-generation at a high twist gives a finite surjection from a finite sum of line bundles with coherent kernel, Serre vanishing makes the section map onto the section module of the image sheaf, and the tail of the finite sum of shifted polynomial tails is finitely generated, so its quotient is too

1.1F2F3

Ampleness and eventual global generation. By [F2] the twisting sheaf $O (1)$ is ample on $P_{A}^{n}$ ; the identity exhibits $P_{A}^{n}$ as closed H-projective over the Noetherian ring $A$ , so [F3] produces $m_{1}$ with $F (m)$ globally generated for all $m \geq m_{1}$ .

2.1F1F4step 1.1

A finite surjection from a finite sum of twists. Since $F (m_{1})$ is globally generated and of finite type on the quasi-compact scheme $P_{A}^{n}$ , [F4] gives a surjection $O^{\oplus N} \to F (m_{1})$ for some finite $N$ . Twisting by $O (- m_{1})$ and using [F1] and its inverse, this yields a surjection $E \to F$ where $E = O (- m_{1})^{\oplus N}$ is a finite direct sum of twists of $O$ .

3.1F5step 2.1

The kernel is coherent. Let $K = \ker (E \to F)$ . Since $A$ is Noetherian, $P_{A}^{n}$ is locally Noetherian, and $K$ is coherent by [F5]; by [F1] every twist $K (m)$ is coherent as well.

4.1F6step 3.1

Serre vanishing for the kernel. Apply [F6] to the coherent module $K$ : there is $m_{2} \geq \max (m_{1}, 0)$ such that $H^{q} (X, K (m)) = 0$ for every $q > 0$ and every $m \geq m_{2}$ ; in particular $H^{1} (X, K (m)) = 0$ for those $m$ .

5.1F1step 4.1

The section maps are surjective. For $m \geq m_{2}$ tensoring $0 \to K \to E \to F \to 0$ by the invertible sheaf $O (m)$ keeps the sequence exact by [F1]. The long exact cohomology sequence of the underlying abelian sheaves (Long exact sequence of sheaf cohomology) contains $Γ (E (m)) \to Γ (F (m)) \to H^{1} (K (m)) = 0$ ; hence the degree- $m$ component $Γ (E (m)) \to Γ (F (m))$ is surjective for every $m \geq m_{2}$ . These maps are compatible with the $S$ -module structure because they are induced by the morphism $E \to F$ and the multiplication maps of the twisting sheaves.

6.1F7F8step 4.1step 5.1

The tail of $E$ is finitely generated. For $m \geq m_{2}$ the module $E (m)$ is the direct sum of $N$ copies of $O (m - m_{1})$ , and by [F7] its sections are identified with the direct sum of $N$ copies of $S_{m - m_{1}}$ ; here $m_{2} - m_{1} \geq 0$ by step 4.1. The resulting tail $⨁_{m \geq m_{2}} Γ (E (m))$ is therefore a finite direct sum of shifted tails of $S$ , each finitely generated by [F8].

7.1F8step 5.1step 6.1

The tail of $F$ is finitely generated. By [step 5.1] the $S$ -module homomorphism of tails $⨁_{m \geq m_{2}} Γ (E (m)) \to ⨁_{m \geq m_{2}} Γ (F (m))$ is surjective, and the source is finitely generated by [step 6.1]; the quotient is finitely generated over the Noetherian ring $S$ by [F8]. This proves the first assertion with $m_{0} = m_{2}$ .

8.1F9step 7.1

Extension by zero from a closed subscheme. Let $i : Y ↪ X$ be a closed subscheme with $X$ locally Noetherian and $G$ coherent on $Y$ . By [F9] the pushforward $i_{*} G$ is coherent on $X$ , so [step 7.1] applied to $F = i_{*} G$ gives $m_{0}$ with $⨁_{m \geq m_{0}} Γ (X, (i_{*} G) (m))$ finitely generated, and by the identification of [F9] this graded module is $⨁_{m \geq m_{0}} H^{0} (Y, G \otimes_{O_{Y}} i^{*} O (m))$ ; this proves the second assertion.

9.1F2F3F5F6F8step 2.1step 7.1cases: zero module and zero ring and n=0∎

Boundary and choice accounting. If $F = 0$ then every $Γ (X, F (m)) = 0$ and the tail is the zero module, which is finitely generated (by the empty family). If $A = 0$ then $S = 0$ and $X = \emptyset$ , all coherent modules are zero and the tail is zero; the Noetherian hypotheses hold for the zero ring and $S$ is Noetherian by [F8]. If $n = 0$ then $X = Spec A$ is affine, $F$ is the associated sheaf of a finitely generated $A$ -module and one checks directly that the tail of $⨁_{m \geq 0} Γ (X, F (m))$ is generated by a finite generating set of $Γ (X, F)$ in degree $m_{0}$ , since the twisting by $O (1)$ is an isomorphism on the single chart; this agrees with the general argument, which also applies because $O (1)$ is ample by [F2]. The Axiom of Choice is consumed through the global-generation theorem [F3], the coherence theorem [F5] and the finiteness theorem [F6]; no chart, resolution or generating family is chosen here beyond the finitely many sections of [step 2.1].

Depends on

Used by

Finite twisted locally free resolutions on projective space Lemma

Dependency tree · two levels

163 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

The Stacks Project, Cohomology of Schemes, Section 30.16 (Tag 01YS) and Section 30.15 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Sections 18.6 and 19.2 (standard reference, not scraped)