LemmaStatement: AI-generatedProof: AI-generated

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Finite global generation on an affine scheme

PDF p. 15: Finite generating surjection onto Q

Statement

Assume the Axiom of Choice. Let $X = Spec A$ be a noetherian affine scheme and let $F$ be a coherent $O_{X}$ -module. There are finitely many global sections $s_{1}, \dots, s_{m}$ such that the morphism $O_{X}^{m} \to F$ sending the standard basis to these sections is surjective. For every open subscheme $U \subseteq X$ , its restriction $O_{U}^{m} \to F ∣_{U}$ is surjective; equivalently, the restricted sections generate $F ∣_{U}$ , whether or not $U$ is affine.

Statement references

Coherent module sheaves

Affine quasi-coherent sheaves are modules

Facts & Assumptions

Given: The Axiom of Choice, a commutative ring $A$ with $1$ , the noetherian affine scheme $X = Spec A$ , and a coherent $O_{X}$ -module $F$ . A morphism of sheaves is called surjective when every target germ has a preimage germ. A family of sections generates a module sheaf when its induced morphism from the corresponding finite free sheaf is surjective.

Goal: Construct finitely many global generating sections and prove that their restrictions generate on every open subscheme.

[F1]

A coherent module is quasi-coherent and of finite type. Under the Axiom of Choice, finite type provides an affine open cover on each member of which finitely many sections generate the restricted sheaf. Coherent module sheaves Finite type and finitely presented module sheaves

[F2]

Under the Axiom of Choice, the canonical map $\tilde{Γ (X, F)} \to F$ is an isomorphism for quasi-coherent $F$ . For any $A$ -module $M$ , its associated sheaf has sections $M_{f}$ on $D (f) = {p : f \notin p}$ , naturally in $M$ and restriction. Affine quasi-coherent sheaves are modules Sections of the associated sheaf on basic opens

[F3]

Distinguished opens form a basis of $X$ , and $X$ is compact under the Axiom of Choice. If finitely many distinguished opens $D (h_{i})$ cover $X$ , with at least one member, then the $h_{i}$ generate the unit ideal. The underlying space of an affine spectrum The prime spectrum is compact in the library's non-Hausdorff sense Distinguished-subset covers detect radicals

[F4]

On an affine scheme, under the Axiom of Choice, a finite family of global sections of an associated module sheaf generates that sheaf exactly when the corresponding elements generate its module. Finite type and finitely presented module sheaves

Proof

technique · Reduce local generation to generation of localised modules, clear denominators, and use a finite distinguished-open cover

1.1F1F2given

Put $M = Γ (X, F)$ . By [F1] and [F2], $F ≅ \tilde{M}$ . If $X$ is empty, then $M = 0$ by the sheaf axiom on the empty cover, so the empty family gives the required surjection; its restriction to every open is also surjective. Henceforth suppose $X$ is nonempty.

2.1F1F3step 1.1

By [F1], $X$ is covered by open sets on each of which finitely many sections generate $F$ . Refine this cover by all distinguished opens contained in its members; these still cover $X$ by [F3]. Compactness gives a finite subcover $D (f_{1}), \dots, D (f_{r})$ , where $r \geq 1$ . Restricting the generating morphism on a member of the original cover to a distinguished open contained in it preserves surjectivity, since the stalks at its points are unchanged. Thus each $F ∣_{D (f_{i})}$ has a finite generating family of sections.

3.1F2F4step 2.1construct

By [F2], each section in the generating family on $D (f_{i})$ is represented by a fraction $m_{i j} / f_{i}^{n_{i j}}$ with $m_{i j} \in M$ and $n_{i j} \geq 0$ . On $D (f_{i})$ the associated sheaf is the associated sheaf of the $A_{f_{i}}$ -module $M_{f_{i}}$ , as follows from the same basic-open section description. By [F4], these fractions generate $M_{f_{i}}$ over $A_{f_{i}}$ . Since $f_{i}$ is invertible in $A_{f_{i}}$ , their numerators $m_{i j} / 1$ also generate $M_{f_{i}}$ . Let $N \subseteq M$ be the submodule generated by all the finitely many $m_{i j}$ . Then $N_{f_{i}} = M_{f_{i}}$ for every $i$ .

4.1F3step 3.1algebra

Let $t \in M$ . For each $i$ , the equality $N_{f_{i}} = M_{f_{i}}$ means that some power $f_{i}^{k_{i}} t$ lies in $N$ : this follows by writing $t / 1$ as a finite linear combination of the $m_{i j} / 1$ and clearing the finitely many denominators and any localisation equality. Choose each $k_{i} \geq 1$ . The opens $D (f_{i}^{k_{i}}) = D (f_{i})$ cover $X$ , so [F3] gives $1 = \sum_{i} a_{i} f_{i}^{k_{i}}$ for suitable $a_{i} \in A$ . Consequently $t = \sum_{i} a_{i} f_{i}^{k_{i}} t \in N$ . As $t$ was arbitrary, $M = N$ .

5.1F1F2F3F4step 1.1step 3.1step 4.1∎

The numerators in step 3.1 are global sections of $F$ under [F2]. Since they generate $M$ by step 4.1, [F4] gives a surjective morphism $O_{X}^{m} \to F$ . For any open subscheme $U$ , the stalk at $x \in U$ of its restriction is the original stalk map at $x$ , hence is surjective. The restricted morphism is therefore surjective and is induced by the restricted global sections. This proves both assertions, including for non-affine $U$ . The noetherian hypothesis is not needed in the argument; the Axiom of Choice is inherited from [F1]–[F4].

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