CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-30

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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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Global sections of projective twists

Statement

Assume the Axiom of Choice as inherited from the cited theorem (The Axiom of Choice). Let $A$ be a commutative ring with $1$ (Commutative ring), let $n \geq 0$ and $d \in Z$ , and let $O_{X} (d)$ be the twisting sheaf on $X = P_{A}^{n}$ (Relative projective space from standard charts, Twisting sheaf on Proj). Then for $n > 0$ , $H^{0} (X, O_{X} (d)) ≅ A [x_{0}, \dots, x_{n}]_{d} (d \geq 0), H^{0} (X, O_{X} (d)) = 0 (d < 0),$ where $A [x_{0}, \dots, x_{n}]_{d}$ is the degree- $d$ graded piece of the polynomial ring (Nonnegatively graded rings and modules, homogeneous elements, and twists), and for $n = 0$ , $H^{0} (P_{A}^{0}, O (d)) ≅ A for every d \in Z .$ The zero ring $A = 0$ is allowed, in which case $X = \emptyset$ and both sides are zero.

Facts & Assumptions

Given: The Axiom of Choice as inherited, a commutative ring $A$ with $1$ , integers $n \geq 0$ and $d$ , and the twisting sheaf $O (d)$ on $P_{A}^{n}$ .

[F1]

Cohomology of twists on projective space: for every commutative ring $A$ , every $n \geq 0$ and every $d \in Z$ , $H^{q} (P_{A}^{n}, O (d)) = 0$ unless $q = 0$ or $q = n$ ; if $n > 0$ then $H^{0} (P_{A}^{n}, O (d)) ≅ A [x_{0}, \dots, x_{n}]_{d}$ for $d \geq 0$ and $H^{0} (P_{A}^{n}, O (d)) = 0$ for $d < 0$ ; and for $n = 0$ , $P_{A}^{0} = Spec A$ with $H^{0} (P_{A}^{0}, O (d)) ≅ A$ for every $d$ and all higher groups zero, with $A = 0$ allowed. (Cohomology of O(d) on projective space)

Proof

technique · direct: read the degree-zero part off the full computation of the cohomology of twists

1.1F1

Apply [F1] to $A$ , $n$ and $d$ . In degree $q = 0$ it states exactly the three assertions: for $n > 0$ the group $H^{0} (P_{A}^{n}, O (d))$ is $A [x_{0}, \dots, x_{n}]_{d}$ when $d \geq 0$ and $0$ when $d < 0$ , while for $n = 0$ it is $A$ for every $d$ ; in the case $A = 0$ the theorem records that $P_{A}^{n} = \emptyset$ and all groups vanish, matching $A [x_{0}, \dots, x_{n}]_{d} = 0$ .

2.1F1∎

Boundaries and choice accounting. The case $d = 0$ is included and gives $H^{0} ≅ A$ for $n \geq 0$ : for $n > 0$ the degree-zero piece $A [x_{0}, \dots, x_{n}]_{0}$ consists of the constant polynomials, and for $n = 0$ the theorem gives $A$ directly. The case $n = 0$ is not deduced from the $n > 0$ formula, which would give $A$ only for $d \geq 0$ , but taken from the theorem's separate clause for every sign of $d$ . The Axiom of Choice is inherited from [F1] and nothing further is selected.

Depends on

Used by

Proper cohomology need not be finite for noncoherent sheaves Counterexample
Coherence is essential for proper finiteness Remark

Dependency tree · two levels

33 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, Lemma 30.8.2 (Tag 01XV) (standard reference, not scraped)
Ravi Vakil, The Rising Sea (29 August 2022), Section 19.2 (standard reference, not scraped)