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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Intermediate cohomology of projective twists vanishes

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 $H^{q} (X, O_{X} (d)) = 0 for every q with 0 < q < n;$ for $n \leq 1$ the index range is empty and the assertion is vacuous. The zero ring $A = 0$ is allowed.

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 $q = n$ . (Cohomology of O(d) on projective space)

Proof

technique · direct: an index range with no integers is vacuous, and otherwise the full computation leaves only degrees zero and n

1.1F1

If $n \leq 1$ there is no integer $q$ with $0 < q < n$ , so the assertion is vacuous. If $n \geq 2$ and $0 < q < n$ , then $q \neq 0$ and $q \neq n$ , so [F1] gives $H^{q} (P_{A}^{n}, O (d)) = 0$ .

2.1F1∎

Boundaries and choice accounting. The cases $n = 0$ and $n = 1$ have empty index range; $q = 0$ and $q = n$ lie outside the range and are not asserted to vanish. The ring $A = 0$ is covered by [F1]. The Axiom of Choice is inherited from [F1] and nothing further is selected.

Depends on

Used by

Nothing in the library uses this result yet.

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)