DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 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.

Twisting sheaf on Proj

Definition

Assume the Axiom of Choice (The Axiom of Choice), inherited from the associated-sheaf construction of Associated sheaf of a graded module on Proj. Let $S = ⨁_{d \geq 0} S_{d}$ be a commutative nonnegatively graded ring, let $X = Proj S$ be the scheme of Proj carries a scheme structure, and for an integer $n$ let $S (n)$ be the graded $S$ -module with $S (n)_{d} = S_{n + d}$ (Graded shift convention for Proj). The twisting sheaf $O_{X} (n)$ is the associated sheaf of the graded module $S (n)$ : $O_{X} (n) = \tilde{S (n)}$ in the sense of Associated sheaf of a graded module on Proj. Thus on a standard open $D_{+} (f)$ with $f$ homogeneous of positive degree, $Γ (D_{+} (f), O_{X} (n)) = S (n)_{(f)} = {a / f^{k} \in S (n) [f^{- 1}] : a \in S (n) homogeneous of degree k d},$ the degree-zero part of the homogeneous localisation of $S (n)$ , with restriction maps induced by homogeneous localisation.

Because $S (0) = S$ and $\tilde{S} = O_{X}$ by construction of the structure sheaf, one has $O_{X} (0) = O_{X}$ . For every pair of integers $m, n$ the multiplication of the graded ring defines sheaf morphisms $O_{X} (m) \otimes_{O_{X}} O_{X} (n) ⟶ O_{X} (m + n),$ induced on $D_{+} (f)$ by the $S_{(f)}$ -bilinear maps $S (m)_{(f)} \times S (n)_{(f)} \to S (m + n)_{(f)}$ , $(a / f^{k}, b / f^{l}) \mapsto a b / f^{k + l}$ , which are compatible with the restriction maps; the induced maps $O_{X} (0) \otimes O_{X} (n) \to O_{X} (n)$ are the canonical identifications.

No invertibility is asserted here. For an arbitrary nonnegatively graded ring the sheaf $O_{X} (n)$ need not be invertible, and the multiplication maps above need not be isomorphisms. The precise positive statement is Invertible twists for degree-one generated rings: if $S$ is generated as an $S_{0}$ -algebra by $S_{1}$ , then every $O_{X} (n)$ is invertible and every such multiplication map is an isomorphism. The sign convention $S (n)_{d} = S_{n + d}$ of Graded shift convention for Proj is used throughout this page.

Remarks

With this convention, the frames in the later example Twist transitions on the projective line transform by $e_{1} = t^{n} e_{0}$ .

Depends on

Used by

Global sections of projective twists Corollary
Intermediate cohomology of projective twists vanishes Corollary
The genus of a smooth plane curve in terms of its degree Corollary
The Picard group of the projective line Corollary
Top cohomology of projective twists Corollary
Global generation does not imply very ampleness Counterexample
h0 differs from the Euler characteristic before vanishing Counterexample
O(-1) has no global generator Counterexample
Hyperelliptic curves and hyperelliptic maps Definition
Relative Proj of a graded quasi-coherent algebra Definition
Twists of a quasi-coherent sheaf Definition
A hyperplane in projective space is effective Cartier with O(H) = O(1) Example
A principal divisor of degree zero on the projective line Example
Adjunction on a smooth plane quartic: the canonical bundle is the hyperplane bundle Example
All twists on the projective line Example
An upper jump of h0 in a flat projective family Example
Generator cocycle for H1 of O(-2) Example
GIT quotients of the projective line for different linearizations Example
Hilbert polynomial of projective space Example
One cocycle carried through the residue realization of Serre duality Example
Projective zero-space over an affine base Example
Serre duality on the projective line, twist by twist Example
The Chow ring of projective space and Bezout degrees Example
The full Riemann-Roch theorem on the projective line, in every degree Example
The intersection pairing on the projective plane Example
The jump l(D+p) - l(D) ranges from zero to the residue degree Example
The twists on the projective line have degree n Example
Twist transitions on the projective line Example
A vector bundle on the projective line has a line subbundle of maximal degree Lemma
Chow groups of projective space Lemma
Cohomology of twists on a smooth hypersurface Lemma
Divisors on the projective line are classified by degree Lemma
Extensions of line bundles on the projective line split after ordering Lemma
Finite twisted locally free resolutions on projective space Lemma
High-degree section module is finite graded Lemma
Hypersurface cohomology sequence Lemma
K-theory of projective space and of projective bundles Lemma
Laurent-monomial decomposition of the projective Cech complex Lemma
Proj is invariant under Veronese regrading Lemma
Proj of a finitely generated graded algebra is projective Lemma

…and 13 more results.

Dependency tree · two levels

8 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, Constructions of Schemes, Section 27.10 (Tag 01MM) (standard reference, not scraped)
Ravi Vakil, The Rising Sea, August 2022 draft, Section 4.5 (standard reference, not scraped)