DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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.

Projective morphisms before Proj

Definition

For an arbitrary base scheme $S$ , a morphism $f : X \to S$ is projective on this page if for some integer $n \geq 0$ it factors over $S$ as $X \overset{i}{\to} P_{S}^{n} ⟶ S,$ where $i$ is a closed immersion and the second arrow is the projection. The relative projective space and its projection are those of Relative projective space from standard charts, and closed immersion has the meaning in Closed immersions of schemes. This is the finite-dimensional H-projective convention of Stacks Definition 29.44.1(2). That source's clause (1) separately uses a projective bundle $P (E)$ ; this page does not conflate the two conventions. The value $n = 0$ is allowed, with $P_{S}^{0} ≅ S$ .

Depends on

Dependency tree · 2 levels

Sources

Stacks Project, Morphisms of Schemes, §29.44 Definition 29.44.1(2)