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

Used by

A proper nonprojective scheme from glued projective spaces Counterexample
Hilbert function and Euler characteristic on a projective scheme Definition
Quasi-projective morphisms before Proj Definition
The empty morphism is finite, proper and projective Example
A projective morphism has a relative Proj presentation Lemma
Chow lemma for proper Noetherian schemes Lemma
Eventual generation of coherent projective twists Lemma
Projective and proper are distinct notions Remark
Coherent higher direct images under proper morphisms Theorem
Degree of the coherent Hilbert polynomial Theorem
Euler characteristic is a Hilbert polynomial Theorem
Projective morphisms are proper Theorem
Serre duality for coherent sheaves on projective space Theorem
Serre vanishing for coherent sheaves and ample twists Theorem

Dependency tree · two levels

10 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

Stacks Project, Morphisms of Schemes, §29.44 Definition 29.44.1(2) (standard reference, not scraped)