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.

Universally closed morphisms

Definition

A morphism of schemes $f : X \to S$ is universally closed if, for every $S$ -scheme $T$ (that is, every morphism $T \to S$ ), the base-changed projection $f_{T} : X_{T} = X \times_{S} T ⟶ T$ is a closed map of topological spaces, where base change is as in Base change of objects, morphisms and properties. Explicitly, for every closed subset $Z \subseteq ∣ X_{T} ∣$ , its image $f_{T} (∣ Z ∣)$ is closed in $∣ T ∣$ . The quantifier ranges over all schemes $T \to S$ ; no finite-type, quasi-compactness, or separatedness hypothesis is part of this definition.

Universally closed morphisms

Definition

Depends on

Dependency tree · 2 levels

Sources