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.

Depends on

Base change of objects, morphisms and properties

Used by

A nonclosed open immersion is not proper Counterexample
The affine line is not proper Counterexample
Proper morphisms Definition
The empty morphism is finite, proper and projective Example
Fpqc descent of properness components Lemma
Projective-space projection is universally closed by finite graded pieces Lemma
Properness is local on the target Lemma
Properness survives arbitrary base change Lemma
Properness survives composition Lemma
Valuation lifts detect universal closedness Lemma
Properness is not a compactness claim on rational points Remark
Proper morphisms are closed Theorem

Dependency tree · two levels

3 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, Schemes, §26.20 Definition 26.20.1 (standard reference, not scraped)