Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)
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.
‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.
Morphisms of schemes
Definition
A morphism of schemes is a morphism of the underlying locally ringed spaces. In particular its maps on stalks are local homomorphisms.
Depends on
Used by
- Closed immersions of schemes Definition
- Locally finite presentation morphisms Definition
- Locally finite type and finite type morphisms Definition
- Open immersions of schemes Definition
- Quasi-compact and quasi-separated morphisms Definition
- Scheme-theoretic image Definition
- Schemes and morphisms over a base Definition
- Morphisms of schemes are local on compatible open covers Lemma
- Open immersions are monomorphisms Lemma
- Morphisms to an affine scheme and global sections Theorem
- Universal property of scheme reduction Theorem
Dependency tree · two levels
5 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, Morphisms of Schemes, Section 1 (standard reference, not scraped)