Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-07
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.
Scheme-theoretic image
Definition
For a morphism , its scheme-theoretic image, when it exists, is the smallest closed subscheme through which factors. Existence is not part of this definition.
Depends on
Used by
- Algebraic group actions, orbit maps, orbit subschemes and scheme-theoretic stabilizers Definition
- Strict transform of a closed subscheme Definition
- The derived subgroup, the derived series and solvable algebraic groups Definition
- The scheme-theoretic image of a dense open immersion Example
- Chow lemma for proper Noetherian schemes Lemma
- Fibres of the orbit map and the scheme-theoretic stabilizer as a closed subgroup scheme Lemma
- Fixed loci are closed and a normal subgroup fixing a point fixes the orbit closure Lemma
- Proper pushforward of cycles and the norm formula Lemma
- Schematic closure and agreement on a dense open Lemma
- Smooth orbits are locally closed and their orbit maps are faithfully flat over every field Lemma
- SNC projective compactifications of smooth quasiprojective varieties Lemma
- A support does not determine its scheme structure Remark
- Line bundles of degree at least 2g+1 are very ample Theorem
- Scheme-theoretic image of a quasi-compact morphism Theorem
- The canonical map: base-point-freeness and the hyperelliptic exception 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, Definition 6.2 (standard reference, not scraped)