DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-09

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.

Images and set-theoretic fibres of classical regular maps

Definition

For a morphism $ϕ : U \to Y$ , its image is $ϕ (U) = {ϕ (x) : x \in U}$ , and its fibre over $y \in Y$ is the set $ϕ^{- 1} ({y}) = {x \in U : ϕ (x) = y}$ . These are set-theoretic notions. A fibre can be empty. No assertion that images are closed is part of this definition; a fibre here is not a scheme fibre or an ideal quotient.

Sources

Source comparison: Milne, Algebraic Geometry, v6.10, §2j pp. 49–50. Conventions here distinguish arbitrary affine algebraic sets from nonempty irreducible varieties.

Depends on

A morphism from an open subset of a classical affine variety to an affine variety

Used by

A classical fibre is base change to a point Lemma

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

J. S. Milne, Algebraic Geometry v6.10, §2j pp. 49–50 (standard reference, not scraped)