DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge 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 fibre

Definition

For a morphism $f : X \to S$ and any point $s \in S$ , its scheme-theoretic fibre is $X_{s} = X \times_{S} Spec κ (s),$ viewed as a $κ (s)$ -scheme. The map $Spec κ (s) \to S$ is the canonical residue-field point from Field-valued points and local-ring points, and the product is base change as in Base change of objects, morphisms and properties. The point $s$ need not be closed. A fibre over a generic point is called a generic fibre. Empty fibres are allowed.

Depends on

Used by

Geometric fibres and geometric points Definition
Fibres after base change Lemma
Coordinate ring of an affine fibre Theorem

Dependency tree · two levels

9 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 26.18.4; Vakil 10.3.2 (standard reference, not scraped)