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.

Geometric fibres and geometric points

Definition

A geometric point of a scheme $S$ is a morphism $Spec Ω \to S$ with $Ω$ algebraically closed. For a morphism $f : X \to S$ and a specified point $s \in S$ , choose an algebraic closure $Ω / κ (s)$ , in the sense of An algebraic closure of a field. In this page the geometric fibre of $f$ at $s$ means $X_{\overset{ˉ}{s}} = X_{s} \times_{Spec κ (s)} Spec Ω .$ Here $X_{s}$ is Scheme-theoretic fibre, and its affine charts extend as in Affine charts after extension of the ground field. The choice includes the embedding of $κ (s)$ ; no preferred algebraic closure or preferred isomorphism between choices is implied.

Depends on

Used by

Geometric properties of fibres Definition
Independence of the chosen algebraic closure Lemma

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

Vakil 10.4.3 and table (Rc), (Ic), (Cc) (standard reference, not scraped)