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.

Fibre product of schemes

Definition

Let $f : X \to S$ and $g : Y \to S$ be morphisms of schemes. A fibre product is a scheme $P$ , with projections $p : P \to X$ and $q : P \to Y$ , such that $f p = g q$ and, for every scheme $T$ and morphisms $a : T \to X$ , $b : T \to Y$ with $f a = g b$ , there is exactly one $h : T \to P$ satisfying $p h = a$ and $q h = b$ . Thus, naturally in every test scheme $T$ , $Hom (T, P) ≅ Hom (T, X) \times_{Hom (T, S)} Hom (T, Y) .$ Write $P = X \times_{S} Y$ . The commutative square with edges $p, q, f, g$ is Cartesian when it has this universal property. Morphisms here are morphisms of locally ringed spaces, as in Morphisms of schemes. No existence assertion is part of the definition.

Fibre product of schemes

Definition

Depends on

Dependency tree · 2 levels

Sources