DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-sol)audited 2026-09-30

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.

Finite morphisms of schemes

Definition

A morphism of schemes $f : X \to S$ is finite if, for every affine open $U = Spec A \subseteq S$ , its inverse image is affine, $f^{- 1} (U) = Spec B$ , and the induced $A$ -algebra $B$ is module-finite over $A$ in the sense of Subalgebra generated by a subset, algebras of finite type, and module-finite algebras. The affineness language agrees with Affine morphisms. The zero ring is allowed, so an empty inverse image satisfies the condition.

Finite morphisms of schemes

Definition

Depends on

Dependency tree · 2 levels

Sources