DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Morphisms of ringed spaces

Let $(X, O_{X})$ and $(Y, O_{Y})$ be ringed spaces. A morphism of ringed spaces

(f, f^{♯}) : (X, O_{X}) ⟶ (Y, O_{Y})

consists of

Equivalently, for every open set $V \subseteq Y$ there are ring homomorphisms

f_{V}^{♯} : O_{Y} (V) ⟶ O_{X} (f^{- 1} (V))

compatible with restriction.

Composition is defined by composing the continuous maps and the corresponding morphisms into the direct images of structure sheaves.

Depends on