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

Definition

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

a continuous map $f : X \to Y$ , and
a morphism of sheaves of rings $f^{♯} : O_{Y} ⟶ f_{*} O_{X}$ on $Y$ .

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

Used by

A morphism of ringed spaces need not be a morphism of locally ringed spaces Counterexample
Morphisms of locally ringed spaces Definition
Pullback of a module along a morphism of ringed spaces Definition

Dependency tree · two levels

7 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

The Stacks Project, Definition 6.25.1 and Section 26.2 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Section 6.2.D (standard reference, not scraped)