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.

A ringed space

Definition

A ringed space is a pair

(X, O_{X})

consisting of a topological space $X$ and a sheaf of commutative rings $O_{X}$ on $X$ .

The sheaf $O_{X}$ is called the structure sheaf of the ringed space.

Depends on

Used by

A locally ringed space Definition
Modules on a ringed space Definition
Morphisms of ringed spaces Definition
Continuous real-valued functions make a space into a locally ringed space Example
Compatible open pieces of ringed or locally ringed spaces glue Theorem
Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories Theorem

Dependency tree · two levels

6 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 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Example 2.2.13 (standard reference, not scraped)