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.

Modules on a ringed space

Definition

Let $(X, O_{X})$ be a ringed space.

An $O_{X}$ -module is a sheaf $F$ of abelian groups on $X$ such that for every open set $U \subseteq X$ the section group $F (U)$ is an $O_{X} (U)$ -module, and for every inclusion $V \subseteq U$ the restriction map

F (U) ⟶ F (V)

is $O_{X} (U)$ -linear after restricting scalars along $O_{X} (U) \to O_{X} (V)$ .

A morphism of $O_{X}$ -modules is a morphism of the underlying sheaves of abelian groups whose components are $O_{X} (U)$ -linear on every open set $U$ .

Depends on

Used by

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, Section 6.26 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, Example 2.2.13 and Section 2.3 (standard reference, not scraped)