DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-06

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.

The underlying space of an affine spectrum

Definition

All rings below are commutative with $1$ . For a ring $A$ , the underlying topological spectrum is the Zariski space whose points are the prime ideals of $A$ ; its basic opens are $D (f) = {p : f \notin p}$ . This notation is deliberately topological until the structure sheaf is constructed. For $A = 0$ there are no proper prime ideals, so this space is empty.

Depends on

Used by

Affine schemes and their coordinate rings Definition
The localization presheaf on distinguished opens Definition
The spectrum of a field is a one-point affine scheme Example
The zero ring has empty spectrum Example
The localization construction extends to the structure sheaf on Spec A Theorem

Dependency tree · two levels

9 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 26.5 (standard reference, not scraped)