LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge 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.

A principal localization identifies its spectrum with a distinguished open

Statement

For $f \in A$ , the morphism induced by $A \to A_{f}$ identifies $Spec (A_{f})$ with the open locally ringed subspace $D (f)$ of $Spec A$ .

Facts & Assumptions

Given: A commutative ring $A$ and $f \in A$ .

[F1]

The map on prime spectra induced by $A \to A_{f}$ is a homeomorphism onto $D (f)$ (The spectrum of a principal localisation is the distinguished open D(f)).

[F2]

The structure-sheaf sections on a distinguished open are the corresponding localizations (Sections and restrictions on distinguished opens of an affine scheme).

Proof

technique · direct

1.1

By [F1], the underlying map is a homeomorphism from $Spec (A_{f})$ onto $D (f)$ .

1.2

On $D (g) \subseteq D (f)$ the relevant section rings are $(A_{f})_{g / 1}$ and $A_{g}$ , canonically isomorphic and compatible with restrictions.

F2algebra

2.1

The basic opens cover $D (f)$ , so the preceding identifications give an isomorphism of locally ringed spaces.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

12 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, Lemma 26.6.6 (standard reference, not scraped)