TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

The stalk of the affine structure sheaf at a prime is A_p

Statement

For $p \in Spec A$ , there is a canonical isomorphism $O_{Spec A, p} ≅ A_{p}$ .

Facts & Assumptions

Given: A prime $p$ and the affine structure sheaf.

Proof

technique · direct

1.1

The opens $D (f)$ with $f \notin p$ are cofinal neighborhoods of $p$ , and their sections are $A_{f}$ .

given

2.1

Hence the stalk is ${\underset{\to}{l i m}}_{f \notin p} A_{f}$ .

step 1.1

3.1

This colimit is $A_{p}$ by the universal property of localization.

step 2.1∎

Depends on

Used by

Spec A with its structure sheaf is a locally ringed space Corollary
The residue field at a point of an affine scheme Definition
The spectrum of a field is a one-point affine scheme Example
The spectrum of a product ring is a disjoint union Example
The stalk maps induced by a ring map are local Lemma

Dependency tree · two levels

18 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 26.5.4 (standard reference, not scraped)