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.

Every irreducible closed subset of an affine spectrum has a unique generic point

Statement

Assume the Axiom of Choice. Every irreducible closed subset of $Spec A$ has a unique generic point. Thus every affine spectrum is sober.

Facts & Assumptions

Given: A commutative ring $A$ , the Axiom of Choice, and an irreducible closed subset $Z$ of $Spec A$ .

[F1]

A nonempty irreducible closed subset is $V (p)$ for a unique prime $p$ , which is its unique generic point (A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point).

Proof

technique · direct

1.1

The irreducible closed subset $Z$ is nonempty, so [F1] gives a prime $p$ generic for $Z$ .

1.2

Any other generic point has the same closure and is equal to $p$ by the uniqueness in [F1].

2.1

Therefore every irreducible closed subset has a unique generic point.

step 1.1step 1.2∎

Depends on

Used by

Spec Z has one generic point and closed prime-number points Example

Dependency tree · two levels

11 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 10.26 (standard reference, not scraped)