Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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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.

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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
1.1F1

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

1.2F1

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

2.1step 1.1step 1.2∎

Therefore every irreducible closed subset has a unique generic point.

Depends on

Used by

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