Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: Literature-sourcedjudge pass (gpt-5.6-terra)audited 2026-09-07
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 closed subset has many scheme structures

Statement refuted

A closed subset of an affine scheme determines its closed subscheme structure.

Facts & Assumptions

Given: A field k.

[F1]

Closed immersions into Spec⁡A are classified, up to unique isomorphism over the target, by quotient rings A/I Closed immersions into affine schemes are quotient spectra.

Counterexample

technique · direct
1.1given

The ideals (t) and (t2) in k[t] have the same radical (t), so their quotient-spectrum closed immersions have the same underlying closed subset V(t).

2.1F1step 1.1∎

The ring k[t]/(t) is reduced, whereas the nonzero class of t in k[t]/(t2) is nilpotent. Their quotient rings are therefore not isomorphic, so [F1] gives distinct closed subscheme structures with the same support.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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