Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedaudited 2026-09-12
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.

The local ring of the affine line at the origin consists of rational functions defined at the origin

Example

Assume the Axiom of Choice, let k be an algebraically closed field, and take X=Ak1 with coordinate t. The origin corresponds to the maximal ideal (t)⊆k[t]. By The local ring at a point of an affine variety is the localization at its maximal ideal, OX,0≅k[t](t)={r∈k(t):r=f(t)g(t) for some f,g∈k[t] with g(0)≠0}.

Thus a rational function is regular at the origin exactly when it admits a presentation f/g with g(0)≠0; another presentation may have a denominator that vanishes at 0. The maximal ideal consists of the elements admitting such a presentation with f(0)=0, and The residue field at a classical point identifies the residue field with k.

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Sources