Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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.

localised polynomial ring regular

Example

Assume the Axiom of Choice (The Axiom of Choice). For a field k and integers 0≤r≤n, the ring R=k[x1,…,xn](x1,…,xr) is regular local of dimension r, with residue field k(xr+1,…,xn) and regular system (x1,…,xr).

Facts & Assumptions

Given: The objects and hypotheses in the example. We work with the Axiom of Choice; cited dependent-choice and resolution-existence hypotheses are retained.

[F1]

localisation and polynomial extension of regular rings: Localizations and finite polynomial extensions of a commutative regular Noetherian ring are regular. Regularity can equivalently be tested at maximal ideals. For every nonzero such ring, gldim⁡R=dim⁡R, allowing infinity. More generally, for a finite module over any commutative Noetherian ring, projective dimension is the supremum of its prime-local projective dimensions. Dedekind domains and their finite polynomial extensions are regular.

[F2]

dimension at most embedding dimension: Every nonzero commutative Noetherian local ring R satisfies dim⁡R≤edim⁡R<∞.

Verification

1.1F1algebra

A field is regular; polynomial extension and localization make R regular. The quotient by the indicated prime inverts every nonzero polynomial in the remaining variables, hence is k(xr+1,…,xn). The maximal ideal is generated by the first r variables.

2.1F2step 1.1algebra∎

The strict coordinate chain (0)⊊(x1)⊊⋯⊊(x1,…,xr) survives in the localization and has length r. The r generators bound cotangent dimension by r, and the embedding bound gives r≤dim⁡R≤edim⁡R≤r. Thus those generators are minimal and form regular parameters. If r=0, the ring is the fraction field; if r=n, the residue field is k, including n=r=0.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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