Alphabeta Math
ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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.

Global dimension of a field and of the integers

Example

Assuming the Axiom of Choice, every vector space is free, giving global dimension zero for a field; pair the cyclic-group witness with the subgroup-freeness argument for global dimension one of Z.

Facts & Assumptions

Given: A field k and the ring Z, under Choice.

Verification

technique · direct
1.1givenconstruct

Every k-module is a vector space and has a basis, hence is free and projective. Therefore every module has projective dimension zero and gl.dim⁡k=0.

2.1step 1.1algebra∎

For every abelian group A, a free presentation has free kernel, so pd⁡ZA≤1 by The integers have global dimension one. The concrete nonzero group Ext⁡Z1(Z/2,Z)≅Z/2 supplies a degree-one witness. Hence gl.dim⁡Z=1.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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