Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge 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.

The Koszul characterization of depth

Statement

Under the hypotheses and notation of lem-koszul-depth-first-nonzero-cohomology, if q=max⁡{j:Hj(K∙(x;M))≠0}, then depth⁡I(M)=min⁡{i:Hi(K∙(x;M))≠0}=n−q. In particular the result is independent of the chosen finite generating sequence of I.

Facts & Assumptions

Given: (R,m) is Noetherian local, 0≠M is finite, and I=(x)⊆m.

Proof

technique · direct
1.1given

lem-koszul-depth-first-nonzero-cohomology proves both displayed equalities.

2.1step 1.1algebra∎

Their common value is depth⁡I(M), which is defined from I and M rather than from the chosen generators. Hence the numerical Koszul expression is generator-independent.

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