ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

betti numbers residue field regular ring

Example

For $R = k [x, y, z]_{(x, y, z)}$ , the residue field has Betti numbers $(1, 3, 3, 1)$ and projective dimension three.

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]

regular local residue field koszul resolution: For a regular local ring $(R, m, k)$ of dimension $d$ , the Koszul complex on any regular system of parameters is a minimal free resolution of $k$ of length $d$ .

[F2]

betti number is rank in minimal resolution: For every minimal degreewise finite free resolution $F_{∙} \to M$ of a finite module over a nonzero Noetherian local ring, $β_{i}^{R} (M) = {rank}_{R} F_{i}$ for all $i \geq 0$ .

[F3]

regular local residue field projective dimension dimension: For a regular local ring $(R, m, k)$ of dimension $d$ , ${pd}_{R} k = d$ and $β_{i}^{R} (k) = (\binom{d}{i})$ for $0 \leq i \leq d$ , with $β_{i}^{R} (k) = 0$ for $i > d$ .

[F4]

dimension at most embedding dimension: Every nonzero commutative Noetherian local ring $R$ satisfies $\dim R \leq edim R < \infty$ .

Verification

1.1

The coordinate chain of prime ideals gives dimension at least three, while the three generators of the maximal ideal give embedding dimension at most three and hence dimension at most three. Thus $R$ is regular of dimension three with parameters $x, y, z$ . Their Koszul complex is a minimal resolution.

F1F4algebra

2.1

The exterior bases have ranks $1, 3, 3, 1$ in degrees zero through three and zero above. The rank and projective-dimension formulas give these Betti numbers and projective dimension three, since the top rank is one.

F2F3step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Theorem 12.33, p.123 (standard reference, not scraped)