ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge 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.

betti numbers from a koszul resolution

Example

For $R = k [x, y]_{(x, y)}$ , the augmented complex $0 \to R \overset{c \mapsto (- y c, x c)}{\to} R^{2} \overset{(a, b) \mapsto x a + y b}{\to} R \to k \to 0$ is a minimal free resolution. Thus $β (k) = (1, 2, 1)$ , with all higher Betti numbers zero.

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]

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.

Verification

1.1

The coordinate local ring is regular of dimension two: the coordinate chain and two generators give the dimension. Its variables are regular parameters. The displayed maps are exactly its two-variable Koszul maps; their composition is $- x y c + y x c = 0$ and the Koszul theorem gives exactness.

F3F1algebra

2.1

Every entry is in $(x, y)$ , so the resolution is minimal. Its ranks in degrees zero, one, two are $1, 2, 1$ and it is zero above two. The rank formula gives the asserted Betti numbers, independently of the characteristic.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

21 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 proof, p.123 (standard reference, not scraped)