ExampleConstruction: Literature-sourcedVerification: 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.

Koszul Resolution Complete Intersection

Example

In $k [x, y]$ , the regular sequence $x^{2}, y^{3}$ gives $0 \to R \to R^{2} \to R \to R / (x^{2}, y^{3}) \to 0$ with ranks $1, 2, 1$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Resolves A Regular Quotient, Complete Intersection Betti Numbers Binomial.

Proof

technique · direct

1.1givenalgebra

$x^{2}$ is regular in $k [x, y]$ , and $y^{3}$ is regular modulo $(x^{2})$ .

2.1step 1.1algebra∎

The Koszul complex resolves the quotient with exterior ranks $1, 2, 1$ , whose alternating sum is zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)