CorollaryStatement: Literature-sourcedProof: 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.

Complete Intersection Betti Numbers Binomial

Statement

For a length- $n$ regular sequence in the maximal ideal of a local ring, the minimal Koszul resolution has $β_{i}^{K} = (\binom{n}{i})$ for $0 \leq i \leq n$ and $0$ otherwise.

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Resolution Minimality Maximal Ideal Sequence, Koszul Betti Numbers Over A Local Ring, Exterior Algebra Basis Monomials.

Proof

technique · direct

1.1givenalgebra

The minimality lemma identifies the Koszul ranks with the Koszul Betti numbers. In degree $i$ the free module has basis indexed by $i$ -subsets of an $n$ -set.

2.1step 1.1algebra∎

There are $(\binom{n}{i})$ such subsets and none in degrees outside $0, \dots, n$ , giving the stated table.

Depends on

Used by

regular local residue field projective dimension dimension Corollary
Koszul Resolution Betti Table Complete Intersection Example
Koszul Resolution Complete Intersection Example

Dependency tree · two levels

7 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)