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.

minimal resolution unit cancellation

Example

Over $R = k [x]_{(x)}$ , the free resolution $0 \to R^{2} \overset{diag (x, 1)}{\to} R^{2} \to R / (x) \to 0$ , with augmentation $(a, b) \mapsto a m o d x$ , contracts to the minimal resolution $0 \to R \overset{x}{\to} R \to R / (x) \to 0$ .

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]

minimal free resolution differentials land in maximal ideal: For an augmented degreewise finite free resolution over a nonzero Noetherian local ring $(R, m)$ , minimality means that every positive differential matrix has entries in $m$ . Equivalently no positive differential admits a unit pivot, or a nonzero two-term identity direct summand. A unit pivot can be cancelled without changing the resolved module.

Verification

1.1

The displayed diagonal map is injective since $R$ is a domain. Its image consists exactly of pairs whose first coordinate lies in $(x)$ , which is the augmentation kernel. Hence the complex is exact.

givenalgebra

2.1

The second coordinates form the two-term identity summand, whose identity homotopy contracts it. Removing it leaves multiplication by $x$ on the first coordinates. Since $x$ belongs to the maximal ideal, this remaining resolution is minimal by the unit-cancellation criterion.

F1step 1.1∎

Depends on

minimal free resolution differentials land in maximal ideal

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

§12.2 minimal-resolution construction and Remark 12.28, pp.120–121 (standard reference, not scraped)