ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

Shifting a three-term complex with all signs

Example

Let $C_{∙}$ be the three-term complex $0 \to Z \overset{2}{\to} Z \overset{0}{\to} Z \to 0,$ with nonzero terms in degrees $2, 1, 0$ . Then $C [1]_{3} = Z, C [1]_{2} = Z, C [1]_{1} = Z,$ and its displayed differentials are $d_{3}^{C [1]} = - 2, d_{2}^{C [1]} = 0.$ Consequently $H_{3} (C [1]) ≅ H_{2} (C), H_{2} (C [1]) ≅ H_{1} (C), H_{1} (C [1]) ≅ H_{0} (C) .$

Facts & Assumptions

Given: The three-term complex $C_{∙}$ above.

[L1]

Shift reindexes terms and multiplies the differential by $(- 1)^{k}$ (The shift of a chain complex).

[L2]

Homology of a shift satisfies $H_{n} (C [1]) ≅ H_{n - 1} (C)$ (Homology of a shift is shifted homology).

[L3]

$A b$ is an abelian category (Abelian groups form an abelian category).

Verification

technique · direct

1.1L1L3givenalgebra

Applying [L1] with $k = 1$ gives the displayed terms of $C [1]$ and the shifted differentials $d_{3}^{C [1]} = - d_{2}^{C} = - 2, d_{2}^{C [1]} = - d_{1}^{C} = 0.$

2.1L2step 1.1algebra∎

The displayed homology identifications are the cases $n = 3, 2, 1$ of [L2]. Thus the example shows every sign and every reindexing explicitly.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)