ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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.

Comparison maps between two resolutions of a cyclic group

Example

Let $P_{∙}$ and $Q_{∙}$ be two displayed copies of the standard resolution $0 \to Z \overset{n}{\to} Z \to Z / n Z \to 0.$ For multiplication by $m$ on $Z / n Z$ , multiplication by $m$ in degrees $0$ and $1$ gives a comparison map $P_{∙} \to Q_{∙}$ .

Facts & Assumptions

Given: Integers $n \geq 1$ and $m$ , and two copies of the standard resolution from A projective resolution of a cyclic abelian group.

[L1]

Projective comparison maps exist (Projective comparison maps exist).

Verification

technique · direct

1.1

Multiplication by $m$ commutes with multiplication by $n$ on $Z$ , so the square with the two differentials commutes. It also commutes with the quotient maps to $Z / n Z$ , because both routes send $x$ to $\overline{m x}$ .

givenalgebra

2.1

Therefore the degreewise multiplication-by- $m$ maps form an augmentation-preserving chain map between the two resolutions. This is the explicit comparison map whose existence is asserted abstractly in [L1].

L1step 1.1∎

Depends on

Used by

An explicit comparison homotopy Example

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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)