ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 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.

The cone of multiplication by m on the integers

Example

Fix a nonzero integer $m$ and let $f : Z [0] \to Z [0]$ be multiplication by $m$ . Then $Cone (f) ≅ (0 \to Z \overset{m}{\to} Z \to 0),$ with the left copy of $Z$ in degree $1$ and the right copy in degree $0$ . Hence $H_{1} (Cone (f)) = 0, H_{0} (Cone (f)) ≅ Z / m Z .$

Facts & Assumptions

Given: A nonzero integer $m$ and the chain map $f = \times m : Z [0] \to Z [0]$ .

[L1]

The cone of a chain map has terms $D_{n} \oplus C_{n - 1}$ and differential $(y, x) \mapsto (d (y) + f (x), - d (x))$ (The mapping cone of a chain map).

[L2]

A chain map is a quasi-isomorphism exactly when its cone is acyclic (A chain map is a quasi-isomorphism exactly when its cone is acyclic).

Verification

technique · direct

1.1L1givenalgebra

Since the source and target are stalk complexes in degree $0$ , [L1] leaves only one nonzero differential, namely $d_{1} : Z \to Z, d_{1} (x) = m x .$

2.1L2step 1.1algebra∎

Because $m \neq 0$ , $\ker (d_{1}) = 0$ and $coker (d_{1}) = Z / m Z$ . Therefore the homology groups are exactly as displayed, and [L2] shows that the cone is acyclic precisely when $m = \pm 1$ .

Depends on

Used by

Relative homology of a map between stalk complexes Example
The three-cone calculation for a composite Example

Dependency tree · two levels

20 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)