ExampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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

L1givenalgebra

2.1

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 $m = \pm 1$ .

L2step 1.1algebra∎

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

15 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)