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

How statement and proof provenance work

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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.

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The cone long exact sequence for multiplication by m

Example

Fix a nonzero integer $m$ and consider the chain map $\times m : Z [0] \to Z [0] .$ Its cone long exact sequence contains the exact segment $0 \to H_{1} (Cone (\times m)) \to Z \overset{\times m}{\to} Z \to H_{0} (Cone (\times m)) \to 0.$ Hence $H_{1} (Cone (\times m)) = 0, H_{0} (Cone (\times m)) ≅ Z / m .$

Facts & Assumptions

Given: A nonzero integer $m$ .

[L1]

The cone of a chain map fits into a long exact homology sequence whose outer maps are the homology maps of the original chain map (The cone long exact sequence).

Verification

technique · direct

1.1L1givenalgebra

Since both source and target are stalk complexes in degree $0$ , their only nonzero homology group is $H_{0} ≅ Z$ . Applying [L1] to $\times m$ gives the displayed exact segment.

2.1L1step 1.1algebra∎

The map in the middle is multiplication by $m$ , so its kernel is $0$ and its cokernel is $Z / m$ . Exactness identifies these with $H_{1} (Cone (\times m))$ and $H_{0} (Cone (\times m))$ respectively.

Depends on

The cone long exact sequence

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)