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

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Relative homology of a composable pair of stalk complexes

Example

Fix nonzero integers $m$ and $n$ . For the composable pair $Z [0] \overset{\times m}{\to} Z [0] \overset{\times n}{\to} Z [0],$ the relative homology groups are $H_{0} (D, C; \times m) ≅ Z / m, H_{0} (E, C; \times n m) ≅ Z / (n m), H_{0} (E, D; \times n) ≅ Z / n,$ and all higher relative homology groups vanish. The long exact sequence of the pair therefore collapses to $0 \to Z / m \overset{[x] \mapsto [n x]_{n m}}{\to} Z / (n m) \to Z / n \to 0.$

Facts & Assumptions

Given: Nonzero integers $m$ and $n$ .

[L1]

A composable pair of chain maps has a long exact sequence of relative homology (The long exact sequence of relative homology for a composable pair).

Verification

technique · direct

1.1L1givenalgebra

Each relative homology group is the homology of a two-term cone complex with differential multiplication by $m$ , $n m$ , or $n$ . Hence the displayed degree- $0$ groups are the corresponding cokernels and all higher groups vanish.

2.1L1step 1.1algebra∎

With only degree- $0$ terms remaining, [L1] collapses to a short exact sequence. The first map is multiplication by $n$ modulo $n m$ , and the second is reduction modulo $n$ , giving the displayed exact sequence.

Depends on

The long exact sequence of relative homology for a composable pair

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)