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

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A six-term cohomology sequence

Example

Fix a nonzero integer $m$ . Let $A^{∙}$ be the cochain complex with $A^{1} = Z$ and all other terms zero, let $C^{∙}$ have $C^{0} = Z$ and all other terms zero, and let $B^{∙}$ have $B^{0} = B^{1} = Z$ with $d_{B}^{0} = \times m$ . Then $0 \to A^{∙} \to B^{∙} \to C^{∙} \to 0$ is the short exact sequence whose component $A^{1} \to B^{1}$ and component $B^{0} \to C^{0}$ are the identity and whose other components are zero. Its associated long exact cohomology sequence collapses to $0 \to H^{0} (A) \to H^{0} (B) \to H^{0} (C) \overset{\partial^{0}}{\to} H^{1} (A) \to H^{1} (B) \to H^{1} (C) \to 0,$ where $\partial^{0}$ is multiplication by $m$ .

Facts & Assumptions

Given: A nonzero integer $m$ and the componentwise maps specified in the Example.

[L1]

Short exact sequences of cochain complexes have long exact cohomology sequences (The long exact sequence in cohomology).

Verification

technique · direct

1.1L1givenalgebra

The three complexes have cohomology only in degrees $0$ and $1$ , and the only nontrivial differential is $d_{B}^{0} = \times m$ . By [L1], there is a long exact cohomology sequence. The groups immediately before $H^{0} (A)$ and after $H^{1} (C)$ are zero, so this long exact sequence collapses to the six displayed terms.

2.1L1step 1.1algebra∎

Here $H^{0} (C) = Z$ and $H^{1} (A) = Z$ . The connecting map sends $1 \in C^{0}$ to the class of $m \in A^{1}$ , so $\partial^{0}$ is multiplication by $m$ . This exhibits the boundary as a degree-raising map.

Depends on

The long exact sequence in cohomology

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)