CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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A degreewise split cone sequence with no chain splitting

Statement refuted

Every degreewise split cone sequence splits as a sequence of complexes.

Facts & Assumptions

Given: The cone sequence attached to $\times 2 : Z [0] \to Z [0]$ .

[L1]

Every cone sequence is degreewise split short exact (The canonical mapping-cone sequence is degreewise split short exact).

[L2]

Homology of a shift is shifted homology (Homology of a shift is shifted homology).

Counterexample

technique · direct

1.1L1givenalgebra

By [L1], the cone sequence for $\times 2$ is degreewise split. Its middle term is the complex $0 \to Z \overset{2}{\to} Z \to 0.$

2.1L2step 1.1algebra∎

If the sequence split as complexes, the middle term would be isomorphic to $Z [0] \oplus Z [1]$ , whose homology is $Z$ in degrees $0$ and $1$ by [L2]. But the middle term from step 1.1 has homology $H_{0} ≅ Z / 2$ and $H_{1} = 0$ . Hence no chain splitting exists, so the displayed sequence is a counterexample.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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)