False statementConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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FALSE: the degreewise splitting of the cone sequence is a chain splitting

Statement

The degreewise splitting of the canonical cone sequence is automatically a chain splitting.

Facts & Assumptions

Given: The chain map $\times 2 : Z [0] \to Z [0]$ between stalk complexes concentrated in degree $0$ .

[A1]

The statement refuted is: the degreewise splitting of the canonical cone sequence is automatically a chain splitting.

[L1]

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

[L2]

The cone of the zero map is the direct sum with a shift (The cone of the zero map is the direct sum with a shift).

[L3]

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

Refutation

technique · direct

1.1A1L1L2givenalgebra

By [L1], the cone sequence for $\times 2$ is degreewise split. If it were a chain splitting as well, then the cone would be isomorphic as a complex to $Z [0] \oplus Z [1]$ as in [L2].

2.1A1L2L3step 1.1algebra∎

But $Cone (\times 2)$ is the two-term complex $0 \to Z \overset{2}{\to} Z \to 0,$ so its homology is $H_{0} ≅ Z / 2$ and $H_{1} = 0$ , whereas $Z [0] \oplus Z [1]$ has $H_{0} ≅ Z$ and $H_{1} ≅ Z$ by [L3]. Hence no chain splitting exists, so [A1] is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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)