TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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The canonical mapping-cone sequence is degreewise split short exact

Statement

For every chain map $f : C_{∙} \to D_{∙}$ in an abelian category, the sequence $0 \to D_{∙} \overset{j}{\to} Cone (f)_{∙} \overset{q}{\to} C [1]_{∙} \to 0$ from The canonical inclusion and projection for a mapping cone is a short exact sequence of complexes, and in each degree it is split short exact in the ambient abelian category.

Facts & Assumptions

Given: A chain map $f : C_{∙} \to D_{∙}$ .

[L1]

The canonical maps are $j_{n} (y) = (y, 0)$ and $q_{n} (y, x) = x$ (The canonical inclusion and projection for a mapping cone).

[L2]

A short exact sequence of complexes is degreewise exact (Short exact sequence of complexes).

[L3]

Finite biproducts of complexes are computed degreewise (Finite biproducts of complexes are computed degreewise).

[L4]

A split short exact sequence in an abelian category is one with a one-sided section or retraction exhibiting the middle object as a biproduct (Split short exact sequence in an abelian category).

Proof

technique · direct

1.1

By [L1], $q j = 0$ . In degree $n$ the sequence is $0 \to D_{n} \overset{y \mapsto (y, 0)}{\to} D_{n} \oplus C_{n - 1} \overset{(y, x) \mapsto x}{\to} C_{n - 1} \to 0,$ which is exact and split by the section $s_{n} (x) : = (0, x)$ . This uses the degreewise biproduct description from [L3] and is exactly the split shape of [L4].

L1L3L4givenalgebra

2.1

The maps $j$ and $q$ are chain maps by [L1], and step 1.1 proves exactness in every degree. Therefore [L2] makes the sequence a short exact sequence of complexes, with the degreewise splittings already exhibited in step 1.1.

L2step 1.1algebra∎

Depends on

Used by

A degreewise split cone sequence with no chain splitting Counterexample
FALSE: the degreewise splitting of the cone sequence is a chain splitting False statement

Dependency tree · two levels

12 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 13.9: Cones and termwise split sequences (standard reference, not scraped)