PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31

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The cone triangle of a null-homotopic map splits in the homotopy category

Statement

If a chain map $f : C_{∙} \to D_{∙}$ is null-homotopic, then its cone triangle is isomorphic in The homotopy category of chain complexes to the split triangle $C_{∙} \overset{0}{\to} D_{∙} \to D_{∙} \oplus C [1]_{∙} \to C [1]_{∙} .$

Facts & Assumptions

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

[L1]

Null-homotopic maps are chain-homotopic to the zero map (A chain homotopy).

[L2]

Homotopic maps have isomorphic mapping cones (Homotopic maps have chain-isomorphic mapping cones).

[L3]

The cone of the zero map is $D \oplus C [1]$ as a complex (The cone of the zero map is the direct sum with a shift).

Proof

technique · direct

1.1

By [L1], the map $f$ is chain-homotopic to $0 : C_{∙} \to D_{∙}$ . Applying [L2] gives a chain isomorphism $Cone (f) ≅ Cone (0) .$

L1L2givenalgebra

2.1

Using [L3], this identifies $Cone (f)$ with $D_{∙} \oplus C [1]_{∙}$ . Therefore the cone triangle of $f$ is isomorphic in the homotopy category to the displayed split triangle.

L3step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

The Stacks Project, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)