LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Homotopic maps have chain-isomorphic mapping cones

Statement

If chain maps $f, g : C_{∙} \to D_{∙}$ are chain-homotopic, then $Cone (f)$ and $Cone (g)$ are isomorphic as chain complexes.

Facts & Assumptions

Given: Chain maps $f, g : C_{∙} \to D_{∙}$ and a chain homotopy $s : f ≃ g$ .

[L1]

A chain homotopy satisfies $f - g = d^{D} s + s d^{C}$ (A chain homotopy).

[L2]

The cone differential is $d (y, x) = (d^{D} (y) + f (x), - d^{C} (x))$ (The mapping cone of a chain map).

Proof

technique · direct

1.1L1L2givenconstructalgebra

Define $Φ_{n} : Cone (f)_{n} \to Cone (g)_{n}$ by $Φ_{n} (y, x) : = (y + s_{n - 1} (x), x) .$ Using [L2], the equality $g - f = - (f - g)$ , and then [L1], one checks $d^{Cone (g)} Φ = Φ d^{Cone (f)} .$

2.1L1step 1.1algebra∎

Replacing $s$ by $- s$ gives the inverse block map $Ψ_{n} (y, x) : = (y - s_{n - 1} (x), x) .$ Thus $Φ$ is a chain isomorphism between the two mapping cones.

Depends on

Used by

FALSE: mapping cone is a functor on the homotopy category with no extra data False statement
The cone triangle of a null-homotopic map splits in the homotopy category Proposition
Whitehead torsion is independent of all auxiliary choices Theorem

Dependency tree · two levels

7 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)