PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 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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Relative homology is invariant under homotopy equivalence of arrows

Statement

If two chain maps are related by a homotopy equivalence of arrows in the sense of Cones preserve chain-homotopy equivalences of arrows, then their relative homology objects are naturally isomorphic in every degree.

Facts & Assumptions

Given: Chain maps $f$ and $g$ together with a homotopy equivalence of arrows from $f$ to $g$ .

[L1]

Relative homology is defined by $H_{n} (D, C; f) = H_{n} (Cone (f))$ (The relative homology of a chain map).

[L2]

The induced map on cones is a chain-homotopy equivalence (Cones preserve chain-homotopy equivalences of arrows).

[L3]

Every chain-homotopy equivalence is a quasi-isomorphism (A chain homotopy equivalence is a quasi-isomorphism).

[L4]

A chain map induces a well-defined map on homology (A chain map induces a well-defined map on homology).

Proof

technique · direct

1.1

By [L2], there is a chain-homotopy equivalence $Φ : Cone (f) \to Cone (g) .$ Then [L3] makes $Φ$ a quasi-isomorphism.

L2L3givenalgebra

2.1

Applying [L4] to $Φ$ yields isomorphisms $H_{n} (Cone (f)) \tilde{\to} H_{n} (Cone (g))$ for all $n$ . Rewriting both sides with [L1] gives the claimed natural isomorphisms of relative homology objects.

L1L4step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

The Stacks Project, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)
Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)