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CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-31
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Relative homology vanishes exactly for quasi-isomorphisms

Statement

For a chain map f:C∙→D∙, the following are equivalent:

  1. f is a quasi-isomorphism.
  2. Hn(D,C;f)=0 for every n.

Facts & Assumptions

Given: A chain map f:C∙→D∙.

[L1]

Relative homology is the homology of the mapping cone (The relative homology of a chain map).

[L2]

A chain map is a quasi-isomorphism exactly when its cone is acyclic (A chain map is a quasi-isomorphism exactly when its cone is acyclic).

[L3]

Acyclic means vanishing homology in every degree (Exactness of a complex at a degree and acyclic complexes).

Proof

technique · direct
1.1

By [L1], the condition Hn(D,C;f)=0 for all n is exactly the condition that all homology objects of Cone⁡(f) vanish.

L1givenalgebra
2.1

By [L3], vanishing of all homology objects means that Cone⁡(f) is acyclic, and then [L2] identifies this with f being a quasi-isomorphism. This proves both directions of the equivalence.

L2L3step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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