PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Isomorphisms of complexes are quasi-isomorphisms

Statement

Every isomorphism in $Ch (A)$ is a quasi-isomorphism.

Facts & Assumptions

Given: An isomorphism of chain complexes $f : C \to D$ with inverse $g : D \to C$ .

[L1]

A quasi-isomorphism is a chain map inducing isomorphisms on all homology objects (Quasi-isomorphism).

[L2]

Homology respects identities and composition (Homology respects identities and composition).

Proof

technique · direct

1.1L2givenalgebra

Since $g f = 1_{C}$ and $f g = 1_{D}$ , [L2] gives $H_{n} (g) H_{n} (f) = 1_{H_{n} (C)}, H_{n} (f) H_{n} (g) = 1_{H_{n} (D)}$ for every $n$ . Thus each $H_{n} (f)$ is an isomorphism.

2.1L1step 1.1∎

By [L1], that means $f$ is a quasi-isomorphism.

Depends on

Used by

FALSE: every quasi-isomorphism is an isomorphism of complexes False statement

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)