PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

Quasi isomorphisms contain identities and are closed under composition

Statement

For complexes in an abelian category, identity maps are quasi-isomorphisms and composites of quasi-isomorphisms are quasi-isomorphisms. We use cochain indexing, so $H^{n} = H_{- n}$ under reindexing.

Facts & Assumptions

Given: For complexes in an abelian category, identity maps are quasi-isomorphisms and composites of quasi-isomorphisms are quasi-isomorphisms. We use cochain indexing, so $H^{n} = H_{- n}$ under reindexing.

[F1]

A quasi-isomorphism is a complex map inducing an isomorphism in every homology degree (Quasi-isomorphism).

Proof

1.1

In each integer degree $H^{n} (1_{X}) = 1_{H^{n} (X)}$ , including $H^{n} (X) = 0$ . Hence identities induce isomorphisms in every degree.

F1algebra

2.1

For quasi-isomorphisms $f : X \to Y, g : Y \to Z$ , functoriality gives $H^{n} (g f) = H^{n} (g) H^{n} (f)$ , an isomorphism with inverse $H^{n} (f)^{- 1} H^{n} (g)^{- 1}$ . This holds for every $n$ .

F1algebra∎

Depends on

Quasi-isomorphism

Used by

Dependency tree · two levels

2 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

13.11.1–13.11.6 (standard reference, not scraped)